Moreover, S*/S'n* is equal to grDe, thus C is a left resolution of the (g,g)-module ...... Now for an arbitrary sheaf S, we use Godement's resolution of C by flasque.
Spherical D-modules and Representations of Reductive Lie Groups Freddric Vincent Bien Licence en mathematiques, Universite Libre de Bruxelles (1983) Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology June 1986 Frederic V. Bien 1986 The author hereby grants to M.I.T. permission to reproduce and to distribute copies of this thesis document in whole or in part.
Signature redacted Signature redacted
Signature of author Certified by
Professor Joseph N. Bernstein Thesip*ugervisor
Signature redacted Accepted b3
p.-
Professor Nesmith C. Ankeny, Chairman Departmental Graduate Committee Department of Mathematics
ssACUSE1TS
ISTUTE
AUG 0 4 1986
Spherical D-modules and Representations of Real Reductive Groups by Fr6ddric Bien Submitted to the Department of Mathematics on May first, 1986 in partial fulfillment of the requirements for the Degree of Doctor of Philisophy in Mathematics ABSTRACT
We study the representations of reductive Lie groups which occur in the space of smooth functions on indefinite symmetric spaces. We characterize these representations in the theory of Dmodules by a condition on the support and a condition on the fibers. This enables us to simplify Oshima-Matsuki's theorem on the discrete series of indefinite symmetric spaces, and to prove an L2 -multiplicity one theorem. We also interpret the C*-multiplicity of standard representations as the dimension of a cohomology space of smooth algebraic varieties. By microlocalization, we exhibit the global sections and localization functors as direct and inverse images of the moment map. We prove that the global sections of irreducible D-modules always break up into components with the same associated variety, and may stay irreducible in some singular cases. Thesis Supervisor: Professor Joseph Bernstein
2
Contents
Page
Introduction
2
Localization theory
8
1.
Basic Notions on D-modules
8
2.
D-modules with a group action
9
3.
Parabolic subgroups and flag spaces
11
4.
Differential operators on flag
13
I.
spaces 5.
Global sections of D
15
6.
D-affine varieties
19
7.
Holonomic modules and
22
Harish-Chandra modules II.
Spherical D-modules
26
1.
K-orbits in a flag space
26
2.
(D,K)-modules with a K-
30
fixed vector 3.
Relations with the analytic theory
31
4.
Going from (D, K)-modules to (D,H)-
36
modules 5.
H-spherical (D,K)-modules
42
6.
D-modules and hyperfunctions
48
7.
Closed orbits and discrete series
54
8.
An algebraic Poisson transform
59
III.
Microlocalization and Singularities 1.
Basic Microlocal notions
62
2.
Microlocal study of the moment map
68
3.
Nilpotent varieties
75
3
Acknowledgements It is a pleasure to express my deep gratitude to J. Bernstein who supervised this thesis. I learned from him the theory of D-modules, and he generously shared with me his understanding of mathematics in a very stimulating atmosphere. Our many conversations were the major inspiration of this work. I thank sincerely D. Vogan for teaching me the foundations of representation theory. He also gave me the benefit of his insight on the subject on many later occasions. I am grateful to B. Kostant and G. Lusztig for several helpful suggestions and for showing interest in this work. This research was facilitated by discussions with several friends. In particular P-Y. Gaillard introduced me to the Poisson transform, K. Vilonen provided my initiation to perverse and microlocal objects and A. Vistoli helped me find my way through singularities. To all of them, I express my warmest thanks. I am also grateful to the MIT Mathematics Department for allowing me to work with J. Bernstein and for its financial support in 1984-1985. I received grants from the Belgian American Educational Foundation in 1983-84 and from the Sloan Foundation in 1985-86. Poincar6 and Fermat Sun workstations did a beautiful job as typesetters. M-E. Butts commanded them skillfully and I thank her for typing two-thirds of this report in a record time. This thesis is dedicated to my former teacher G. Hirsch who will certainly be pleased with the pervasive influence of topology on these pages.
4
Introduction. The goal of this work is to study the representations of reductive Lie groups which occur in the space of smooth functions on an indefinite symmetric space. The representations realized by square integrable functions were constructed by Flensted-Jensen, Oshima and Matsuki. We prove that the discrete series has multiplicity one, and we present a cohomological formula for the multiplicities of standard representations. We have chosen to study these representations by Beilinson and Bernstein's theory of differential operators on complex flag manifolds. We find a canonical map between certain D-modules and the sheaf of hyperfunctions along a real flag manifold. Combined with previous work of Helgason and Flensted-Jensen, this exhibits a natural intertwining operator between these D-modules and functions on symmetric spaces. By microlocalization, we also make a finer study of the moment map, and decide the irreducibility of the global sections of certain D-modules considered as representations of Lie algebras. More precisely, let G be a complex connected reductive linear algebraic group, and let GR be a real form of G. Consider the fixed point subgroup H of an involution a of G and its corresponding real form HR. Then GR/HR is a symmetric space. The problem is to find which representations of GR can be imbedded in CO(GR/HR), which ones can be imbedded in L 2 (GR/HR) and when they can be so imbedded, in how many different ways. To formulate the answer, one should first understand that the building blocks of C(GR/HR) are generally not the irreducible representations of GR. The standard representations are better designed for this purpose but not all of them will appear.A smooth representation V of GR which does appear is called HR-spherical and is characterized by: HomHR (V, C) 4 0, i.e. its continuous dual V* contains an HRfixed functional vo. If we want V to be imbedded into C*(GR/HR), then vo must generate V*. The duality involved is important because in general V itself does not contain any HR-invariant vector.
By a result of Casselman and [Wallach],
one can replace the H-spherical condition by Hom(h,KH)(V
0,
C) : 0 where V 0
is the Harish-Chandra module of V. Using the inductive version of Zuckerman's 5
functor, this is equivalent to HOmH(L V 0 , C) : 0, i.e. (L VO)H : 0 since H acts
semisimply on this space. We pass to the theory of D-modules. The Helgason isomorphism for the Riemannian symmetric spaces shows that the spherical representations have a particular shape: they can be realized on a partial flag variety. The Iwasawa decomposition associates to G and H a complex flag space X and a real flag space XR imbedded in X. To fix the ideas, consider a real form Gd of G for which Hd = H n Gd is a maximal compact subgroup. Let PR be a minimal parabolic subgroup of Gd with complexification P. Then X ~ G/P and XR ~ GR/PR. We also need the fixed point subgroup K of an involution 0 of G commuting with a, such that KR = K n GR is a maximal compact subgroup of GR. Let Tp = P/(P,P) be the Cartan factor of P. To every A E t*,, one can associate a G-sheaf DX of twisted differential operators on X. To define the notion of H-spherical (DA, K)-modules, we will use the functor L' introduced by Bernstein: it maps (DA, K)-modules to (PA, H)-modules in a fashion analogous to the inductive Zuckerman's functor. Assume henceforth that X E t* is dominant, so that the global section functor is exact. We say that a (DA, K)-module is H-spherical if I'(X, 'CHM)H 7 0 The standard (DA, K)-modules on X are constructed as follows. Take a K-orbit Y with a K-homogeneous line bundle Oy (r) such that (A - pp, r) is a module for
the Harish-Chandra pair (py, Ku), y E Y. To simplify the exposition we suppose in this introduction that i : Y c-+ X is a closed imbedding; a more general case is worked out in the text. Let i, be the direct image in the category of DX-modules. Then M (Y, r) = i. Oy (r) is a standard (DA, K)-module. Now let X0 denote the open H-orbit in X and let L be the Levi factor of P. Theorem 1: Let A + PL be B-dominant and regular. M (Y, r) is H-spherical if and 0 and (h,, K n H,) acts trivially on the fiber Oy (r)y, y E Y n X0
.
only if Y n X 0
Let us call H-square-integrable a (DA, K)-module whose global sections can be
imbedded as a discrete summand of L'
fn(GR/HR).
Using [Oshima and Matsuki)
L 2 -estimates, we can simplify their result on the discrete series of GR/HR. Suppose
6
M(Y,-r) is H-spherical and G is semisimple. Theorem 2: M (Y, r) is H-square integrable if and only if A E t* is dominant regular and rank G/H = rank K/H n K. Let t be the canonical Cartan space of G. If A + pt E t* is dominant, then r (X, MN(Y, r)) is an irreducible
(9, K)-module.
But in general there is a small strip
of dominant A E * for which A + pt is not dominant. D. Vogan has checked by coherent continuation that P(X, M(Y, r)) remains irreducible in this strip. This hard result has a pretty consequence. Theorem 3: If G is a classical group or G 2 , the discrete series in L 2 (GR/HR) is
multiplicity free. In the case of compact groups, this is the well-known fact that (GR, HR) is a Gelfand pair. Let us note that one should distinguish between the multiplicity of an irreducible representation V of G as subquotient of C*(GR/HR, X) and as submodule, here X denotes an eigencharacter of the algebra of invariant differential operators on the symmetric space. The former can be quite big although finite, cf. [van den Ban] and we do not know a good way to determine it. The latter is dim (V*)H. Theorem 3 does not affirm that this dimension is 1 for V = r(X, m(Y, r)) as above. If this were true, it would imply easily the irreducibility of V. In fact the search for some multiplicity one theorem motivates this whole thesis. We are able to interpret (V*)H as the cohomology space of a smooth algebraic variety with values in a local system. This formula holds for all standard modules obtained from affinely imbedded K-orbits. If V is reducible, it simply determines the dimension of the (G/H)). For any x in the open H-orbit X0
,
space of morphisms: HomoR (V, C
let M be the isotropy group of z in H. The invertible sheaf Oy (r) gives rise to a
n X0
.
K n H-homogeneous local system C(A, r) on Y
Theorem 4: Suppose that A + PL is B-dominant and regular. Then r (X, Z J.4(Y,
r))H
~ HO(Y n XO, C(A, r))KnH
where s = dimY + dimM - dimK n H. 7
This formula gives easily the uniqueness of the HR-invariant functional in many nontrivial cases. It also shows that the even principal series for SL 2 (R) admits generically two R -invariant functionals. Now we describe the content of the chapters. In Chapter 1, we summarize Beilinson-Bernstein theory of D-modules on flag spaces and its relations to the representations of g. We take a slightly more general viewpoint than is usual, for we work over the variety X of parabolic subgroups of G conjugate to a given one, say P. The vanishing theorem for DA-modules is true as long as A is dominant in t*. But P (X, DA) need not be generated by the enveloping algebra U (g) for all A E t,, as it is for the space of Borel subgroups. This is due to the fact that the closure of the Richardson orbit Cp C g* of P need not be a normal variety. However, if A + PL E t* is dominant, then U (g) generates P (X, DA). If in addition A + PL is regular, the category of (DA, K)-modules on X is equivalent to the category of (g, K)-modules with infinitesimal character A + PL and associated variety contained in ~Cp. To study P(X, DA), we generalize a method of Militi6. The non-vanishing of the global sections of a (Dx, K)-module depends on the Borel-Weil theorem. To include entirely this classical result in the theory of D-modules, one should consider modules over a sheaf of matrix differential oeprators. One could generalize the theory much further than we have done here. In Chapter 2, after recalling some properties of the K-orbits in X, we study the case K = H of Riemannian symmetric space. We classify the K-spherical (DA, K)modules: they all come from trivial line bundles on the open K-orbit. In particular, there is no square integrable module and they all have a unique K-invariant vector. These results have been known since [Kostant]. In Section 2.3 we explain what is the foundation of our approach for the reader with an analytic background. Next we describe Bernstein's definition of Zuckerman's functor for D-modules. At this point we are ready to study the H-spherical (DA, K)-modules. The idea is again that they are related to the open H-orbit and to homogeneous line bundles which are trivial for (h, K n H). We also explain how to deal with the spaces GR/HR. We prove the formula giving the number of (0, H)-morphisms from Ox into 8
Ki.M(Y,
r)
for a standard (DA, K)-module .M(Y, r), in terms of the cohomology of Y n X 0 . A better understanding of this formula would give a purely algebraic classification, but at the present time we are obliged to use as an intermediate step the real flag variety XR of G' which is imbedded in the middle of X0 . In some sense this is fortunate because we can construct a bijection between K-orbits Y in X such that Y n XR 7 0 and Norm(G , K)-orbits in XR. Then we answer a question raised by Flensted-Jensen in a lecture at Utrecht in August 1985: what is the relation between the square-integrable standard H-spherical (DA)-modules and the sheaf of hyperfunctions sections of a line bundle on XR? It turns out that the first can be imbedded into the second, thanks to the existence of a canonical morphism of functions
f*
f! 0
-+
w[-rdf] which exists for any map f. Let us remark that
tracing back through Helgason and Flensted-Jensen isomorphisms, we obtain the unique 'imbedding' of some square integrable standard H-spherical into
L 2 (GR/HR).
(DP,
K) module
This shows that the D-module realization of a Harish-Chandra
module is in fact quite natural to study harmonic analysis on symmetric spaces. -
Then we focus on closed K-orbits in the equal rank case, and using a deep L2
estimate of Oshima and Matsuki, we prove that the discrete series of GR/HR has multiplicity one. Finally, we define an algebraic Poisson transform from (DX, K)module on G/P to
(Dx,, K)-modules
on G/H. [De Concini and Procesi] have con-
structed a very nice compactification of G/H which exhibits G/P as a piece of the boundary of G/H. I expect that the nearby cycle functor is the inverse of the Poisson transform. I hope to be able to prove it soon. In Chapter 3, we solidify the bridge between g-modules and DA-modules on an arbitrary flag variety X. Except for its last section, chapter 3 is independent of chapter 2. It consists of an attempt to understand when does the global section functor preserves irreducibility. Let Ux be the quotient of U(g) by the ideal determined by A E t*. The image of the moment map A : T*X -+ g* is the closure of the Richardson orbit Cp C g* of P. Following a method introduced by Gabber and [Ginzburg], we define the formal microlocalizations ex, Ux of the algebras DA, Ux. eA is a G-sheaf of twisted formal microdifferential operators on T*X, and Ux is a G-sheaf supported on Cp. The microlocalization is an exact and faithful functor 9
which sends a D-module M to an e-module whose support is the characteristic variety CharM of M. The functors r and A of global sections and localizations between DA- and Ux-modules become the functors images between
AX
1
. and i* of direct and inverse
and Vx-modules. One result of this approach is the following.
Proposition 1: Suppose M is an irreducible DX-module on X and A is dominant in t*,. Then all the irreducible (g, K)-submodules of r(X, M) have the same associated variety. Concerning irreducibility itself we obtain the following improvement when the moment map g is birational. Proposition 2: Let M be an irreducible (Dx, K) module with A dominant in t* Suppose pt(CharM) contains at least one normal point of
'.
Then P(X, M) is
(g, K) irreducible (or zero). [Kraft and Procesi] have essentially determined when Cp is a normal variety for the classical groups. Combining their results with the above proposition proves the irreducibility of certain (g, K) modules-known under the name of Ap(A)-for some non-trivial cases, like for SPn(R). The microlocal approach and the study of examples suggests that the unibranchness of an open dense subset of pt(CharM) should suffice to imply the irreduciblity of r(X, M) over (g, K) when M is (DA, K)-irreducible and 1z birational. The reader will find in Section 111.3 a formula to detect the unibranchness of a point in Op. More precisely, using [Borho-MacPherson] theory, we show that the number of connected components in
1r(z), z E Cp, is the multiplicity of the special Weyl group
representation attached to Cp, in a cohomology space of the Springer fiber of z. To express a quantity related to X in terms of the full flag variety - as this formula does - simplifies the problem. Still, this multiplicity remains mysterious to us.
10
Chapter I. Localization Theory 1. Basic notions on P-modules Let X be a smooth complex algebraic manifold. Let 0 = Ox be the sheaf of regular functions on X and D = DX be the sheaf of differential operators on X. A D-module M is a sheaf on X which is quasi-coherent as an 0-module and which has a structure of module over D. Let M(D) denote the category of left D-modules. We shall deal with some sheaves of rings slightly more general than D. Consider the category of pairs (A, iA) where A is a sheaf on X of C-algebras and ig : 0 -+ A is a
morphism of C-algebras. The pair (D, i) where i : 0 -+ D is the natural inclusion will be called the standard pair. Definition 1.1: A sheaf of twisted differential operators on X (tdo for short) is any pair (A, iA) which locally isomorphic to the standard pair. Lemma 1.2: The group of automorphisms of (i : 0 -+ D) is naturally isomorphic to ZI(X) the group of closed 1-forms on X. The set of isomorphism classes of tdo's on X is in bijection with the 6ech cohomology space H1(X, Z1). In particular tdo's form a linear space; for a proper variety X H'(X, Z') is the C-subspace of Hd (X) generated by algebraic cycles. Example 1.3: Let L be an invertible sheaf of 0-modules on X,then the sheaf D (L) of differential operators from
into itself is a tdo. Let
corresponds to the
element c in the Picard group H'(X, 0*), then D(L) corresponds to the logarithmic differential of c, i.e. dc/c in H1(X, Z'). Example 1.4: If X = P" is the projective n-space, then H1(X, 0*) = Z, hence, the invertible sheaves on P' form a lattice. But H1(P", Z1) = C: the sheaves of twisted differential operators on P" form a vector space. Remark 1.5: If V is a locally free sheaf of 0-modules on X of rank bigger than one, then the ring of differential operators of V into itself is not a tdo. It consists of matrix differential opreators. Let
f
: X -+
Y be a morphism of smooth complex algebraic varieties. 11
Let
Oy, Ox be the sheaves of regular functions on Y and X. We will denote by f*, f" the inverse and direct images functors between the categories of 0-modules. There is an induced map f* : H1 (Y, Z') -+ H'(X, Z'). Starting from a tdo (Dy,A, i) on
E
Y corresponding to w
H1 (Y, Z 1 ), we can construct the tdo. (Dx,pA, f*i) on X
which by definition is associated to the data f*w E H'(X, Z'). Let M(Dy,) (resp. M(Dxf .A)) be the category of Dyx-modules on Y (resp. Dx,p x-modules on X). Then one can define two functors : * inverse image: * direct image:
f f.
: M(Dy,A) -+ M(Dx,pA)
: M(Dx,-x) -+ M(Dy,x)
In case f is not affine, the direct image functor has good properties only between the derived categories of bounded complexes of D-modules. In fact at the level of derived categories, the functor f' is simply the inverse image of 0-modules shifted by the relative dimension of f: rd(f) = dimY - dimX. One can transform a left Dmodule into a right one by tensorization with 0 the sheaf of top degree differential forms. Then the direct image of a right Dx,..A-module M in the derived category is the direct image as 0-module of .M &DX
fA.
f*Dy,,%. For material of this section we
refer to [Bernstein] and [Beilinson - Bernstein 19831.
2. D-modules with a group action Let G be a complex algebraic group with Lie algebra g, and suppose G acts on a smooth variety X. Let Y be a sheaf on X. By definition, a weak action of G on 7 is an action of G on Y which extends the action of G on X. When .7 is a Dx-module, there is however more structure involved. If a denote the action of G on X, then we have a morphism da : g -+ r(X, Dx) and P(X, Dx) acts on itself by the commutator action ad. So there is a map ad -da : g -+ Endr(X, Dx). On the other hand Dx has obviously a weak action of G, say P, which yields the map d,8 : g -+ EndP(X, Dx). It is natural to require that the map da be G-equivariant and that ad - da coincide with dp. More generally, let (D, i) be a sheaf of twisted differential operators on X. Definition 2.1: An action of G on D is a weak action 3 together with a morphism 12
7r
(X, D) such that:
: g -+ 1.
7r
is G-equivariant with respect to Ad on g and P on P(X, D),
2. for
E g : ad7r( ) = dfl( ).
Suppose now that D is a tdo with a G-action given by P and
7r.
Definition 2.2: A (D, G)-module .7 on X is a quasi-coherent Ox -module with a structure -y of D-module and a weak action b of G such that: 1. -y is G-equivariant with respect to 3 on D and b on 7, 2. for e G g : -)r(e) = db(e) The difference between a weak action and an action can also be illustrated as follows. Consider the diagram
X
-G x X
x
(g, X) -+ a(g)X
1. In fact in this case, V is a module over a sheaf of matrix differential operators, as in remark 1.5. We could enlarge the theory of D-modules to include the case of matrix differential operators acting on matrix valued functions. But we will not need this generality here. 13
3. Parabolic subgroups and flag spaces The structure theory of linear algebraic groups is very clearly explained in Springer's book, so here we will only review certain facts from a point of view suited to the later developments. The unipotent radical RuG of a linear algebraic group G is the largest closed, connected, unipotent normal subgroup of G. G is called reductive if RuG = {e}. Henceforth, G shall denote a complex, connected, reductive, linear algebraic group. The connectedness assumption is not at all necessary for the study of D-modules but it simplifies the exposition here. A parabolic subgroup P of G is a closed subgroup such that the quotient variety G/P is complete. A Borel subgroup B is a connected and solvable subgroup of G which is maximal for these properties.
One proves that a closed subgroup of G
is parabolic if and only if it contains a Borel subgroup. Since G is connected, a parabolic subgroup is its own normalizer. We prefer to view the flag space X G/P as the as the variety of all parabolic subgroups of G conjugate to P. Let P be a parabolic subgroup of G and N = Np its unipotent radical. A Levi subgroup L of P is a closed subgroup such that the product map LxN -+ P is an isomorphism of varieties. Then L is reductive, normalizes N and is the centralizer in G of a connected torus in P. We shall also say that L is a Levi subgroup of G. If T is a maximal torus in P, there is a unique Levi subgroup of P containing T. All Levi subgroups of P are conjugate. The projection P -+ P/N gives an isomorphism of any Levi subgroup of P with P/N. Therefore we call P/N the Levi factor of P and we denote it by Lp; it is canonically attached to P. The Levi factors Lp and Lp, of two conjugate parabolic subgroups P and P' are not necessarily canonically conjugate because we can twist a given isomorphim by an inner automorphism of Lp or LP,.
.
Let P 1 := (P,P) denote the commutator subgroup of P and put Tp = P/P1
Tp is isomorphic to a maximal torus in the center of Lp and we call it the Cartan factor of P. For two conjugate parabolic subgroups P and P', the Cartan factors Tp 14
and Tt are canonically conjugate. When P is a Borel subgroup, its Cartan factors coincides with its Levi factor and is isomorphic to a Cartan subgroup of G. This special case is good to bear in mind for understanding the sequel easily. Let us denote by boldface letters the Lie algebras of the groups considered. To define the set of roots of t, in g, we use the following trick. Choose a Levi subgroup L of P and let C be the connected component of its center. Then C acts by the adjoint action on g and we obtain the roots of C in g. Note that these roots do ,
not always form a root system. Now C is isomorphic to Tp by the map P -+ P/P 1
hence to every root of C in g corresponds a unique linear form on t,. We call these linear forms the roots of t, in g; they are independent of the choice of L. Let R(t,) c t* be the set of roots of t, in g. R(t,) is naturally divided into the set of roots whose root spaces are contained in n and its complement. Let R+(tP) be the set of roots of t, in g/p. If a is a root of t, in g, the corresponding root space g, need not have dimension 1; dim g, is called the multiplicity of a. Let pp be the half sum of the roots contained in R+(t,) counted with their multiplicities. Let B be a Borel subgroup of G, contained in P. The map B/B1 -+ P/P1 gives a canonical surjective homomorphism TB -- Tp which dualizes to an inclusion t* tb. Hence we may think of R(t,) as a subset of t.
R(t,) is always a root system.
If we had chosen a invariant bilinear form on g*, then R(t,) could be viewed as the projection of R(tb) into t*, and the multiplicity of a E R(t,) would be the number of roots in R(tb) which project onto a. We can also define the roots in the Levi factor
4, as follows.
Choose a maximal
torus T in B. There is a unique Levi subgroup L of P containing T, so we have the roots of t in E. From the canonical isomorphism t
4, which
R(tep) of roots of t,
in
R(tb) c t*. Set R+(tb,
4,) =
roots in R+(tb,ep). We have p,
-~+
t*, we obtain the set
is independent of the choice of T. R(t,4,)
g
R+(tb) n R(tb,4,) and let p, be the half sum of the = Pb -
Pp
The outcome of this stylistic exercise is that we have a canonical comparison between the various root systems considered.
To a Borel subgroup B of G, we
can associate the triple (tb,R+(tp),R(tb)) where tb = b/b and R+(t,) c t. 15
For
different Borel subgroups, these triples are canonically conjugate. So we identify them with one abstract triple (t,R+,R) called the Cartantriple of G. The reductivity of G implies that R = R+ - R+. Similarly to a pair B C P we can associate the
quadruple (t,t,,R+(t,,), R+(t,)) where t, = p/pi, R+(t,) C t* '-+ t* D R+(te,). Note that there is a canonical element p, in t. Let D(t) be the set of simple roots in R+(t). The set D(e,) = D(t) n R(t,ep) of simple roots of L characterizes the conjugacy class of the parabolic subgroup P. Via this correspondence, the subsets of D(t) are in bijection with the G-conjugacy classes of parabolic subgroups of G. If P is a parabolic subgroup of G corresponding to the subset I of D(t), let X XI the space of all subgroups of G conjugate to P. X is called the flag space of G of type P or of type I; we will also use the words flag manifold or flag variety of type P. When P is a Borel subgroup of G, we call X the full flag variety of G. X is always a complete smooth complex projective algebraic variety on which G acts transitively by conjugation. If ICJ are two subsets of D(t), and if P is a parabolic subgroup of G of type I, there exists a unique parabolic subgroup that P C
Q.
flag variety of
Q of type
J such
This yields a smooth fibration Xr --+Xr with fiber isomorphic to the
Q of type
P.
4. Differential operators on flag spaces Let X be the flag space of G of type P. Let 0 = Ox be the structure sheaf of X and TX be its tangent sheaf. We denote by a:
g -+ TX the morphism of
Lie algebras defined by the action of G on X. Let U be the enveloping algebra of
g. Put U* := OfcU and endow this sheaf with a multiplication extending the ring structure of U, and the structure of 0-module on U*. Explicitely for f, g E 0, A,B E U: [f 0 A,g 0 BI = fg
[A,B+ fa(A)g 0 B - ga(B)f 0 A
A direct computation shows that this bracket induces a Lie algebra structure 16
on go := 00cg C U*. Set: p0 = Ker(a : g* --+ Tx) = {Eg*
E px VX E X}
Here p. is the parabolic subalgebra corresponding to the point z in X and 2 is the value of the local section e at the point x. Since p0 is the kernel of a morphism of Lie algebras, it is an ideal in g*, and hence p* is also an ideal in g*. Moreover the restriction of the bracket to p0 is 0-linear and p*/p* ~ Ooct,. We can use proposition 2.3 to classify the sheaves of twisted differential operators on X which have a G-action. They correspond precisely to the linear forms A :p --+ C trivial on pi, i.e. to the elements of t.
As is customary the center of
symmetry of the picture is not o but ppEt*. So to avoid further normalizations, we set D.X to be the tdo on X corresponding to the weight A - pp E t.
We can describe
more explicitly the tdo DA. Every weight A Et* determines a morphism A* : p* Ox. Let I.X be the ideal of U* generated by the elements (
-
-+
(A - p,)o(e) where e
is a local section of p0 . Then DA = U*/Ih. Example 4.1: Let A Et*, i.e. A EHom(Tp,C*), and let 0 (A) be the corresponding invertible G-sheaf of 0-modules. Then Diff 0 (A) = DA+p,. In particular DX = Dp. Let us denote by DA the global sections on X of Dx. The center of DA is C: the constant functions. We have a morphism 7r : g -+ DA, since DX is a tdo. It extends to a morphism 7r : U -+ DA which must send the center Z of U to CED by a certain character 8. On the other hand the Harish-Chandra isomorphism
4
identifies Z with the ring Z(t)' of polynomials on t invariant by the Weyl group W of G. One should think of 0 first as a collection of isomorphisms, one for every point factors of all Borel subgroups of G with t. By transposition, we obtain b* : SpecZ : A
'-4
-
of the full flag space, which turns out to be constant once we identify the Cartan
XX. It follows that XA = X1 if and only if A = wit for some w E W.
Lemma 4.2: Let X be the flag space of type P with Levi factor L, and let A E t*. Then the character 0 :Z -+ CC DX coincides with XA+p,.
The proof of this lemma is a standard gymnastic exercise with p-shift. The shift by p, reflects the difference in the action of T on the volume forms of the full flag 17
variety and of X.
5. Global sections of 9A Our goal in this section is to describe '(X, DA) in terms of the enveloping algebra U. In the case of the full flag variety, this question has a nice answer, but for partial flag varieties the general situation is not clear yet. We shall first construct some bigger sheaves of algebras D, and Dt, which will give us some insight in the problem. Recall that po = {
I
Ego I e,, Ep., Vx E X}, and similarly define n* = {e Ego
e En,, Vx E X}. These are both G-invariant subsheaves of g*, hence these are
ideals in g*. Definition 5.1:
Pt,
DP = U*/Uon*
=
U*/U 0 p*
The center of D is isomorphic to the center Z(f) of U(e), while the center of Ae, is isomorphic to Z(tp) = U(t,). We can view these algebras as living respectively on SpecZ(f) = t*/WL and SpecZ(t,) = t.
into t.
Of course t* = (t*"L imbeds naturally
D, is the specialization of De along the subvariety t*, and for A Et*, DA is
the specialization of Dt, at the point A. Geometrically we can interpret the sheaves D and Dt, as follows. Identify X with G/P and let P=LN be a Levi decomposition of P. We have to fibrations 7r =
G/N -+ G/P and r = G/P1 -+ G/P. Then P1
=
?ro(DG/N)L is the direct image
in the category of 0-modules of the sheaf of differential operators on G/N which commute with the right action of L on the fibers of 7r. Similarly, Dt, = Tr(DG/PIp)C
where C ne Tp is the connected component of the center of L. We have the Harish-Chandra isomorphism for L : '/4 : Z(e) --+ Z(t)W" which involves only a shift by pl. We also have a homomorphism 0 : Z -+ Z(e) which involves a shift by pp. Let us denote by H the space of harmonic polynomials on t for the action of the Weyl group W of g; dim H(g) =
#
holds for L and H(e)_;H. Chevalley's theorem asserts that: Z(t) = H 0 Z
Z(t) = H(t) ® Z(t) 18
W. A similar notation
#
It is readily seen that Z(e)= HWL®Z, so that Z(t) is a free Z-algebra on
W/WL generators. Indeed H is isomorphic to the regular representation of W and HwL is isomorphic to the space of functions on W/WL. Definition 5.2: Ut = U OzZ( E) There is a natural map Z(t) -- D1, because although p*/n* is not necessarily a trivial bundle over X, the ambiguity dissappears if we consider the center of the enveloping algebra of 1,. Moreover the restriction this map to Z coincides with the restriction of U -+ De to Z. Hence we obtain a well-defined morphism U1 --+ (X, DA). Lemma 5.3: U, -+
r (X, DI) is an isomorphism.
H'(X, D) = 0
for i > 0.
The proof we will sketch generalizes MilitiI's proof for the full flag variety,
De is an isomorphism
cf. Milicic's forthcoming book. First observe that Z(e)
-+
onto the center of DP. Next to show that
~+ I(X,.), it suffices to
U ®z Z(e)
prove that it is an isomorphism at the graded level. Z. Put Y = G/N , X =G/P,
7r
Set S = S(g), then SG =
: T*Y -+ X and let OTky be the sheaf of regular
functions on T*Y which are right invariant under the action of L on Y and which are homogenous in the fiber variables of the projection T*Y-+Y. Then grD -
7.
Oz
and F(X, 7r0 O#.y) = R(T*Y)L is the ring of regular functions on T*Y which are right L-invariant. Since there is a natural inclusion grr(X, Di) c
I(X,grDj), it
suffices to prove that: S 9z Z(1) -+
R(T*Y)L
To prove this, one resolves the sheaf 7r 0 .f
A
by a Koszul complex C = S* ®
no. C has an obvious structure of left g-module, but it has also a structure r of
right g -module via the formula: r(x)(u 9 v) = -ux 0 v +u 9 [x,v] for x eg, u e So and v e An 0. C is endowed with the usual derivation which preserves the (g,g)-module structure. One proves that this C complex is acyclic. 19
Moreover, S*/S'n* is equal to grDe, thus C is a left resolution of the (g,g)-module DP. There is a third quadrant spectral sequence whose term E"'- is HP(X, S* ®0
Aq n"*C) and which abuts to its term E;
=HP-(X,grDj).
Now we can compute HP(X, S 0 ®o Aq n*) identify AqnO with
r4j
S ®HP(X, Aq no). Indeed we can
=
the sheaf of holomorphic q-forms on X via the Killing form on
g. By Dolbeault's theorem HP(X, 11q) ~H!' (X), and since X is a compact Kihler smooth manifold, the Hodge theorem implies that H"(X, C) ~ ep+q=nHa'P(X). On
the other hand, the cohomology of a flag variety is generated by the fundamental classes of the Schubert cycles. Algebraic cycles live only in degrees p=q. There is a natural length function on the quotient W/WL; let us denote by 1, the number of elements of length p in W/WL. Then we obtain: HP(X,Aqno) =0
ifp : q
dimHP(X, APn*) = 1, This implies that the spectral sequence degenerates, and that: E,= Hn(X, grD) = 0
forn 7 0
grEo, = grr(X,grD) = S®H*(X,C) H* (X, C) is nothing else than the space of WL-invariant harmonic polynomials on t* for the full Weyl group W. It follows, using Chevalley's theorem recalled above, that S ®z grZ(f) is isomorphic to grr(X,grD). And since the gradations correspond, we obtain the desired result.
E I have not found a simple description of F(X,DP,,), but at least the following can be said. Let X1 = G/P1 , and let D(X1 ) denote the algebra of algebraic invariant differential operators on X1 . The action of G on X1 gives a morphism op : U -* D(X1 ) called the operator representation of U on X1 . Let I(X1 ) denote its kernel. Proposition 5.4: (Borho-Brylinski)
I(X1 ) = Ann(U ®U(p) C) = 20
f
Ann(U ®u(p) CA)
The map 7r : X1 -+
X is a G-equivariant Tp-fibration.
The ring 7r 0OX, is
graded by the lattice of characters of Tp acting on the right of X1 and we have the corresponding gradation on 7rDx,. The zero component is just
Dt,, and since
the G-action commutes with the right Tp-action, roop(U) lies in L(X,Dt,). On the other hand the right action of Tp on every fiber of 7r gives a monomorphism r:Z(tp) -+
D(X1 ), and by the commutativity of Tp, irerZ(t,) lies also in L(X,Dt,). Now composing the Harish-Chandra homomorphism V) :Z -+Z(t) with the natu-
ral projection Z(t)-+Z(t,), we can view Z(tp) as a Z-module. Then it is not difficult to see that there is a well-defined monomorphism:
U/I(X) (
Z
Z(t,)
Note that when X is the full flag variety, I(X1 ) = 0, and the above map is an isomorphism by proposition 5.2. Now we examine the global sections of D), A Et*. Recall that A determines a character XX:Z-C.
Proposition 5.5: (Beilinson-Bernstein-Brylinski-Kashiwara)
If X is the full flag
variety, for any A Et*, we have:
U/U.Kerxx ~+ P(X, Dx)
For a proof, see [Militie]. Proposition 5.6: Let X be a flag space of type P and A Et*. If A+ PL is dominant in t, then U -* P(X,DA) is surjective with kernel I(X 1) + U.KerXA. Proof: Consider the map 7r : Y -+ X where Y is the full flag variety.
r'Dis a
DY,A+PL-module on X. By applying Dy,,\+p, to the section 7r'(1), we get a surjective map of sheaves Dy,A+,p, -- 7r*D,\. Now the functor F(Y,-) of global sections is exact because A + PL is dominant, c.f. theorem 1.6.3. Hence F(Y,Dy,A+P,)-+r(Y,y,A+P,) is still surjective. But U surjects onto the first algebra by the previous result, and
r(y, 7rDA) = r(X,DA). Thus U surjects onto L(X,DA). The assertion on the kernel is clear from the discussion above.
21
Remark 5.7: If A + pi is not dominant in t, surjective, even though A is dominant in
t.
then U -- F(X,DA) may not be
A example with unitary highest weight
modules for SP 4 (R) (8 x 8 matrices)-communicated to me by D.Vogan- exhibits this phenomenon. However it is always true that U/(I(X + U.Ker xA) injects into J(X,DA).
6. P-affine varieties The theory of Beilinson and Bernstein works as well over any algebraically closed field k of characteristic zero. So let X be a scheme over k. Define an Oxring R to be a sheaf of rings on X together with a ring morphism Ox --+ R such that R is quasicoherent as a left Ox-module. An R-module is then a sheaf of left R-modules, quasicoherent as a sheaf of Ox-modules. Denote by M(k) the category of R-modules. F : M(R)
Put R := L(X, k). There are natural adjoint functors
MOk): A where P(M) := P(X, M) are the global sections of M and
A(N) := R OR N is called the localization of N. r is left exact, A is right exact and we have the derived functors Rr and LA. Definition 6.1: We say that X is R-affine if r and A are (mutually inverse) equivalence of categories.
Here is a criterion for R-affinity. Proposition 6.2: If every R-module is generated by its global sections and H'(X, .M) =
0 for i > 0, then X is R-affine. This proposition says that if r is exact and faithful, then it is an equivalence
of categories. It is clear by Serre's theorem that any affine variety is R-affine. Dx is an Ox-ring; we shall see that any flag space X is Dx-affine. A Et,, of twisted differential operators on a flag space X of type P for G. For simplicity we consider only the case k = C. The dA for A E Mor(Tp, CX) define a lattice in t* and hence a real structure. We shall say that A Et* is P-dominant if for any root a E R+(t,) we have < A, av >$ 0, -1, -2,....
We shall say that A Et* is P-regularif for any root
a E R+(tp), we have < A, av >$ 0. Recall that the positive roots are those which
22
are in g/p. Via the inclusion t, c
t*, we view the elements of t* as elements of
t*, and there is a well-defined element pt=
Pb - Pp
Et*.
Theorem 6.3: (Beilinson-Bernstein) 1. If A Et* is P-dominant, then the functor
r : M(A)
-+ M(DA) is exact.
2. If A Et* is P-dominant and A+ peE t* is B-regular, then the functor
r
is also
faithful. Thus under the conditions of the theorem, X is DA-affine. The case of the full flag variety is explained in [Beilinson-Bernstein 1981] and this theorem can be proved in an similar way. We will only describe the changes for the key lemma. Set
0
=0x. Let F be an irreducible G-module, Y = 0 0c F the corresponding
(.T),
G-sheaf. Let
for i = 1,... , k, be a filtration of 7 by G-sheaves of 0-modules
such that the quotients .j/.-
1
~ V (a,) correspond to irreducible representations
of the Levi factor L of P with highest weight Ii Et*. If IL Vt*, V(tt) is not a
D-module strictly speaking because it is not an invertible sheaf. It is the sheaf of sections of a homogenous vector bundle over X and what really acts on V(It) are twisted matrix differential operators. The ring of these operators is still generated locally by 0 and the enveloping algebra U. Hence
(I)
is a U*-module and the
center Z of U acts on it by the character Xm+,'b. Let us call L-weights of F the weights 1L Et* which appear in the Jordan-H6lder series of F. Let 1L be the highest weight of 7 and v be the highest weight of F*. Set 7(pt) = V(g) Oo Y . Let i:
71(v) -- F(v) and p : F -+
i
idm : F1(v)®)9 -+ F®l (v) and pm =p9idM : 1 9M -* M{(IL). Observe that if
Fk/7-1
~ V(ts). For any 0-module M, put iM =
V Et*, then 71(v) ~ 0, and iM : M --+ 7
M (v). Now let M be a DA-module, then
all the sheaves considered above have a structure of U*-modules by the Leibnitz formula and ijM, pm are morphisms of U*-modules. Lemma 6.4:
1. Take V Et*. If A Et* is P-dominant, then iM has a right inverse
jM (unique) in the category of U*-modules. 2. Take
,
Et *. If in addition A + pt Et* is B-regular, then pm has a right inverse
qg (unique) in the category of U*-modules. Proof: Consider the filtration .5 0 M(v) of 7 0 M(v). It is easy to check that 23
the subquotients .F E M(v)/..1 ® .(v)
= .M(A + v) are U*-modules on which Z
acts by the characters Xi = Xx+A+v+p.
Claim: The weight A + Pt = A + Al + v + pt E t* is not conjugate by the Weyl group of G to any weight A + It, + v + p, for i > 1. To prove such a statement, we can assume that A is dominant in the analytic sense,i.e. < ReA, av >;> 0 for all a E R+(t,). For the Bernstein-Gelfand principle will make the passage from the analytic notion of dominance to the algebraic one, cf. appendix in [Bernstein-Gelfand]. Now suppose that there exists some element w E W such that w(A + pe) = A + Ai + v + pt, for some i, Since si Et* is the lowest weight of F with respect to t, -it positive roots. Let us introduce a norm product and let resp. t-',
1
+ Ai is a sum of
on t*coming from a W-invariant scalar
I - IN,resp. - L denote the composition of the projections onto t*,
followed by the norm. We have I
I A 1=1 A IN and I Pt
1=1
1=
IN + IXIL for
x Et. Then
Pt IL. Since W acts by isometries, if A + pt is conjugate to
A + jyi + v + pt, they must have the same norm. But
i+v+p IN unless ui =kti,
I A + PIIL
i .M such that
i'(f) = idM. Denote Imf by i!.M. If M is irreducible, then i!.M is the unique 25
irreducible submodule of i.M and the unique irreducible quotient of fiM (and the unique irreducible subquotient of any of these D-modules whose restriction to Y is non-zero). The modules i.M, i.M are called standard modules or also respectively maximal and minimal extension of M, while i .M is called the irreducible module corresponding to (Y, M) or also the middle extension of M. By definition a D-module is holonomic if it has finite length and all its JordanH6lder components are irreducible modules of the type constructed above. One says that a holonomic D-module (on compact X in the twisted case) has regular singularities (RS for short) if all its components originate in bundles with regular singularities at infinity. The basic property of holonomic modules is that the corresponding derived category of complexes with holonomic cohomology is stable under the functors of type
f 1, f.;
if M is holonomic then * M is also holonomic. The same
applies to holonomic RS. Let us return to representations of g. We are going to study (g,K)-modules for certain algebraic groups K such that the connected component K' is a subgroup of G. First we say that (g,K) is a Harish-Chandrapair if g is a complex Lie algebra, K is a complex linear algebraic group (possibly disconnected) such that k is a subalgebra of g and there is a compatible map Ad: K -- Intg. A (g,K)-module M is by definition a representation of g and an algebraic representation of K on the same linear space M such that the representations coincide on
k
and the map g
xM -+ M is K-equivariant. (g,K)-modules correspond via the functor of localization to (D,K)-modules, i.e. D-modules M with an action of K such that the imbedding
k -+D
k
acts via
and the map D x M --+ M is K-equivariant. The case where K
does not act by inner automorphisms on g is also interesting. To include it in this framework, it suffices to assume that the group G' generated by G and the outer automorphisms of g given by K, acts on the flag space X and on the tdo D. To get an interesting theory one needs sufficiently large groups K. Say that K is admissible if K acts on the full flag variety of G with finitely many orbits or, equivalently, if k is transverse to some Borel subalgebra. Fix an admissible K. It is not hard to see that any coherent (D,K)-module is smooth along the stratification 26
given by the orbits of K, so is holonomic and has regular singularities. The irreducible (D,K)-modules are in bijective correspondence via the if, construction with the irreducible smooth (D(y),K)-modules on the various affinely imbedded K-orbits Y, and these smooth modules are charaterized by representations of the stabilizers of points. This readily gives a classification of irreducible (D,K)-modules and so of (g,K)-modules. For any K-orbit Y, put Ty = K n Py/K n (Py, Py) 9 Tp where y E Y (note that Ty does not depend on y E Y). Ty is the product of the torus Ty and the finite abelian group Ty/Ty. Theorem 7.1: (Beilinson-Bernstein) For A Et,* the irreducible (Dx, K)-modules are in bijective correspondence with the set of pairs (Y,ry) where Y is a K-orbit in X and Ty is an irreducible (t,Ty)-module on which t acts by A - p,. If X is the full flag variety and if A Et is dominant regular, this is also the classification of irreducible (g,K)-modules with infinitesimal character xA. On a partial flag variety X of type P, under the hypothesis that A E t,*, is P-dominant and A + p E t* is B-regular, then we obtain the classification of (g,K)modules with infinitesimal character Xx+p, and associated variety contained in the closure of the Richardson orbit of P. To work clearly with the standard modules one has to suppose that the Korbits Y are affinely imbedded. We see that the standard and irreducible modules corresponding to an orbit Y form families with dim(tp/ty) continuous parameters and dimTy discrete parameters. All standard modules are irreducible for generic values of the parameters. If they are irreducible for all values of the parameters then Y is a closed orbit. The groups K we will consider are the fixed points of involutions of G. This corresponds to the Harish-Chandra modules or representations of real reductive groups. Then the standard modules i. correspond to the standard representations, cf.[Vogan]. One could also take K=N or B where B is a Borel subgroup of G and N is its unipotent radical. This corresponds to representations of g with highest weights. These cases can be reduced to the first - although in general one prefers to go the other way - thanks to the facts that N-orbits are equal to B-orbits and 27
B-orbits on X are in bijection with G-orbits on XxX. This yields the equivalence between representations with highest weights and representations of complex reductive groups. In general the hypothesis in the theorem 6.3.2. can be weakened. For a (DX,K)module M with K is reductive, if A Et* is P-dominant and A + pt Et* is regular with respect to the roots of K, then I(X, x) 5 0. This follows from the Borel-Weil theorem. In the following chapters, we will only consider coherent (D,K)-modules and finitely generated (g,K)-modules. So we add this hypothesis to the definitions. For any map f : Y -+ X between smooth algebraic varieties, define the functor
*f!*.
28
f*
=
Chapter II. Spherical D-modules 1. K-orbits in a flag space. The results of this section are known to specialists. Let G be a complex connected reductive linear algebraic group. Let B be the variety of Borel subgroups of G, and P the variety of subgroups of G conjugate to a fixed parabolic subgroup P. Let K be an algebraic subgroup of G; it acts on B and P. If the number of K-orbits is finite, then there is a Zariski open K-orbit which is automatically unique and dense. Conversely: 1.1 Lemma:(Brion) If an algebraic group acts on a flag space with an open orbit, then the number of orbits is finite. Two subgroups K and B of G are said to be transversal if k + b = g. The existence of an open K-orbit on B is equivalent to the existence of a Borel subgroup B transversal to K. We have a G-equivariant fibration 7r : B -+ P which assigns to a Borel subgroup B the parabolic subgroup P E P containing B. Therefore the finiteness of the number of K-orbits in B implies this finiteness in P. The fiber of 7r over P is the variety of Borel subgroups of P. Note that if Y is the closure of one K-orbit in P, then 7r- 1 Y is a closed K-stable subset of B with the same number of components as Y hence, it is the closure of one K-orbit in B. Let 0 be an involution of G and put K = Ge. Then any Iwasawa decomposition of g with respect to k shows that K acts with finitely many orbits on B. 1.2 Lemma: The orbits of K are affinely imbedded in B. Proof: Consider the map h : B -+ B x B : B -+ (B,OB). The diagonal action of G on B x B decomposes this variety into #W orbits of G where W is the Weyl group of G. To w E W corresponds the G-orbit C, of (B, wB) where B is any Borel subgroup of G. Claim 1: The G-orbits C, is affinely imbedded in B x B for any w E W. Indeed, let us consider the projections pi and P2 : B x B --+ B on each factor. For simplicity we fix a Borel subgroup B of G. The B-orbits in B are called the 29
Bruhat cells: they are indexed by W and we denote by B. the B-orbit of wB. B,,, is an affine space of dimension (w). Let wO be the longest element of W and B the corresponding cell. Then B is open in B. Consider the open subset V = piI(B0) n pjl(B) C B x B; it is isomorphic Since B =
to A21(wo).
U
wB*, the subsets
VW1,W
2
= p 1 (wiB0) nl p21 (w 2 B0), for
wEW
wi, w 2 E W, form an open cover of B x B by affine spaces. It suffices to check that Cw n VW 1 ,W is affine for any w 1 , w 1 , w 2 E W. Consider the map pi : Cw n pi(B*) -+ B* ~ A(wo). It is surjective since C. = G - (B, wB) and the fiber over w 0B E B is 2
{(w,B, wObwB) I b E B} ~ B, ~ A-(w). Since it is B equivariant, it is a fibration over an affine space with affine fibers. Restricting B* to a smaller open subset U of B, if necessary, we see that Cw n pi 1(U) is affine. Now the complement of BO in B is the set of orbits of non-maximal dimension; it is a connected hypersurface H of B. Hence C, n pi 1 (U) n pil(B ) is the subset of B x B obtain from C,,, n pi 1 (U) by removing the points of B x H. But C n pi I(U) n (B x H) is a connected hypersurface in C n pj1(u). Hence C, n piL(U) n pi 1 (B ) is still affine. A similar
.
arguments applies to the other open sets VW,,W 2 We continue the proof of the lemma.
Every K-orbit in B is mapped by h into a single G-orbit. Consider one G-orbit Y in B x B, h-1Y is a K-stable subset of B, which may be disconnected. Claim 2: The K-orbits in h-'Y are all open subsets of h-1 Y. Let us first show that this claim implies the lemma. h-1Y
B
-+
h
Y BxB
This is a Cartesian square and j is an affine morphism by claim 1. Since base change preserves affinity, i is also an affine morphism. Claim 2 says that a K-orbit in h-1 Y is a union of connected components of h'1Y. Hence it is affinely embedded in B. 30
To prove claim 2, consider two points B and B' in h-1 Y which are close to each other, so that we may write B' = exp x B for some x Eg.
Then h(B') =
(expxB,O(expx - B)) and h(B) = (B,OB). Since h(B) and h(B) belong to the same G orbit Y, for B and B' close enough, there exists y Eg such that (exp x B, O(exp x B)) = (exp yB,expy OB). But a Borel subgroup is its own normalizer, hence exp(-x) exp y E B and exp(-x) exp Gy E B. Since z and y are small, (exp)~ 1 exp y ~ exp(-x + y) and exp(-x) exp Oy = exp(-x + x - y Eb and x - Oy Eb. Therefore x -
Dy).
Hence
8 Eb and this implies
B' = exp (
2
B
Obviously exp(Yev) E K. Thus B' is in the same K-orbit as B. This proves the claim and finishes the proof the Lemma 1.2. RNow let us consider the K-orbit on a general flag space P. The finiteness of the number of orbits follows from that for B. But these K-orbits need no longer be affinely embedded. 1.3 Example: G=G1 3
P= 0
*
*
P=P 2 (C)
0~ O = Ad diag (1, -1, -1)
so that K = G1I x Gl 2 =
0)* 0*
*.
Then K has three
orbits on P: the point 0 corresponding to the line Iz, 0, olin C' and the hyperplane H ~ P1 consisting of the lines contained in {[0, X 2 , Xs1 I
X 2 ,xi,
E C} and the
complement C of these two first orbits. C = P 2 \ {p1 U 0} = A 2 \ {0} is not an affine variety. Note that P has two orbits on P: {O} and P 2 \ {0}. The latter is again not affine. If we analyze the proof of lemma 1.2 we first have to replace B x B by P x6 P where 'P denotes the set of parabolic subgroups of G conjugate to OP. Note that although there always exist 0 stable Borel subgroups, OP need not be conjugate to P.
(Take for example GL 3 and K = 03). The G orbits on P x' P are still 31
paramelrized by W/WL where L is the Levi factor of P and so are the B-orbits on P. PxIP=
I
G-(P,6(wP)) P=
I
BwP
WEW/WL
WEW/WL
The B-orbits on P are affine spaces. Let Bw,P = P0 be the big cell and {C' =
G -(P,0(wP))}. Then Cf n p-1(P*)
--
P0 is a fibration but the fiber over wP is
isomorphic to P - 6(wP), i.e. a P-orbit on OP. The trouble is that the P orbits on P or 'P need not be affinely embedded, cf. example above. The second part of the proof of lemma 1.2 extends easily to the general case. The fact that a K-orbit Y is affinely embedded means essentially that the boundary 8Y of Y has codimension 1 in Y. However, a closed subset of an algebraic variety is always affinely imbedded.
Hence the closed K-orbits in P are affinely
imbedded and smooth. The K-orbit of a Borel subgroup B is closed if and only if B is 8-stable, [Matsuki, Springer]. Similarly: 1.4 Lemma: The K-orbit of P in P is closed if only if P contains a 6-stable Borel subgroup. Proof: Let Y = K - P and
7r
: B -*
P. If Y is closed then ir-'Y is the
closure of one K-orbit. Hence r- 1 Y contains a closed K-orbit say K - B in B. By Matsuki-Springer's characterization, B is 0-stable and since 7r(B) E Y, there is a K-conjugate of P which contains B. Hence P contains a 0 stable Borel subgroup. Conversely, if P contains a 8-stable Borel subgroup B, then Y = r(K-B). Again by Matsuki-Springer, K -B is closed, hence compact. Therefore Y is compact. [I Note that if K - P is closed, then K n P is parabolic in K and K- P is isomorphic to the flag space of K of type P n K. The map h - B --+ B x B : B e-* (B,8B) can easily be related to Springer's parametrization of K-orbits on B. Choose a 8-stable Borel subgroup B and a 0stable cartan subgroup T in B. Let A = {g E G I g 1 8g E NG(T)}. K acts on the left of A and T on the right. Put V = K \ A/T. 1.5 Proposition:(Springer) B = f K - vB vEV
32
On the other hand B x B = ]I G- (B - wB). Since h maps K-orbits into wEW
G-orbits, it induces a map h : V -+ W
n
'-4
n-1 n. The image of h consists of
0-twisted involutions, i.e. elements w E W such that w - O(w) = 1.
II.2. (D, K)-modules with a K-fixed vector Let g=keaen be an Iwasawa decomposition of g with respect to 0. It is not unique but all choices are conjugate by K. At the group level KAN is only an open dense subset of G. Let L = CentG(A) and P = LN; this is a Levi decomposition of the parabolic subgroup P. The G-conjugacy class of P is uniquely determined by 0, and we say that P is associated to K. The corresponding flag variety X = P is also called associated to K and K - P is the open K orbit in X. Note that if G(0, R) denotes a real form of G whose Cartan involution is 6 and if a is chosen to be defined over R, then P is the complexification of a minimal parabolic subgroup of G(O, R). An (g, K)-module V is called K-spherical if the space VK of K-invariant vectors in V is nonzero. As will be explained in the next section, the irreducible K-spherical (g, K)-modules are those which are realizable as functions over the real symmetric space G(0, R)/K(R). Kostant classified the irreducible K-spherical representations of G(0, R): they all are quotients of principal series representations induced from one-dimensional representations eA ®1N, A E t,, of a minimal parabolic subgroup P(0, R) = L(, R)N(0, R) of G(0, R) cf. [Kostant]. Let D be a tdo on X. We shall say that a (D, K)-module M on X is K-spherical if Hom(O,K)(0, 4) 0. This notion is interesting mainly for simple or standard (D, K)-modules. We also say that M has trivial K-isotropy if for every K-orbit Y in the support of M, the isotropy group Ky acts trivially on the fiber of M at y E Y. 2.1 Theorem: Let M be a simple or a standard (D, K) module on X. Then M is K-spherical if and only if supp M = X and M has trivial isotropy. Moreover M has at most one K-invariant section, up to scalar. Proof: Hom(OK)(0, M) is a vector space and we want to compute its dimension. One can verify that the only K-orbit on a flag space which can support a 33
standard (D, K)-module containing the trivial representation of K is the open Korbit. The sections of 0 are determined by their restriction to the open K-orbit X*. Le i :
c-
X0 be the inclusion of a point in X*. By K-equivariance, it suffices
to compute dim HomK, (1, i'M) where K. is the stabilizer of x in K and 1 is the trivial module C. This expression shows that if M is K-spherical, then x E suppM, and by K-equivariance X 0 C suppM, hence X = supp). Let j : X 0 c+ X. Since M is simple or standard it can be written as jl, j.L or jI.L for some line bundle L on X0 corresponding to a representation r of K. Now i'M = i' dim hom(1, C,)
=
1 if r is trivial
=
0
= C,. Hence
otherwise.
The converse assertion is clear from the above discussion. LI 2.2 Remark: Harish Chandra has proved that if 6 is an irreducible representation of K and V is an irreducible quasisimple Banach space representation of G(O, R), then mtp(6,V) < dim(6) cf. [Godement]. In particular if 6 = 1 then this implies mtp(1, V) E 1, as in the above proposition.
II.3. Relations with the analytic theory. It is inspiring to bear in mind the relations between the D-module picture and the analytic picture on symmetric spaces. In this section we adopt the notation of the analysts. Let G denote a real reductive Lie group obtained as follows. Consider the complex connected reductive algebraic group G, (that we previously denoted by G) and let 0 be an involution of G, with fixed point group K, Take G = G,(6, R) to be a real form of G, such that 0 is a Cartan involution of G, i.e. K = KC(R) = K,(0, R) = K, n G is a maximal compact subgroup of G. Consider another involution a of G, which commutes with 0. Let H, be the fixed point set of a in GC and let H be H, n G. Using a we can also define another real form of Ge, namely Gd := G, (a, R) and Kd := Kn Gd. Hd := Hn Gd. Observe 34
that by definition Hd is a maximal compact subgroup of Gd. The symmetric space Gd/Hd is Riemannian and is called the dual of G/H.
Consider an Iwasawa decomposition of g, with respect to a : gc = hc E a e n, and let W(a) be the Weyl group of a in g,. Let D(G/H) and D(Gd/Gd) be the algebras of invariant differential operators on G/H and Gd/Hd respectively. They are naturally via holomorphic differential operators on GC/H and there is an isomorphism D(G,/H) ~ S(a)w(") or if you prefer a*/W(a) ~ MaxD(G,/H,) where a shift by p(a, n) is included. In particular these algebras are commutative and every A E a* defines a character XX of both algebras. We are going to study the irreducible representations V of G which occur in C**(G/H); as is customary we assume that V is quasi-simple, i.e. the center Z of U acts by scalars on V. Note that Z is the algebra of left and right G-invariance differential operators on G, hence there is a natural map Z -+ D(G/H). Thus we reduce the problem to study the irreducible representations of G which occur in C* (G/H, xX) = the XX-eigenspace of D(G/H) in C (G/H). Next by a result of [Casselman - Wallach] there is a natural C* topology on any irreducible Harish-Chandra module coming from a canonical globalization of M which is an irreducible smooth representation M'" of G. In particular one can deduce from their result that every K-invariant functional on M extends continuously to M', i.e. HoMK (M,
1)
= HoMK (M , IL) where the second Hom contains
only continuous functionals. This result implies that the study of the irreducible G-submodules of C**(G/H) is equivalent to the study of the irreducible (g, K)submodules of C**(G/H) := the space of K-finite differentiable functions on G/H. Assembling these two observations, the problem is reduced to the decomposition of the (g, K)-modules of C'*(G/H; Xx). Because of the K-finiteness these eigenfunctions are automatically real analytic. Thus, the object of interest is really the space A(G/H; Xx) of real analytic K-finite functions on G/H which are eigenfunctions of D(G/H) for the eigencharacter XX. 3.1 Lemma: Let V be an irreducible representationof G with infinitesimal character XX. Then the multiplicity of V as submodule of C*(G/H) is finite. 35
Observe that this multiplicity is the dimension of Homc(V, C*(G/H)) = HomH(V,
by Fr5benius reciprocity.
1)
Let VH be the space of H-coinvariants of V.
VH =
L1HV , V/hV surjects onto VH. and (VH)* = HomH(V, 11) = (V*)H. By the above
result of Casselman-Wallach, V/hV = V*/hV* where V* is the Harish-Chandra module of V. V 0 /hV 0 can be considered as a space of functions on G which are eigenvectors of Z, left K-finite and right invariant by H . These conditions form a holonomic system of differential equations with regular singularities. It is known that its solution space is finite dimensional. Let us mention that the multiplicity of V in C* (G/H) as submodule i.e. dimVH can be quite smaller than the multiplicity of V has subquotient. However, they are both finite and thanks to the following result we have an upper bound on the multiplicity of V as subquotient. Let W be the complex Weyl group of g.
Let
g = h e r = k e s; let a, be a maximal abelian subspace of r n s and t = cent(a; g). Let W1 be the complex Weyl group of t. 3.2 Proposition: [van den Ban] Let 6 be an irreducible representationof K, and A E a*. Then:
mtp(6; AK(G/H; X,\)) HxHnKX a
X
where p is the projection on the second factor and q is the quotient map given by the diagonal action of H
n K on the right of H and on X, and a is the action morphism
of H on X. 4.2 Lemma:Let Y be a smooth variety on which the group B acts freely.
Put
q: Y-> Y/B. Then q* : DYIB - mod -+ (Dy, B) - mod is an equivalence of categories.
Recall that q* is the inverse image in the category of 0-modules. Denote by q+ the inverse of q*. It is related but in general different from the direct image functor.
4.3 Definition:Let M be a (D, K) module on X.Set LHMX K:= a~q+pr FKnH Ffl(M) Kp
Ki M
is a well-defined (D, H)-module for D has a G-action. q+ enters the formula
because we want an H-action on LHM which is as compatible as possible with the
K-action on M. We will only need the case where H/K n H is an affine variety, then a. is well-defined without recourse to derived categories. 4.4 Theorem: 2H o A)
A, o LK
for A dominant in t*,.
The proof of this result is explained below. Proof of Proposition 4.1:
L-' ~ rd-i 0 Ad(k/b)
Let M be a (g, B)-module and V a (g,K)-module. We have to relate Hom(g,K)(V, PM) and Hom(g,K) (V, LM). A (g, K) module V is just a K-module with a map g
®
V -+ V compatible with
the representations of K on both sides. r and L commute with tensoring by the
42
finite dimensional representation g of K. So it suffices to relate the spaces HomK(E,IM) and HomK(E,LM) where E is now a finite dimensional representation of K. Moreover HomK (E, PM) = HomK (1, E* o FM)and HomK(E, LM) = HomK(11, E* ® LM). Again, because r and L commute with tensoring by algebraic representations of K, it suffices to relate the spaces. HomK(11,
M) and HOmK (IL, LM)
Now HomK(1, PM) = Hom(k,B) (11, M) and the right derivatives of this module are the spaces H*(k, B; M). On the other hand HomK(I, L,M) = IL(kB)M and the left derivatives of this module are the spaces H. (k, B; M). Let I be the ideal generated by b in the exterior algebra In
At k.
A* k
and put I =
Cohomology is computed using the standard complex A* k/I' with the
usual differential, which gives an acyclic (k, B) resolution of the trivial module 1. Then the i-th homology group of the complex Hom(kB)(A k/I', M) is H'(k, B; M).
Homology is computed using the same standard complex. The i-th homology group of the complex A* k/I'®(k,B)M is Hi (k, B; M). Let b' be the orthogonal of b in k*; b = (k/b)*. Then Hom(k,B)(Ak/I', M) ~ A'b'
0
M.
(kB)
The top degree of A* k/I* is d = dim(k/b).
Moreover as (k, B) modules:
A' k/I' ~ A'(k/b). Now we have an identification of (k, B) modules.
ty:A'(k/b)-+ A d-ibL (9 Ad(k/b) because Ad-i bJ- is dual to Ad-i (k/b) and we can view A'(k/b) as the space of linear maps from Adi (k/b) to Ad(k/b). Thus we obtain Poincare duality d
Hi (k, B; M) ~- H
i (kB;M0(k/b))
43
The left hand side is L-'HomK(Il, LM) while the right hand side is Rd-iHomK(1t, PM ® This proves the assertion. F1 Proof of Theorem 4.4:
Xio = Ax o Lfor
A E t, dominant.
Let M be a (Dx, K) module on the flag variety X. Since A is dominant, it suffices to prove that:
r(x, 'K)
=
Lr(x,
m)
2CHM = a~q+p0 .M X - H x X Xq H xKnH X + X
Let us consider
'CH.M
has and (0, H)-module first: F(H xX,p'M) = R(H)0 T(X, m)
where R(H) denotes the ring of regular functions on H. Also P(H x Kf Finally, r(X, a,q+p
X,q+p*M) = R(H)OKnHr(X,M)
) = R(H)
0
P(X, m). So at the level of H-modules
h,KnH
the functor CH commutes with F(X,.). Now we examine the g-action using the functorial properties of L. Built in the definition of L, there is a projection formula LKnH(V ®M) = V ® LnH(M) for any (g, H) module V and (g, K n H) module M. A (g, H) module M is just an (h, H) module together with a map g 0 M -+ M of (h, H) modules. The projection formula gives a map g O LH(M) = L4(g 0 M) --+ L(M). Thus L4(M) is a (g, H) module, when M is a (g, K) module. Z'(M) comes naturally equipped with a DA-module structure. But we can view it in the same functorial way as above, because the t.d.o.Dx has a G-action. If V is a
(DA, H) module on X and M is a (DA, K n H) module on X, we have the projection formula: LHfl(V O.M) =V 44
ZeHflH(A)
In particular if we take 1 = 0 and if M is a (Dx, K) module on X, we obtain a map DA
L' ()
(M)
-+
which give the same structure of Dx-module on L'(M) as the one given by the direct definition of
L.
The forgetful functor F obviously commutes with L(X,.). By parallelism, it is easy to see that the following diagram commutes: (DX, K) - mod
r(x')
(g,K) - mod
(DA, H) - mod
r(x,.)
(g, H) - mod
This proves that L commutes with r(X,.), By adjointness, the same is true for L and A. Indeed for any (g, K)-module M and any (DX, H)-module M, we have Hom(D,,H)(AxL H(M), IM) = Homg,H)(LH(M), r(X, m))
= Hom(g,KnH)(M, r(X, FM))
=
Hom(g,KnH)(M, FP(X, .))
=
Hom(A,KnH)(AA(M), FM) = Hom(DKnH)(Ax(M), FM) =
Hom(p,H)(ef AA(M), .).
n
II.5 H-spherical (D, K)-modules. We continue with the complex connected reductive linear algebraic group G. Suppose a and 0 are two commuting involutions of G with respective fixed point sets H and K. Let P be a parabolic subgroup of G attached to H by an Iwasawa decomposition, and let X be the flag space of type P. Put X* = H - P; it is the unique open H-orbit in X. Let D, be a sheaf of twisted differential operators on X, A E t*, and consider a (Dx, K)-module M. Recall the functor L from the previous section; we will only use 'CH = LnH
KnH.
ince H/KnH is an affine variety, the
action morphism a : HXKnHX -+ X is affine. In case LHM is located in several 45
degrees we consider only its zero component. Because of theorems 1.6.3 and 11.4.4, we will assume in this section that A E t* is dominant. 5.1 Definition: M is H-spherical if L(X, Since H acts semisimply on r(X,
HM),
spherical if HomH(F(X, LH.M), C) : 0. But
CKM) 0
0.
it is equivalent to say that M is HH
commutes with r(X,.), so this is
still equivalent to: HomH(LHr(X, m), C) = Hom (h,KnH)(r(X, m), C) $ 0.
Thus our definition of H spherical module is compatible with the natural one for (g, K)-modules. For a point x in X, and a subgroup R of G, we denote by R, the isotropy group of x in R; R. = R n P,. 5.2 Definition: M has trivial H-isotropy if for every point x E suppM, j : x -+ X, the isotropy group K n H,, acts trivially on the fiber j'M of M at x, and A - pp = 0 on h,.
Let a,, = t,1 /h, n t,. Then if M has trivial H-isotropy, A A
(Dx, K)-module
E a*
at the point x.
M is called standard if it is the maximal extension i.C of
an invertible K-sheaf C on an affinely embedded K-orbit i : Y -* X, it is called costandard if it is the minimal extension iL, and M is simple if it is the middle extension i4.2. Of course L corresponds to a representation r : K, -+ C' , y E Y, such that dr coincides with A - p, on k. n t,, so we will often denote C by Oy (A, r). We have also the sequence i,
-+ ii.L -
i.L.
5.3 Theorem: Suppose that A+ PL is B-dominant and regular. Let M be a standard (D,, K) -module on X corresponding to the data (Y, r). If M is H-spherical then Y n X* 0 0 and M has trivial H-isotropy. Proof: As for (D, K)-modules, one can easily see that the open H-orbit X* is the only H-orbit which may give rise to a standard (D, H)-module containing the trivial representation of H. dimHomH,(C, j
H(M))
By H-equivariance dimHom(O,H)(O,
where j : x c
KM))
=
X is any point in the open H-orbit X*. 46
jO((M) is simply the fiber of L'(M) at x.
But Y n X0
$ is equivalent to Y n X0
$
suppJ2')(.) = H -suppM = H -.
It is non-zero if and only if x E
because X* is open. When this condition is satisfied, there may exists a non-trivial map C
-+
joC(M)
only if H, acts trivially on
jo L(M).
By the definition of
LH
this is equivalent to r being trivial on (he, K n H,). [I The converse assertion is likely to be true but we cannot quite prove it. The difficulty lies in the fact that in general j'C'M 0 Cizj'.M.
Let us examine the
simplest example. 5.4 Example: SL2 (R)/R*
G = SL 2
H = diagC*
a K ={(b
b 2 2 a I a -b21
X = P1 with homogeneous coordinates [zo, zi]. K has three orbits: Y1 {[1, -1]}
=
{[1, 1]}, Y- 1
and the complement Y.. H has also three orbits: [1,0], [0,1] and the com-
plement X*. Take x = [1, 2] E Yo n X 0 , j : X C-+ X. Take M = i.Oy for i : Yo
Cx. 51
In 5.5, one replaces KnfH-invariantsby the e-isotropic subspace of H*(YnX 0 ; C(r)) for the action of K n H. 5.8 One more example: Let us remark that if .M = i.Oy(r) is a standard H-
spherical (DX, K)-module, .R = il.Oy(r) may not be H-spherical. This happens for example if GR/HR ~ SL 2 (R), i.e. G = SL 2
x
SL 2 ,H = diagSL2 ,K = S0 2
X
S0 2 , X = P1 x P1. H has two orbits X0 and X' in X and K has nine orbits. Let Y be a K-orbit consisting of a line minus a point, then i.Oy is H-spherical but ii. is not. (Compare with theorem 6.9).
11.6 D-modules and hyperfunctions. The goal of this section is to show that a standard (D, K) modules on the flag space X can be mapped into the sheaf BxR of hyperfunctions on X supported on a real analytic subvariety XR.
The map we will obtain generalize the classical
imbedding Ox c-+ BXR First let us pass the bridge going to the analytic set-up. If Xa" is the complex analytic variety corresponding to X, let A be the sheaf of holomorphic functions on Xan, and set Dan = AooD. Since A is flat over 0, the functor an: D-mod -+ Da"-
mod is exact and obviously faithful. Moreover X is a projective variety, so we can apply Serre's GAGA principle to deduce that a" is an equivalence of categories, and it commutes with all functors of direct and inverse images. Let a be an involutionof G with fixed point set H and let G(u, R) be a real form of G which admits a as a Cartan involution. Let XR be the real analytic variety of minimal parabolic subgroups in G(C, R) and let X be its complexification. Thus X ~ G/P where P is associated to H by an Iwasawa decomposition. XR is naturally a real analytic subvariety of X and n = dimRXR = dimcXc. The restriction of the sheaf A to XR is the sheaf of real analytic functions on XR we write AR := j* A where j : XR -+ X. Let PR be the functor of local sections with support in XR: it maps sheaves on X to sheaves on X with support in XR. Let WR be the orientation sheaf of the normal bundle of XR in X. 52
6.1 Definition: The sheaf of hyperfunctions an X supported along XR is
B = MX(A 9WR) Sato's
theorem
says
that
0 WR)
IXR,(A
=
0
for
i
$
n.
Another way to think of this sheaf is B = j.(j'A ® WR)[ n] where J. and j! are functors in the category of sheaves. 6.2 Remark: The functors jI and j. used here are functors between DR-modules on XR an Danmodules on X"". DR is simply the restriction of Dan to XR ; this is well defined because XR is a real form of X. In the case at hand, one can give a direct description of DR as a quotient of AROcU(g) by an ideal JAR determined by the character A - pp, as is done in the complex case, cf. 1.4. In this way one can develop a theory of real D-modules; in particular the shift used in the definition of jI is [dimXR - dimR(X)] = [-n] Suppose as before that K and H are subgroups of G defined by two commuting involutions 0 and
-. Let KR = K n G(o, R) and HR = H n G(a, R). HR is a
compact Lie group acting transitively on XR. 6.3 Definition: K1 = Norm, (G(a,R),K) K1 is a finite extension of KR ; K1 = KR if KR is compact. 6.4 Lemma: Let Y be a K-orbit in X. XR. The correspondence Y
'-+
Then YR = Y n XR is a K1 -orbit in
YR is bijective between K-orbits Y in X such that
Y n XR :5 q and K1 -orbits in XR; moreover dimCY = dimRYR.
This bijection
preserves the inclusion order. Proof: One can compare the explicit parametrizations of K-orbits in X and KR-orbits in XR given in [Matsuki). A K-orbit is determined by a 0-stable, a-stable Cartan subspace a and a set of positive roots R+(a), module conjugation by K, i.e. we should take R+ (a) module WK (a) the Weyl group of a in K. For KR, we have to take only the 0-stable a-split Cartan subspaces a; but WKR (a) may be smaller 53
than WK(a). This difference is corrected by replacing KR by K 1 . It is then clear that Y n YR is a single K1 -orbit, possibly empty if (a, R+ (a)) is not defined over R relatively to G(U,R), i.e. if a is not a-split. The inverse map YR
-+
Y = K -YR is injective thanks to the above parametriza-
tion, because two a-split Cartan subspaces are conjugate by K if and only if they are conjugate by KR. So the bijection is established. It clearly preserves the inclusion order. If Y and YR correspond to each other, then YR is a real form of Y in the sense that it consists of the parabolic subgroups-i.e. points in Y, which are the complexifications of minimal parabolic subgroups of G(a, R), hence dimCY = dimRYR if YR 5 0. L 6.5 Example: Take G = SL 2 (C), H = S0 2 (C), K = diagC*. Then X = CP1 and XR = RP'. We have G(u,R) = SL 2 (R). H = S0 2 ,K = diagR* and K1 as R* n iR*. In homogeneous coordinates, z E K acts on X or XR by [x 0 , X 1 ] -+ [zx 0 , z- 1 x] for xi E C or R. Therefore K has three orbits: the points [1, 0], [0, 1] and their complement Co. Similarly K1 has three orbits: the points [1, 0], [0, 1] and their complement R. which is disconnected. Observe that R. splits into two orbits for the action of K.
f
: Y -+ X be a map between two topological manifolds; set rdf = dimY
-
Let
dimX: the relative dimension of
f.
We shall denote by wyIx = wy/f 'wx the
orientation sheaf of Y relatively to X. This is the sheaf of sections of the line bundle on Y whose transition functions are given by the sign of the Jacobian determinant for the transition functions of the vector bundle Ker(df : TY -+ TX) if f is a submersion, and Nx(Y) = the normal bundle of Y in X if
f
is an immersion. For a
general map, use the factorization into a cofibration followed by a fibration. Recall that the functors f* inverse image, f' inverse image with proper support are defined between the derived categories of complexes of C-sheaves on X and Y, having a constructible cohomology. The next result can be seen as a generalization of the Thom isomorphism. 6.6 Property: There is a canonical morphism of functors:
f* -+ fo
wYlX[-rdf] 54
which is an isomorphism when f is smooth. Proof: We will only need this property for closed imbeddings. To do a little better, we prove it for smooth maps and closed imbeddings. The general case follows by showing that the above morphism is independent of the factorization of
f.
Let S be a C-sheaf on X and assume f is smooth. Then f behaves locally like the projection of a vector bundle p : E -+ X. The Thom ismorphism H'v(E, p* S) ~ HI-~d'(X, S ®9 ptwElX) is given by integration along the fibers, and its inverse is the multiplication with the Thom class of E, cf. [Bott and Tu p. 88]. In the derived category we have: fif*S
~
fl(f1S 0 wEIX)[-rdf]
S 9 fiwEIx[-rdf]
and there is a unique isomorphism 0 : f* S -+ f'1S the identity on S
®
wEIx[-rdf] such that f!(o) is
f*wEIX[-rdf].
If f is a closed imbedding, then
f*S
is quasi-isomorphic to the cohomology of
S restricted to Y, while f'S is quasi-isomorphic to the local cohomology ofS along Y. Let us first take S = C. Then H (X; C) ~e H.'(Nx Y,; C) This is seen by working in a slice of X transversal to Y and by using the fact that, for a point y E X, H (X, C) = Hs(ByIBy; C) = H,(TyX; C) where By is a small ball around y. The Thom isomorphism for the vector bundle p: NxY --+ Y yields HI(Y,wyix) ~ HC,(N Xy, C) Since tensorization by wYIx is clearly involutive, we get: f*C ~> f'C 0 wyx[-rdf] 55
Now for an arbitrary sheaf S, we use Godement's resolution of C by flasque sheaves:
0
injective. 0
-+
-+
C
-+
S
-+
.9
-+
S
-+
- - - Since we work over a field, the g9
are even
9 0 -+ S ®91 -+ - - - is a flasque resolution of S. Let Fy
be the functor of local sections with support in Y, then there is a natural map: s
D rygi
-+
ry(s
D gi).
S
Dy(9.)
is quasi-isomorphic to
S ® Iy (C), so we get
a natural map. f*S D f'c
f's.
Using the result established for C, we obtain the desired morphism. f*S = f*S ® f*C ~+ f*S ® f'C 0 wylx[-rdf| 6.7 Proposition: Let i : Y -+
X,j : Z -+
f'S ®wylx[-rdf.
l
X be two maps between topologial
manifolds. Let S be a complex of C-sheaves an X having constructible cohomology. Then the above morphism yields a canonical map of sheaves: iii'S -+ j*(j'S 0wzlx)[-rdj] Proof: By adjointness we have a map iii'S -- S. Apply j* to get j*i S
-+
j* S.
On the other hand, the previous property gives as a map j*S - j'S ® wzlx[-rdf]. By composition we get an element of Hom(j*iii'S,j'S ® wyx[-rdf|). Using adjointness of j* and j, we obtain the desired map.I The Riemann-Hilbert correspondence translates these results into identical statements for D-modules and hyperfunctions on X. The group K1 acts on B an let us consider the subsheaf BK on which K1 acts algebraically, i.e. the local sections of B which transforms under K1 according to the restriction of an algebraic representation of K. The elements of BK are automatically distributions. Let Y be a K-orbit in X such that YR = Y n XR 0 0, and i : Y -+ X is affine. Let A E t* be integral and consider the standard (DA, K)module .M(Y, A) = ifi'A(A). We can define similarly BK 1 (A) = subsheaf of B whose global sections are KI-algebraic. 6.8 Theorem: There is natural morphism: M (Y, A) -+ BK (A) which respects the action of (DR, K1 ), and is injective if Y is closed. 56
Proof: This follows from 6.6 and 6.7. Indeed the global sections of M (Y, A) are -
K-algebraic, hence they are mapped to K-algebraic elements. rd(f) = dimRXR dimRX = -n. Note that with the shift, both modules live in degree zero. If Y is closed, then .M(Y, A) is irreducible over (Dx, K). But by K-equivariance the elements of M (Y, A) are determined by their restriction to YR. Hence this map must be injective. LI In fact, if Y is closed the image of the (DA, K) module .M(Y, A) consists precisely of the hyperfunctions in BK1 (A) which are supported in YR. Using this result we can refine the classification of H-spherical representations. Assume A is dominant integral. 6.9 Theorem: Let M = i Oy(r) be a (DX, K)-module with Y a closed K-orbit, YnX 0, (A, r) is trivial on (h,, K n H,) and A + PL is B-dominant regular, then M is H-spherical. Proof: One can prove that Y n X* letter].
$
0 is equivalent to Y n XR, cf. [Matsuki,
By the above result, there is a non-trivial map
f
: i y(A) -+
BK(A).
Moreover il. Oy (A) is the unique irreducible sub quotient of if Oy (A) whose restriction to Y is non-zero. Hence by the flabbiness of B, f(i Oy (A)) is a submodule of BK (A)1 Now using Helgason's and Flensted-Jensen's isomorphisms, the (g, K)-module M corresponding to iOy (A) appears as a submodule of AKR (GR/HR). Thus Hom(h,KflH)(M, 1)
$
0
i.e. HomH(L H(M), 1) 0 0 or also HomH (1, LH(M)) = Hom(o)(O, CH) $ 0. In general if we require the morphism f : i Oy (A) -+
E]
BK. (A) to be only KR-
equivariant, then there may exist several maps including some injective ones.
11.7 Closed orbits and discrete series. It is known that when KR is a maximal to compact subgroup of GR, the closed K-orbits on the full flag variety of G support the fundamental series of representations in C* (G). When rank G = rank Kx., this fundamental series fills in the eigenspaces of the discrete spectrum of Z acting on L2(G), and is called the discrete series of G, (for a proof see [Mili~in]). Oshima and Matsuki have shown that similar 57
statements hold for symmetric spaces when one consider real flag varieties and this result can be translated to complex flag varieties where one should consider only some particular closed K-orbits, see [Oshima]. In this section we want to recover Oshima-Matsuki's theorem from our formalism, so as to explain the choice of these particular K-orbits. We will also prove that the discrete series of a symmetric space has multiplicity one. As before take G with two involutions a an 0 whose fixed point sets are H and K. Let A be a maximally a-split torus in G, i.e. a torus on which a acts by -id and which is maximal for this property. Let L = Cent(A; G), then L = MA where M = Cent(A; H). A may be much smaller than the center C of L, but the adjoint action of A on g yields a genuine root system R(a) while in general R(c) is not a root system. Take a set of positive roots R+ (a) and let N be the nilpotent subgroup of G whose Lie algebra is spanned by the root spaces corresponding to R+ (a). Then P = LN is a parabolic subgroup of G and the flag variety X of parabolic subgroups of G conjugate to P is said to be associated to H by the Iwasawa decomposition. 7.1 Definition: rank GH = dim A = t We say that we are in the equal rank case if rank GH = rank K/H n K, i.e. if we can choose A C K. Let GR be a real form of G such that KR = K n GR is compact and put HR = H n GR. For A E a*, let L' (G/H ; X) be the space of real analytic KR-finite functions on GR/HR which are square integrable and eigenfunctions of D(GR/HR) for the eigencharacter X = xA. We want to describe this space as a Harish-Chandra module. Let us first recall a result due to Flensted-Jensen, Oshima and Matsuki. Theorem: LI(G/H; X) # 0 for some X if and only if rank GH = rank K/HnK. If A is singular, L'(G/H; X) = 0 Therefore we will focus on the equal rank case. Since X0 = H-P is the open H-orbit in X and HnP = M is the stabilizer of P = x E X. Since A C K, Y = K.x is a closed K-orbit in X. WG(A) = WH(A). Choose 58
representatives w,..., WM for the cosets in W(H/K n H) := WH(A)/WKnH(A). Then all the closed K-orbit in X whose intersection with X* is not empty are of
the form Y = KwjP, as j = 1,..., m. Thanks to theorem 5.2, we can describe all the H-spherical standard (D, K)modules supported closed K-orbits in X.
They must be supported on some Y
and have trivial H-isotropy. It suffices to do the construction for Y = K - x. Let t, = p/p be the Cartan factor of p. Then a ~ a, = p/pi + m and we have a canonical inclusion a*
c
,.
The isotropy group of x in K is connected because Y
is a flag space for K; K n H. contains the group A n M of elements of order 2 in A. Let i : Y -+
X. Let r : K, -- CX be such that dr coincides with A - pp on
kx n t,. Let Oy(r) = indK (r) and set .M(Y, r) = i.Oy(r). 7.3 Proposition: * M(Y,r) is an H-spherical (DX,K)-module if and only if A trivial on K
E a* and r is
n Hx.
" When j runs through 1 to m and r through all possibilities, these modules M(Y,,r)
constitute all the irreducible H-spherical (DP, K)-modules on X,
supported on closed K-orbits.
This is clear by theorem 5.2 and the fact that il = i..
Put M(Y, A) =
I'(X, m)(Y, A)). Now we can formulate Oshima-Matsuki's result. 7.4 Theorem: In the equal rank case for A dominant and regular in a* there is an isomorphism of (g, K)-modules:
0@M(Y, A)~ L2 (G/H; x). j=1
Proof: A - p, is integral in t* since it is a character of A and T, surjects on A. Hence there is a line bundle L and X such that i'L = Oy(r). Let BK1 (LC) be the sheaf of K1 -algebraic hyperfunctions along XR with values in the line bundle L, i.e. BK(1
)==(,n(L WR), cf.
6. Let YR, = Y n XR and put BKI (yj,) be
59
the space of hyperfunctions in BK1 (,C) supported in YR,,. By theorem 6.8, there is a bijective morphism of (DA,K1)-modules:
.M(Yj,,r) -+ BK 1 (Yj,T)Since A is dominant, the global section functor is exact and this bijective morphism carries over to global sections. It is not difficult to see that YRJ is a single KR-orbit. Indeed A C K and by Matsuki's parametrization of KR-orbits on GR/PR, there are m = #W(H/K n H) closed KR-orbits on G/PR. X* is a complex neighborhood of XR, hence if a closed K-orbit does not intersect X*, it cannot intersect XR. We have found that there are m closed K-orbit which intersect X*. Thus it suffices to prove that if a closed K-orbit intersects X*, it also intersects XR. This was done in [Matsuki, letter]. As a result , we do not need to use K 1, but only KR. Again by the dominance of A, the Poisson transform is an isomorphism. Combined with Flensted-Jensen's isomorphism this gives an imbedding of (g, KR)modules m
eM(Y,,A) j=1
c-+
AK(G/H;xA)
where AK (G/H; x\) is the space of real analytic KR-finite functions on GR/HR with eigencharacter Xx for D(G/H). Now the deep result proved in [Oshima-Matsuki is that a function in AK(G/H; x,\) is square integrable if and only if it is the image of a hyperfunction in BKR (L) supported on some Y n XR.
Thus the image of the above imbedding is precisely
L2K(G/H; XA). El 7.5 Corollary: Suppose that A + PL is B-dominant. Let V be an irreducible Hilbert space representationof G with infinitesimal characterA+PL. Then dim Homc(V, L 2 (GH)) 1. Proof: Let us denote by the same letter V, the Harish-Chandra module of V. By theorem 7.4, V imbeds into some I'(X, i.Oy (A) where A E a* is P-dominant and i is the inclusion of a closed K-orbit Y = K -p into X. If A + pt E t* is B-dominant, 60
then the enveloping algebra U of g generates r (X, Dj), cf. 111.2.2.2.
Therefore
r (X, i. Oy (A)) is an irreducible (U,K)-module and for distinct orbits Y's,they are inequivalent because the
(DA,
K)-modules i.Oy(A) are inequivalent.
Hence V =
P(X, i.0(A)) and occurs with multiplicity one in L' (G/H, X). This gives the result for almost all infinitesimal characters, in particular the regular ones.Fj Even if A is P-dominant, it may happen when A is small enough that A + pt is not B-dominant. D.Vogan has checked by coherent continuation that the modules r(X, i.oy(A)) - which can also be described as some Ap(A), cf. [Zuckerman] and 111.2.3.2 - are irreducible (U,K)-modules. as long as A E a* is P-dominant. By theorem 7.4, to prove multiplicity one, it suffices to prove that these modules are inequivalent for distinct orbits.
In chapter III, we will prove this property for
classical groups and G 2 . We have not found a way to handle the case of groups of type E, and F in this limit range of A. The point is that the following description of r (X, i. 0 (A)) as K-module is not explicit enough. 7.6 Proposition (Blattner Formula): For A E t* dominant and Y a closed K-orbit, there is an isomorphism of K-modules: F(Xi*i'O(A)) = EY2.O(-1)Hj(Y, O(S(n n s) ®CN_, 9 A d(n n s))) where n is the nilpotent radical of some p E Y, g = k + s and d = dim (n n s). Proof: There is a K-invariant gradation on i. Oy (A) given by the Euler operator, radial to Y in X and for which gri.0 (A) ~ S(Tx/Ty) 9 0(A) 0 fl-'(A)
The shift by f1Xj-Iy (A) is there because to define i.Oy (A), one has to transform Oy (A) into the right D-module of top degree differential forms apply i4 and finally multiply back by U'(A).
Oy (A)
on Y, then
As a K-sheaf on Y, fl-1(A)
~
0 (Ad(n n s)). To prove the identity 7.6 it suffice to apply the functor of global sections in the derived category to the above equality of graded modules and to compute the Euler 61
characteristics. Hi(X, i. Oy(A)) = 0 for j > 0 because A E a* is P-dominant. So the left hand side is simply r(X,i.Oy(A)). L From the Borel-Weil-Bott theorem it follows that the representations of K which may occur in r(X, i*i'O(A)) have an extremal weight of the form V = A + Pn - Pc + EaER(nns)nara E
tk
Here t* is the dual of a Cartan subalgebra of k such that a* C t*; pn := p(nns), p, p(n n k) and Pn + P = p. In particular if A + pn - P, is K-dominant, then A + pn - pc
will be the smallest K-type of r(x,ii'o(A)). Given integral weight A E a*, it defines a character Xx of D(G/H) , there is a (g, K) isomorphism L._(G/H; where ij
x)
~WEW(H|KnH)P(X, ii
o(A))
is the inclusion of the closed K-orbit 'Y into X. Restricting to K, we
have by Blattner's formula: top
r(x, itiw!O(A)) = Ei;>o(-1))Hi(wY, Oy(S(wn n s) o Cw(A Thus the set of possible K-types is w(A+pn-PCEaER(-nns)n
P) 9 A(wn
n s))
-a), where w E WH(a)
represents a coset in W(H/K n H) and w makes A + p, - pn is K-dominant. Since Wp is not conjugate to p by K, those sets of K-type are different. Taking some large K-dominant K-types U, of highest weight 1i = A
+
pn - Pc + EaER(nfl)naa
which occurs in
r(x, i~i'Ox(A)), we have, Hi(Y,0y(p))
=
U,
j=0
=
0
otherwise
Since yi and wpt are not K-conjugate, U, and Uwvp are inequivalent.
Hence if
we know that both these K-types occur, the (g,K-modules F(X,i*i'Ox(A)) and
F(X,iiw!Ox(A)) will be inequivalent. and all the constituents of L'(G/H; XA) 62
will be inequivalent. Unfortunately there can many cancellations in Blattner's formula, and in general it is very difficult to say if a possible K-type occurs or not.
11.8 An algebraic Poisson transform. The purpose of this section is to describe a functor which transforms a (D, K) module on X into a G sheaf SK(M) on G/K. This functor is analogous to the Poisson transform in its effect on modules. Its definition could be formulated in terms of real flag varieties and real symmetric spaces, and the definitions are related by a restriction morphism, as in section 6. As for the real case, the inverse functor should be a kind of boundary value map using the compactification of G/K described in [Springer] and due to Oshima, De Concini and Procesi. First we review what is the Poisson transform. Consider a real form GR of G, and a spherical principal series representation of GR induced from a minimal parabolic subgroup PR Ind (A) ~ {f : GR -+ C
I. f(gp)
= eA-(p)f(g) g E GR, p E PR. f E C** (GR)
}
where A E a* and AR is a maximal R-split terms of PR. GR acts on Ind(A) by left translation, denoted
e.
Let KR be a maximal compact subgroup of GR. Then
we can decompose GR = KR - AR -NR and PR
=
MR ' AR - NR. This representation
contains exactly one K-invariant vector, say vA, because by Fr5benius reciprocity, we have: HomKR (1, Ind(A)) = HomKR
(1, C(KR
/MR)) = HomMR(11
C.
vx can be explicitely written.
v, : BR = KRARNR -+ C: kan
i-+
e ~(a)
There is a G-invariant pairing between Indp (A) and its contragredient IndG(-A) given by integration on K.
(u,v) =
u(k)v(k)dh for u E Ind 63
(A), v E Ind
(-G)
Hence every vector v E Indp (-A) defines a GR equivariant map. S, : Indp(A) -- C* (GR)
Sv(u)(g) = (t(g-')uv) g E GRU E IndpG(A) Since vx is K-equivariant and equal to 1 in KR, we obtain an intertwining operator S := SV_' S: Indp(A) -- C*O(GR/KR)
S(u)(g) = ((g- 1 ))u, v-) = which is called the Poisson transform.
u(gk)dk
When A is dominant (recall that the roots
of n are negative). Indp(A) has a unique irreducible submodule J(A). This is true also when A is singular because the unitary spherical principal series is irreducible
[Kostant]. J(A) is the socle of Indp(A) and it consists of the functions which have the smaller growth at infinity.
From Langlands-Militie study of exponents, we
obtain that v,\ E J(A). Dually Ind (-A) has a unique irreducible quotient J'(-A). v-, can be viewed as a nonzero vector in J'(-A). Moreover v\ is cyclic for Indp(A) in the sense that G - vA is dense. It follows that when A is dominant, the Poisson transform S is injective.
8.1 Observation: A representation of G may occur as a subquotient in C**(G/K) although it has no K-invariant. For example, take GR = SL 2 (R), KR = SO(2), P = B,
A = p = 1. IndG(p) e C(RP 1 ) vA(kan) = 1. The G-module structure of C*(RP1 ) is one irreducible submodule C = the constant functions and two irreducible quotients D 2 and D2 which are square-integrable representations.
For
f
E C*(RP), (Sf)(gK)
= fSO( 2 )f(g-'k)dk. The image
by S of a finite Fourier series E2=,cnein on RP1 is the harmonic polynominal = 0 cnz"
+
n'-n
n on Dr
SL 2 (R)/SO(2). For K-finite vectors, we have
S(C)
=C
S(D2)
=
S(D2)
=2C[2
zC[z]
64
(C)K
C
(D2)K
0
(D 2 )K
0
In fact not only Ho(k, D2 ) = 0, but also H,(k, D2 ) = 0 for all i, since H1 (k, D2 ) Ho(k, D2 )*.
=
Note that the obvious K-invariant functional on C* (D2), namely
6. =evalution at the origin, is zero on S(D 2 ) and S(D2). In conclusion, we observe that the KR-spherical irreducible representations of GR give only a small part of C*(GR/KR), but the Poisson transforms gives a whole eigenspace of the ring of differential operators on GR/KR. This is the general situation as was proved by Helgason for K-finite vectors and by Kashiwara, Oshima et al. for representations of GR, see [Schlichtkrull]. 8.2 Theorem: S : B(GR/PR;LA)Z4A(GR/KR;xX)
is a GR-isomorphism when A
is dominant, with inverse a boundary value map 3. Oshima has constructed a compactification of GR/KR for which GR/PR appears as a piece of the boundary (dimGR/KR = dim GR/PR + dimAR). We pass to complex algebraic groups G, K, A, P, M. We already know how to go from (D, K)-modules on G/P to hyperfunctions on GR/PR. Now we want to transform a (D, K) module on G/P into a (D, K)-module on G/K. By analogy with the real situation, we should work with the diagram: G/P+'-G/M-F+G/ K. n is affine but not proper: its fibers are isomorphic to the open K-orbit in G/P. Since we want to integrate along the fibers of x, this may look annoying, but it is rather difficult to ask a smooth map to be at the same time affine and proper (and different from the identity). The advantage of an affine map is that the direct image functor is exact. T M is a (DG/K, K)-module Let M be a (DA, K)-module, then .c7r on G/K with
eigencharacter XX with respect to the image of Z in D(G/K): the algebra of Ginvariant differential operators on G/K. 8.3 Definition: Kv7r' is the algebraic Poisson transform of G associated to K and P. De Concini and Procesi have constructed a compactification C of G/K' where K' is the normalizer of K in G, which exhibits G/P as a piece of the boundary, 65
cf.
[De Concini-Procesil and [Springer].
spherical vector in a representation of G.
It is the closure of the orbit of a KIt is a smooth projective variety on
which G acts with a single open orbit: G/K and a single closed orbit: G/P. An analog for D-modules of the boundary value map is the nearby cycle functor whose monodromy invariant part is: i*j. where i and j are the inclusions of G/P and G/K in the compactification of G/K. It involves a choice of A corresponding to the infinitesimal character Xx. Since C is constructed using only a maximally 0-split Cartan subgroup A, it is natural to focus on the case A E a* . By theorem 2.1, this is even necessary to deal with K-spherical representations. Then for a close of A dominant, one would expect the nearby cycle functor to be the inverse of the algebraic Poisson transform 1cgr'; I hope to be able to prove it soon. GR/KR is real analytic submanifold of G/K. We can restrict K.j'.M to GR/KR. For K-spherical modules, we should obtain a subsheaf of the sheaf of real analytic K-finite functions on GR/KR which are eigenfunctions of D(G/K). So that this algebraic Poisson transform and the real Poisson transform. An application of these considerations would be to define globalizations of (D, K)-modules as W. Schmid does it for Harish-Chandra modules. Finally, let us observe that in the same way, we can transport (D, K) modules on G/P to (D, K) modules on G/H. This is useful for G/H may be the complexification of an indefinite symmetric space, and the real Poisson transform is not defined in general when HR is not compact. Since the fibers of G/MH -+ G/H are isomorphic to the open H-orbit X* in G/P, this explains in another way the condition in 5.3 on the support of an H-spherical (D, K)-module.
66
Chapter III. Microlocalization and Singularities III.1 Basic Microlocal Notions 1.1 Constructions and Definitions Given a smooth complex algebraic variety X, one defines on the cotangent bundle T*X of X the sheaf of Ex microdifferential operators, see [Kashiwara], or [Schapira]. The sheaf ex is in some sense the localization of Dx, just as the sheaf of holomorphic functions is a localization of the sheaf of polynomials. More precisely, a microdifferential operator is invertible wherever its principal symbol does not vanish, [Schapira, 1.1.3.4]. The sheaf ex is a coherent, noetherian sheaf of rings with has a Zariskian filtration. If p= (x, 0) E T*X, then gr(ex,p)
Oxx[
1,.-
, en]
where n = dim X. If p = (n, e) E T*X, e 5 0, then gr(ex,p) ~_O,.x,p[ T -1, T
where P*X is the projective cotangent bundle of X, and P is the image of p in P*X. Let ex-mod denote the category of sheaves of &x-modules on T*X which are quasi-coherent over OT*x. The coherent ex-modules form an abelian category. Let 7r : T*X -
X be the projection ; then &X is flat over r-lDx. Therefore the
microlocalization functor mic is exact: mic : Dx - mod -+ ex - mod
M '-4 ex
r,-1D,
7r~ 1 m
Moreover the functor mic preserves coherency and its image is the subcategory of x-modules defined on all of T*X. Since ex
I T*X
~ Dx, its inverse functor on this
subcategory is simply the restriction to the zero section. The support of an Ex-module M is also called its characteristic variety and is denoted by charM. If M = mic N, char M is simply the characteristic variety of N in the sense of D-modules. The variety char M is a closed analytic subset of T*X, 67
stable for the action of C' on T*X and involutive, i.e. (char M)' C charA. One can also define the characteristic cycle of M, denoted [charM]. An x-module M is called holonomic if it is coherent over Ex and char M is a lagrangean subvariety of T*X, i.e. (char M)' = char M. The subcategory of holonomic ex-mod is denoted by ex-modh. There is a natural duality on ex-modh which exchanges left and right Cx-modules: * : X-+ ext" (M, ex[n])
The property exte (M, ex) = 0 for j # n = dim X characterizes the holonomic x-modules among all coherent ones. The sheaf &x operates on the sheaf C, of micro-functions on X. One can also defines similar sheaves Am, BM, Cm, DM, em for a real C'-manifold M. Then CM is a
sheaf on T*M whose corresponding presheaf associates to an open set V C T*M the quotient of BM(M) by the space of hyperfunctions on M which are micro-analytic at every point of M, [Kashiwara, p. 14]. As an example, take X = C, Dx = C[z, a], then ex contains the element defined away from the zero secton of T*C and (a - 1)-' = (a-' + 8-2 + a-3 +-
a-' ).
The infinite order differential operators do not belong to ex.
1.2 Microlocalization of a non-commulative ring. Let A = UjA; be a filtered unitary ring such that A- 1 = 0 - this implies that A is complete with respect to the topology induced by this increasing filtration. Suppose that grA is a commutative noetherian ring. Then 0. Gabber has defined the localization of A with respect to a multiplicatively closed subset S C grA. We follow the presentation of [Ginzburg]. Consider the category of homomorphisms f : A -- B such a that
(i)
B is a complete Z-filtered unitary ring.
(ii) f preserves the filtration. (iii) the elements of (grf)(S) are invertible in grB. 68
There is a universal object is : A -+ As in this category; i.e. for any morphism f : A -+ B in this category, there is a commutative diagram. A
B
As In particular, As is a complete and separated Z-filtered ring. The map is : A -+ As is strongly compatible with the filtrations and S-1 grA.
gr(As) If OVS, then As k 0. The map a : A
-
grA is called the principal symbol map.
We have defined As to have the following property: if the principal symbol of a E A belongs to S then is(a) is invertible in As. Finally, As is a flat A-module. 1.2.1 Remark: For any conic open subset V C SpecgrA, let S = S(V) be the multiplicative set of elements of grA invertible on V, i.e. which do not vanish at any point of V.
Then V -+
As(v) is a presheaf defined on the conic open
subsets of SpecgrA. We denote the corresponding sheaf by A = mic(A), and call it the formal microlocalization of A. This terminology agrees with 1 in the sense that 6x consists of the elements in mic(Dx) which satisfy a certain convergence property. Also if M is a coherent noetherian A-module, then the sheaf associated to V
'-+
As(,)
OA
M is a coherent sheaf of micA-modules denoted micM and is
called the formal microlocalization of M. The functor mic: A - modc
M
-
A - modc
A®AM
is exact. The support of mic M is the characteristic variety of M. 1.2.2. Construction As: Let t be a transcendental over A. Consider the graded ring FA = Et'Ai where A = UiA, is the Z-filtration on A.
It is a subring of
A[t, t 1 ]. When the ground field is C, FA should be regarded as a bundle of rings over C* whose fiber at z E C* is A ~ FA/(t - z)FA and which specializes to grA = FA/ + FA when t -+ 0. If b E A is a locally ad-nilpotent element, i.e. for 69
every a E A, there is an n E N such that (adb)"(a) = 0 where adb(a) = ba - ab,
then the localization Ab of A with respect to the multiplicative set {bn I n E N} is well-defined. For example, the localization of the ring FA at t is isomorphic to A[t, t-1]. Now consider the multiplicatively closed subset S of grA we had at the beginning.
Let FS C FA be the set of homogenous elements in tkAI - for some
~ At/At_1 c grA belong to S. Set
k- such that their images in tkAk/t k-AkI
A(k) = FA/tkFA and S(M) = FS/tkFA n FS. Using the commutativity of grA, it
is easy to check that every u E AMk) is ad-nilpotent, (in fact (ada)k+l I AM') = 0). This guarantees the existence of the non-commutative localization (S(k))-1A(). The projective system -+ FA/t3 FA --+ FA/t2 FA --+ FA/tFA ~ grA
combined with the universal property of localization gives rise to the projective system. -+
(S( 3 )) -'A( 3) _ (S( 2 ))-'A(
2
)
-+
(S( 1))-'A(1 ) ~ S-1grA.
Set B = lim(S')-1 A('). This ring is the localized version of FA. The equality A = FA/(t - 1)FA suggests the definition.
As =
B/(t - 1)B =lim lim(S')~1A('). mult. by t
This presentation of As is not practical for applications, but the important things to remember are the properties of As.
1.3 Microlocalization of DA We shall need twisted sheaves of formal micro-differential operators in the same way as DA is a twisted version of Dx. Let X be the complex flag space of type P of the group G and let A + t* be as in chapter 1. Set ?r : T*X -+ X. We have defined the twisted sheaf of differential operators DA on X, cf. 1.4.3. The sheaf DA has a natural filtration by degree, and there is a symbol isomorphism o
Applying 7r-
1
gr
~+ 'r.OTx.
we get an injection & gr(7r-DA) -+ OT.X which maps gr(7r-lDA) onto 70
the sheaf of germs of regular functions on T*X which are polynominal in the fiber variables. Fix an affine open subset U in X, then T*U C T*X can be identified with Spec(grDA(U)). For a conic open subset V of T*U such that 7rV = U, let S(V) be the multiplicatively closed set of elements of gr(7r-DA)(V) = (grDx)(U) which are invertible on V. Then as in section 2, taking (grDx(U) for A, we can define x(V) :
(gr7r-DA)(V)s(v). It is easily seen that the open sets V of T*X which
project onto affine open sets of X form a base of the Zariski topology of T*X. Hence we obtain a presheaf V
ex(V) defined on the set of conic open subsets of T*X.
'-f
By definition, the corresponding sheaf ex is the formal microlocalization of Dx. 1.3.1 Definition: The sheaf ex on T*X constructed above is called a twisted sheaf of formal micro-differential operators - tmo for short. The sheaf 7r-'Dx is subsheaf of ex. If D E Dx, 7r-'D is invertible in ex wherever its symbol does not vanish. We also have a symbol isomorphism a : gre
~>
EjEZOT'X(j) where OTrX(i) denotes the sheaf of germs of homogenous functions of
degree j in the fiber variables. Moreover OT-X is faithfully flat over eEZ 0 T-x (j), [Schapira p. 77], so we will often identify a graded gr(ex)-module M with the module OrT*X,grrM. From the general properties of microlocalization in section 2, we have the following result. 1.3.2.
The functor mic
Property:
Dx - modc
x - modc :
-e
e4 0,-i1, M is exact and faithful. Its image consists of the ex modules which are defined on all of T*X; on this subcategory, the left inverse functor is the restriction to the zero section of T*X. 1.3.3. Remark: Strictly speaking, microdifferential operators are formal microdifferential operators which satisfy a certain convergence property, [Schapira, p. 11]. If E-ooijsrMP, (x, ) is the expression in local coordinates of a microdifferential operator D defined over an open set U c T*X and pj is homogenous of degree j in the variable
= (, ....
,). Then D E ex, i.e. D is a microdifferential op-
erator over U if for every compact subset K c U, there exists c > 0 such that 63p
EO
) (,8EK
(P-
( j is finite. So we want the terms
su
p. ,(x, () j to
be the coefficients of a Taylor series on C which converges in some disk of radius e. 71
In this approach, it would be natural to define a sheaf of twisted micro-differential operators as a pair (, i :
7r~'x
-- ex) locally isomorphic to the standard pair
(ex, ix).
1.4 Microlocalization of Ux We adopt again the notation of chapter I. Let Z be the center of the enveloping algebra U = U(g) and pick X E Max Z. Define Ix = {z - X(z)
Iz E
Z}U
Let p be a parabolic subalgebra of g and define: J, = Annu(U ®[p,p] C). I and Jp are both two sided ideals in U. Set
UX := U/(Ix + Jp) The natural filtration on U induces a filtration on Ux. 1.4.1. Lemma: C := Spec grUx = closure of the Richardson nilpotent conjugacy class of p in g*. Proof: [Borho-Brylinski I, p. 455 & 4561E) C is a closed cone in g* and taking Ux for A in
2, we can define the formal
microlocalization UX of Ux on C. We regard it as a sheaf on g* supported on C. For any conic open subset V of C, UL(V) consists of Ux together with the inverses of the elements of Ux whose principal symbol do not vanish of any point of V
n C.
In particular, if V contains 0 E g*, then C C V and Ux(V) = Ux.
111.2 Microlocal study of the moment map. 2.1 I and A as i. and i. Let G be a complex connected reductive algebraic group and P a parabolic subgroup of G. Let X be the flag space of G of type P. The cotangent bundle T*X is isomorphic to the set of pairs (p, x) such that p is a parabolic subalgebra of g 72
conjugate by G to Lie P and x E p C g*. G acts on T*X by conjugating the pairs (p, x) and it preserves the canonical symplectic structure of T*X. This action gives rise to a so-called moment map u which in our case is just the projection u : T*X -+ g* : (p,X) -+ x it is G-equivariant with respect to the coadjoint action on g*. The image of tz is C = Gpi- = the closure of the Richardson nilpotent conjugacy class of P in g*. Our reference for the basic properties of the moment map and their implications for the relations between D-modules and g-modules is [Borho-Brylinsky I & III]. Now consider an admissible subgroup K of G, that is a subgroup having only finitely many orbits on the full flag variety of G, and let k be its Lie algebra. Let M be a coherent (DA, K) module on X for any A E t*, cf. Chapter 1; and let M = I'(X, M) be the corresponding coherent (g, K) module. If A + PL is dominant and regular, then M generates M over DA, cf. 1.6. Let Char M C T*X be the characteristic variety of M. The support of M - denoted Supp M, also called the associated variety of M - is the subvariety of g* defined by the annihilator of grM, for a good filtration on M. Finally let W be the set of K-orbits on X. 2.1.1. Lemma: a) Char M C UYEwTjX
b) p(TjX) = K(pJL n k-I) for all p E Y c) If A + p(u) is dominant and regular, then pL(Char M) = Supp M. Proof: These results are proved for A = pe in [Borho-Brylinski III] see 2.5 for a), 2.4 for b), 1.9 for c). a) and b) are independent of A. The translation functors give equivalence of categories inside the dominant chamber; this implies c). E Let X = Xx be the infinitesimal character defined by A E t and the HarishChandra morphism p : t --* Spec Z. Recall the algebra Ux = U/Ix+J, introduced in 1.3. Then M is naturally a (Ux, K)-module. We also have the sheaf of algebras Ux = mic Ux on C c g*. micM is a sheaf of (Ux, K)-modules whose support is contained in C C g*. Let i. and it* denote the direct and inverse images between calEA-modules on T*X and Ux-modules on g*. 73
2.1.2 Proposition: The following diagram commutes. (&Ax, K)
-modc on T* X
.
mic T (Dx, K)
(Ux, K)
-modc on g* -_T
T mic -mode on X+ 2P
(Ux, K)
-mode on e E G
2.1.3. Remark: For M E Dx-mod, F.M is first a module over TDP. Since DA is a t.d.o with a G-action, we have a map U -- FDA which factors through Ux. Hence by "restriction of scalars" we may view FM as a Ux-module. Proof: If we call v : X -+ e the map of X onto the point e, then r is v, and AX is v*. By naturality of the construction of the functor mic, the following diagram also commutes.
T*X4 X
g*
4e
We get the projection formula for (Dx, K)-module F .(tI*UX
-Dyr - 1,7)
=
Ux (& u(X,
7)
2.2 Associated variety of a submodule of I'.m First let us prove a simple lemma. Let D be a t.d.o. on X. 2.2.1. Lemma: Let D =P(X, D). if r: D - modc -+ D - modc is exact, then it sends simple objects to simple ones, or to zero. Proof: HomD (M, TN) = Homp(AM, N) where A is the localization functor with respect to D, i.e. AM = D OD M. Let N be an irreducible D-module, and suppose M is a D-submodule of TN. By adjointness, the nonzero map M -- TN gives a nonzero map AM -+ N which therefore must be epi because N is irreducible. Applying r we get a map rAM --+ N which is still epi or zero by the exactness of r. But there is a natural map M --+ AM, adjoint to the identity on AM. The composite map M -- TN is our original map, hence it is epi. This implies the irreducibility of TN. L] 74
Recall that by restriction of scalars, any rDx module on X is a Ux module for X = xA E Max Z. We are interested in the following problems: let X be an irreducible DA-module. When is 1 M irreducible over Ux ? When is
.M completely
reducible over Ux? 2.2.2 Proposition: a) If the moment map A : T*X -+ C c g* is birational and has a normal image, then rDA = Ux, for every A E t*; and the filtrations coincides. b) If A + pt is dominant then Ux
+FDx is surjective.
Proof: Part a) is due to Beilinson, Bernstein. The result follows from the fact that the map S(g)/grix+grjp-F(X,grDA) is an isomorphism, which is independent of A. Part b) is due to Borho-Brylinski. It follows by applying the translation principle to the isomorphism Ux, ~+ (X, Dx). Note that when C is not normal, or y4 not birational, the natural filtration on Ux, may be different from the operator filtration on r(X, Dx).
1
This proposition answers our problem in almost all cases. Unfortunately, some examples which escape are discrete series of symmetric spaces. So it would be a shame to disregard them. Applying the technique of microlocalization, we can go one step further. Let us consider a closed conic Lagrangean subvariety is the set of K-orbits in X. Put Z = u(A)
g
A of UYEW Ty X
where W
C. Recall that the generic point Z*
of a scheme Z is any point whose closure is Z. One can think of Z* as being an open subset of Z whose closure is Z. We say that the moment map pL : T*X is an isomorphism in a neighborhood of
A'
if there is a neighborhood U of
A*
C in
T*X and a neighborhood V of Z* in C such that it : U -+ V is an isomorphism. Let M be an 6x (resp Ux) -module on T*X (resp. C) with support We say that M is irreducible at its generic point if M over x
I A*
I A*
A
(resp. Z).
(resp. Z*) is irreducible
(resp Ux | Z*). It is clear that if M is irreducible then it is irreducible
at its generic point. Let M be an irreducible ex-module whose support is 75
A and
suppose p.M
#
0.
0
2.2.3 Lemma: If i is the isomorphism in a neighborhood of A and if A is Pdominant, then 1z,,{ is irreducible at its generic point. Proof: The first hypothesis means that it is birational and that Z contains an open dense subset of normal points of C. We can take Z to be this open set and let
A
S
A'
00
= -+
p- 1 (Z). Let
A'
and
A'
be their neighborhoods in T*X and C such that
Z' is an isomorphism. Then 4(A') ~ Ux(Z'). Moreover since A is P-
dominant, A. is exact with inverse p*. Hence (p*.M) I Z ~ over (it*S)
IZ
~ Ux
,(.M
I A)
is irreducible
| Z.
The associated variety Ass(M) of a Harish-Chandra module M is the zero set of the ideal Ann(grM) in S(g) for a good filtration on M. It is clear that when M has an infinitesimal character Ass(M) = supp(mic M). 2.2.4 Theorem: Let M be an irreducible DX-module on X A > 0. Let N -+ P.M be a g submodule of P.M. Then: Ass(N) = Ass(P.m)
Proof: Hom(g,K)(N, r, m) = Hom(D,K)(AN, M).
Thus we get a nonzero map: AN --+ M, which must be surjective since M is irreducible. By hypothesis P is exact, hence PAN -+ PM is still surjective and the composite map N -+ PAN --+ .M is the original inclusion. Looking at associated varieties we get: A,,(N) 9 A,,(Pm) 9 A,,(PAN). (1) Now A.,N = supp(mic N) and A,.(JTAN) = supp(ji.pi*mic N). But supp(u*mic N) p~'(suppmic N)and supp(pj) 9 j(supp7) for any ex-module for T*X. Hence supp(po~*mic N) C ppc1 (suppmic N) = supp(mic N). Comparing this inclusion with (1), we get
A.,(N) = A..(P.m) = A.,.PAN.L Remark: CharM D char(AP.M) and they may be different. The following consequence of the above theorem is clear in light of 2.3.2 and 2.3.4.b.
Corollary: Suppose that A E t* is P-dominant. If a (g, K)-module of 76
the type Ap(X) is reducible, then all its composition factors have same associated variety. Now let M be a simple (DA, K)-module on X with characteristic variety
A.
Suppose A is dominant and P(X, m) $ 0. 0
2.2.5 Proposition: If i is an isomorphism in a neighborhood of A, then every proper quotient of p.(mic M) is supported on a subset of Z = /(A)
of smaller
dimension.
Proof: By 2.2.3. we know that p.(mic M) is irreducible at its genenic point. Let L be a (Ux,K)-submodule of tii(micM). By 2.2.4 and the above remark, the support of L is Z. Thus if we take L simple, the L is unique and (it*.(mic
))/L)
IZ
is zero. This implies that supp (p.*(mic M)/L) is contained in the closed subset 0
Z - Z which is of dimension smaller than Z. l
2.2.6 Corollary: If It is an isomorphism in a neighborhood of A and if P.M is a semisimple (g, K)-module, then P.M is irreducible. Proof: Given an exact sequence 0 -+ L -*
M -+
N -+ 0 of (Ux, K)-modules,
we have suppM = suppL U suppN. Since P m is semi-simple, every submodule L of P.M is also a quotient. By 2.2.4 L should have support equal to Z and by 2.2.5, L should have support strictly smaller than Z. Hence, either L
=
LM or L = 0. L
If P.M is a unitary Harish-Chandra module, then it is semi-simple. In the next
section we give a criterion for unitarity. The examples suggest that the number of possible submodules of p.(mic M) should be bounded by the number of connected components of pt' (z) n A, for 0
z E Z.
By an argument on characteristic varieties, this can be proved in very
special cases, see 3.2 step 3.
2.3. Vogan's complete reducibility criterion. It is known and it can be seen with the functor L introduced in section II.4 that the global sections of standard (D, K)-modules on the full flag variety are canonically
the K-finite duals of the standard (g, K)-modules constructed in Vogan's book by cohomological parabolic induction.
We recall Vogan's result which says that
77
cohomological induction from a 8-stable parabolic subgroup preserves unitarity; hence, the (g, K)-modules induced in this fashion from a unitary representation are completely reducible. The 0-stable assumption can be relaxed to the hypothesis that the 'K-orbit of the parabolic subgroup is closed, but we won't need this generality here. For the study of D-modules, it suffices to know the result for induction from 1-dimensional representations. Let Z be a closed K-orbit in the flag space X of type P. We have a smooth fibration r : B -+ X. 7r- 1 Z is a K-stable closed subvariety of B; thus it is the closure of one K-orbit, say Y. Observe that F =
ir'Z
is smooth by construction. If 0 is
the involution of G defining K, then the K-orbit of P E X is closed if and only if P contained a 0-stable Borel subgroup. If P itself is 0-stable, then P contains a 0-stable Borel subgroup. For A E g* a linear form fixed by 0 and purely imaginary with respect to the real form g(0, R) (such that kng(0, R) is a compact subalgebra), let L(A) = Cent(A; G) and U(A) be a 0-stable unipotent subgroup of G such that P(A) = L(A) - N(A) is parabolic. Then P(A) is clearly 0-stable. Assume P is of this form and let p(u) be the half sum of the roots in the unipotent radical U of P. Suppose A + p(u) is the differential of a character of P. Consider the associated line bundle 0A on P and pull it back to 0A on B. If i : Y c-* B is the inclusion and c = codimBY, then the global sections of the standard (DA, K)-module i. 6y(A) form the space H (B, 6A). We view it as a (g, K)-module. 2.3.1. Lemma: H (B, A) ~ Ap(A) For the definition of Ap(A) see [Vogan 1984). This lemma appears in [Zuckerman] without the bar over Y. If j : Z c
P is the inclusion of Z = K.P, then we
have of course H (P, 0A) = Hp'(B, 6A), by base change.
2.3.2. Lemma: Hz(P, 0A) ~ AP(A) as (g, K) modules. This lemma is not difficult to prove using the functor L. AP(A
=
AAPA(A) =AALA,Ln
I', L
gpro Ln)(C
K(C
P(u)) KLnK AA
A
C
~ Lk, flind
LnK) (CA
78
C p(u))
(C
0 C,))
Consider (D, L n K)-module CA+p(U) over the point P and let i, : P
o for all a E R(u,h), then Ap(A) is a unitary (g,K)-module. This is 7.1 e) together with 8.5 in [Vogan 19841. In fact the final remark in Vogan's article shows that in some cases unitarity is still true in some strip "below 0". For instance, if P = P'
and G = SLn+1 (C), t* ~ C, and the unique root a
in R(u, t,) is n + 1, so that 2p = n(n + 1). Then for A E C such that ReA > -n, the above unitarity result is still true, and in fact the vanishing theorem 1.6.2. also holds; compare with [Vogan 7.1.dl. 2.3.4.
Corollary: a) If Re(a, A + p(t)) > 0 for all a E R(u,h), and if CA+p(u) is
a unitary (t, L n K)-module, then Ap(A) is an irreducible unitary (g, K)-module. 79
b) If Re(a, A) ;> 0 for all a E R(u,h), and if CA+p(u) is a unitary representation of (t, L n K), then '(P,iz* C) is a completely reducible (g, K)-module. Proof: b) follows from 2.3.2 and 2.3.3. a) follows from 2.2.2.b), 2.3.2 and 2.3.3., using the fact that i,.OA is an irreducible (DA-lp(e), K) module.
111.3 Nilpotent varieties In paragraph 2.2 we saw that the properties of the moment map lead to results about the irreducibility of DA-modules considered as g-modules. In this section we pursue this approach and we describe some known results about the singularities of the image of A. Then we deduce some consequences for the reducibility of standard H-spherical (g, K)-modules. We have seen in 2.2.2 that if the moment map It is birational and has normal image, then the global section functor r sends an irreducible D-modules M to an irreducible g-modules. The study of examples suggests that it should suffice to assume that it is birational and that an open dense subset of pu(CharM) is unibranch. We have not been able to prove this assertion, but we can at least give a criterion for unibranchness in terms of multiplicities of Weyl group representations.
111.3.1 Normality and unibranchness First we recall a proposition due to Springer for a) and Richardson for b). 3.1.1. Proposition: Let X be a complex flag space of type P for the connected reductive algebraic group G and let i : T*X -+ g* be the moment map a) i is a proper map and Impz is the closure of the Richardson nilpotent conjugacy class C, of P in g* b) i is finite over C, of degree b,= Cent(x,G)/Cent(x,P) I for any x E C,. See [Steinberg, 4.2]. In particular, since T*X is smooth, when b, = 1, g is a resolution of the singularities of its image, and if all fibers of 1L are connected, then the image of A is a normal variety. This is the case when B is a Borel subgroup, then X is the full 80
flag variety, and Imp is the whole nilpotent cone )* in g*. Indeed Kostant proved directly that )*
is normal by showing that )*
is a local complete intersection,
smooth in codimension 1. In fact .1* is even a complete intersection whose defining ideal is the set of G-invariant polynomials on g* without constant term and the only singular point of )*
is the origin.
To study the nilpotent element of g*, we may assume that of g* is semisimple and also we can identify g with g* via the Killing form. It should be said also all results that we shall review could be formulated for the unipotent elements of the grou itself, since the exponential maps - over C - is a diffeomorphism between the set of nilpotent elements in g and the set of unipotent elements. When X is the full flag variety, we denote it B; otherwise P. So in general the moment map pt sends T*P into M. Let us put P. =
- 1 (x) for x E M. The fiber can be described easily:
Pz = {p E P I x E n(p)} where n(p) is the nilpotent radical of p. P are called Spaltenstein varieties. Unless P = B, the varieties P, are different from P. the fixed point set of the flow generated by x on P: P' = {p E P Ix E p} Both are projective varieties, but the following result is false for P, in general. 3.1.2. Lemma: The varieties P. are connected. In fact, as we see from 3.1.1. b), when x is a Richardson element for p. P consists of bp points. Now we will investigate the normality of a nilpotent variety, i.e. the closure of a nilpotent conjugacy class. Since a product variety is normal if and only if every factor is normal, we may assume that g is a simple Lie algebra. 3.1.3.
Theorem: (Kraft-Procesi) If g = stn, then every nilpotent variety is
normal. If g = spa, on, then every nilpotent variety is semi-normal and it is normal if and only if it is normal in codimension 2. By g = On is meant g = so, but G = On.
The fact that conjugation is
taken under On affects only the 'very even' classes which appear only when n is a 81
multiple of 4. A variety X is semi-normal if every homeomorphism Z -+ X is an isomorphism; so roughly it means that the nilpotent variety in the classical groups have no cusp singularities. In fact Kraft and Procesi have determined that the only type of abnormal singularity which can appear is the product of an affine space with two singularities of type A, glued together at one point. A point x of a variety X is called unibranch if the preimage of x in the normalization of X is connected. An example of singular unibranch point is ; the typical example of non-unibranch point is . A unibranch point is also called topologically normal. If an algebraic variety X is seminormal and all its points are unibranch, then X is normal. Normality (and seminormality) is a local, open condition. Hence the set of abnormal points is always closed. But the set of multibranch points need not be closed. In the following picture, all points on the z-axis are abnormal, but 0 is unibranch.
Let us denote by IH,"(X) the n-th local intersection cohomology group of X at x with coefficients in Z. An equivalent way to express the unibranchness of a point x in a complex variety X is to ask that IH2(X) = C. Assume that the moment map i : T*P -+ U, is birational, where C,, is the nilpotent variety determined by the Richardson class of P. From the theory of Borho and MacPherson, the dimension of IH2(Z,, should be expressible in terms of multiplicities of Weyl group representations. We describe the answer which is quite elegant but unfortunately still hard to compute. It generalizes to arbitrary G a formula in [Borho-MacPherson, p. 710] proved for SLn. All conjugacy classes in g have even dimensions (over C). For z E R, let Cx 82
be the conjugacy class of x and 2d. = codim (C,,: X). One has d. = dimB2 . Let d, be the codimension in R of C,. Then dimP.
d2 - d,. Let p, be the special
representation of the Weyl group W of G associated to the Richardson class C, by Springer's correspondence. For any x E R, W acts on H'(B.). Let C, = R,. 3.1.5. dimIH.,,)
Proposition:
When it is birational, z E NP, we have dimH*(P,) =
= mtp(pp, HzP(Bz))
Proof: Let A* = R*A*Qi'.p. If t is an irreducible representation of 7r- 1 C.,x E i,, let V(,, be a Q-vector space of dimension equal to the multiplicity of p in the monodromy representation of 7rC2 on the top cohomology group of P. Let i* : C. c-+ N, be the inclusion, and L, the local system on C2 with monodromy representation p. We shall say that the pair (C., p) is relevant to u if dimP = d2 - d, and V(,,) is a semismall map - i.e. dim(fiber) 5
$
0. Since y
dim(stratum) - we can apply a result of
Borho-MacPherson to deduce for z E A,
H'(Pz) = e(c2,X,)IH:2(dz-dp)(Cz; L~,) ®V(2 ,,) where the sum runs over the pairs (C2 , P) relevant to yi and such taht z E Ox g R,. A similar formula holds on the full flag variety
H'(Bz) = e(.z,)IHi-2d (Ox; L,) ® Vx But here the Weyl group W acts on both sides.
Borho and MacPherson have
shown that the right hand side is the decomposition of H'(B,) into irreducible representations: W acts on V(x,,,) by an irreducible representation p(,,) and (1) dimIHi-2d (0., L,) = mtp(p(3 ,,), HII(Bz)). In this way, one obtains a bijection called the Springer correspondence: (2)
{(x, p) I x E R/G,
P E ( 7riCz)^, p appears in H2d.(B2 )}
+-+
{P(,,)} = W^
If we plug formula (2) into (1), we obtain:
E
(3) dim H'(Pz) = z
2 mtp(p(, 2 1 ,), H+ dp(B,)) - dim V(,,)
EC.9)/p
83
®
(4) H*(Pz) =
IH;2 (dz-p)(O, LV)
® V(Zj)
2EOn CXp
Since C C C, we have d, ;> d,. Hence for -2(d. - d) to be non-negative, we must have d. = d, and so x E C,. On the other hand for relevant (x, P): dimV(,w,) = mtp(p , H2(dz.dp)(PZ)) When x E C,, P. is a point by the birationality of it, therefore: dimV(
=
1
if
=
0
otherwise
=
L
Collecting this information, and since p, = p(z,), (3) and (4) become respectively dimH*(Pz) = mtp(p,, H 2dp (B)) H*(Pz) = IH,(-A,,) F1 Note two particular cases of this proposition: -if z E C, , then z is a normal point of n,, hence H*(Pz) = C. H 2dp(B,) is the top cohomology group of Bz, and the identity 3.15 reduces to Springer's result: the special Weyl group representation pp occurs with multiplicity one in H2, (B,). -if z = 0, then 0 is unibranch because i-'(0) = P. B0 = B, H*(B) is isomorphic to the space of W-harmonic polynomials on t*, so H* (B) is equivalent to the regular representation of W. The identity 3.1.5. reduces to Joseph's result: p, occurs with multiplicity one in the space of W-harmonic polynomials on t* of degree d,.
III 3.2. Spherical Representations and Characteristic Cycles. We will concentrate our attention to the flag spaces X of type P where P is a parabolic subgroup of G associated by an Iwasawa decomposition to the centralizer H of an involution o- of G.. These are the flag spaces which appear in chapter II. There exists a Richardson nilpotent element x for P which is regular in a real form G(o-, R) of G such that H n G(o-, R) is compact. Therefore, the centralizer of x is 84
contained in some parabolic subgroup conjugate to P, cf. [Kostant-Rallis], and by 3.1.1.b. the moment map t : T*P --
p, is birational.
To decide whether the global sections of an irreducible (DA, K)-module M on P form an irreducible (g, K)-module or not, we shall do the following:
1. If .A, is normal, then 2.2.2. a) applies and gives a positive answer.
2. If )p is not normal, we look at the characteristic variety of M. If its image by 1 contains some normal points of M,, then 2.2.4. applies and gives a positive answer.
3. If g(CharM) does ot contain any normal point of n,, then we look at the characteristic cycle of M and Pm. If the coefficients of the cycle of largest dimension are equal, then we can again conclude that P.M is irreducible.
The application we have in mind is a direct proof of the irreducibility of discrete series representations of symmetric spaces.
Hence, after step 1.
, we will only
consider (DX, K)-modules which are supported on closed K-orbits in P. Also because very little seems to be known about the normality of nilpotent varieties in the (Note: For G 2
,
exceptional groups, we will only consider the classical groups.
irreducibility is preserved because the only nilpotent variety which occur are either the whole cone or the point zero.)
Step 1: The weighted Dynkin diagrams of the nilpotent orbits which are regular in the real form of G corresponding to H, are obtained as follows. Consider the Satake diagram of this real group. Put labels 0 on the black dots and labels 2 on the white dots. Note that these Richardson classes are always even. To translate these Dynkin diagrams into partitions we apply the inverse of the rule described in [Springer-Steinberg Ch. IV]. The answer is, with the obvious restrictions on p and q. 85
h
sin
s(gl,
sin
son
sln (R)
sl2n
'P2n
s2n
s02n+1
sop X s02n+l-,
sop, 2n,+ 1-
s02n
Sop X SO2n-p
sop,2n-p
S 0 4n
g12n
son
s 0 4n+2
gl 2n+1
sn+2
--
-
spn
SPq X SPnq
sPq,n-q
-
4
spn
gln
spn(R)
g(R) x gln_,)
so, xso-,
Satake diagram
sup,n-p
-
--.
Partition
e-
e
-O
s
(2p + 1, 1"-
p-
1
e(n)
(n, n)
. e-o
2
)
g
e=~0
O-
-
(2p + 1, 12(n-p)) (2p + 1,
K
12(n-p)-1)
(2n, 2n)
-
o-.
(2n + 1, 2n + 1) ((2q + 1)2,
1 2(n-2q-1))
(2n)
N.B.: When 2p = n in case su,,, or so,,, or q = n/2 in case sp,,, the corresponding partitions are simply (n) or (n, n). In light of [Kraft-Procesi], we can draw the following conclusions, which were first observed by D. Vogan: 1. if g = stn all nilpotent varieties are normal. 2. if g
=
son, all nilpotent varieties considered here are normal.
3. if g = spn, the nilpotent varieties considered here are normal if and only if
h = gen or sp, x sp,(n = 2q) or sPq,q+l(n = 2q + 1) The conjugacy class (2n,2n) in s04n is very even. This partition describes one
04n-conjugacy class which splits into two disjoint SO 4 n-conjugacy classes.
The
image of the moment map is the closure of one component. Kraft and Procesi have
proved [Prop. 1.5.4] that this SO 4 n-nilpotent variety is normal. For the other cases, it suffices to apply 3.1.3 and the fact that these partitions have no degenerations of type (2n - 2n) -
(2n - 1, 2n - 1,1, 1), cf. [Kraft-Procesi].
However, for the symptectic groups, the partition ((2q + 1)2,12(n-2q-1)) show that the corresponding conjugacy class is not normal. In codimension 2, it has a singularity which looks like two singularities of type Aq glued at one point , times
an affine line. So for q = 1, the picture 3.1.4. give a heuristical idea of the kind of singularity we have in hands.
86
Using 2.2 a) one obtains:
3.2.1. Proposition: For a flag variety X of type P with P minimal in some real form of G, U surjects onto P(X, D\) for all A E t*, if g =.s4 1 , son, G2 , and if g = sp, with gR = spn(R), SPq,, or SPq,,+. Step 2: Take g = sp, h = sp, x sp,,. The group K will determine the real form of g and h. First we consider K = GL, so that GR = SP,(R) is the split real form and HR = SPt(R) x SPe
(R). The rank of the symmetric space GR/HR
is min(e, n - f). After a few calculations, one finds that there is a unique closed K-orbit Y which interests the open H-orbit. Moreover the image p(TjX) C g* contains nilpotent elements associated to the partition ((2 + 1)2,j2(n-2t-2)); these are generic in C, = pi(T*X), hence normal points. Using 2.2.5. we conclude that if M is a standard (Dl, K)-module supported by the closed K-orbit Y with A E t* then T(X, M) is an irreducible (g, K)-module. However U(g) does not surject onto
P(X, DA) for certain dominant A such that A + pt E t* is not B-dominant. Now take K = SPk x SPnk. Then GR = SP(k,n-k) and HR = SP(r,e-r) x SP(n-e+k+r,k-r) for some r < k. (We suppose t < n-t). In the equal rank case, one must have t < r and 2t < k. It suffices to consider the situation r = k > 2. After a few calculations, we find again taht there is a unique closed K-orbit Y which intersects the open H-orbit. But the most regular elements of p(T.X) correspond to the partition (3 2t, 12(n-3t)). Comparing it with the partition for the Richardson class in C, we see that if the rank t of G/HR is 1, then we can conclude that irreducibility is preserved for dominant A and DA, K)-modules supported on Y. In higher rank we cannot conclude anything. However, observe that the centralizer of a generic element x of pz(T X) is always connected. It would be interesting to know if this implies that x is unibranch, i.e. u-'(x) is connected. However using 11.7.5 and D. Vogan's result asserting that the (g, K)-modules P(X, i. oy (A)) which give rise to square integrable representations of G on GR/HR are irreducible, we obtain the following multiplicity one property: 3.2.2.
Property:If G is a classical group or G 2 , then the discrete series of
L 2 (CR/ HR) is multiplicity free, for any real form G 87
of G and any symmetric
subgroup H of G. Step 3: Or last resort is to compute the coefficients of the characteristic cycle of F.M (also called the associated cycle of P.M). For Y a closed K-orbit and M a standard (Dx, K) module on M. The characteristic cycle of M is simply [A] = [Tj X], because M is irreducible. The characteristic cycle of Fm is the direct image in Ktheory of [A]. So we can write: [Char P.M] =
tz. (A) =
X(Pz,
0(A))[i(A)] +
lower dimensional terms
Here X(Ps, O(A)) is the Euler characteristic of the variety, x is generic in p(TjX) and 0(A) is the restriction to P of the sheaf 0(A) that we have extended from Y to X. If x(P, 0(A)) equals 1 as it is the case if P, is a normal point (and it is birational), then P.M is irreducible at its generic point. Indeed, the coefficient of the variety Z of largest dimension in Char (rPM) bounds the number of consitutents of P.M whose associated variety is Z.
88
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