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RICHARD MELROSE. Abstract. The calculus of Heisenberg operators on a contact manifold is re- called and described in terms of parabolic compacti cation ...
HOMOLOGY AND THE HEISENBERG ALGEBRA AT ECOLE POLYTECHNIQUE, MAY 20 1997 JOINT WORK WITH C. EPSTEIN AND G. MENDOZA RICHARD MELROSE Abstract. The calculus of Heisenberg operators on a contact manifold is re-

called and described in terms of parabolic compacti cation and blow up. The Hochschild homology is computed for its ideals and those of the related extended Heisenberg calculus. These computations are used to reduce the index formula for these algebras to a combination of the standard Atiyah-Singer theorem and the index theorem of Boutet de Monvel for Toplitz operators.

1. Compactifications Let V be a vector space, of nite dimension over R and let L  V be a subspace. Choosing a complement W to L, V = L  W, we can consider the homotheity (1)

t : V ! V ; t (l; !) = (t2 l; t!) ; t > 0 ; in which the variables in L are `parabolic'. The space of orbits in V n f0g is a sphere denoted LSV: The disjoint union, the L-parabolic compacti cation of V , L (2) V = V t L SV has a natural structure as a compact C 1 manifold with boundary (a ball) where the C 1 structure near the boundary (in nity) is generated by the smooth functions on V n f0g which are homogeneous of non-positive integral orders (0; ?1; ?2 suce) under . Lemma 1. The Lie group of linear transformations of V preserving L lifts to act as C 1 di eomorphisms of LV preserving the two submanifolds of the boundary @ L = L L \ L SV and Z = W \ L SV : Here L L and W are the closures of L and W in LV . As a result of this lemma LV  with its C 1 structure is independent of the choice of W (as is Z | see Figure 1 for another complement W 0 ). Example 1. The radial compacti cation of V corresponds to L = f0g; we write f0gV = V . Example 2. The case L = V is   V  V = V ; @ V ; N @ V in terms of parabolic blow up of the boundary described in [4]. With these conventions, @ V V  @ V and the notation above, L L and W = f0g W is consistent. We are most interested in the case dimL = 1. Then the complement of Z = W \ LSV in the boundary, LSV , is LSV nZ = U+ t U? 1

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RICHARD MELROSE U+

L

W’ Z W

V

U_

Figure 1. The L-parabolic compacti cation.

where  correspond to the orientations of L. Lemma 2. In case dimL = 1, U have natural structures as vector spaces and there is a group of natural isomorphisms to V=L, l : Usgn(l) ?! V=L ; l 2 Ln f0g ; l0  ?l 1 = t1=2  Id ; l0 = tl ; t > 0 : We may identify U+ and U? , and hence regard then as the same vector space U; using the isomorphism M = (? Id)  ?l  ?l 1 : U+ ! U? for any l > 0: Then M is a di eomorphism of L SV which xes Z pointwise. The closure of U+ in LSV is L SV n U? and this is canonically identi ed with its U+ parabolic compacti cation. Thus L SV may be viewed as the sphere made from two copies of the U-parabolic compacti cation of U with the unique C 1 structure making the identi cation map of the two copies into a di eomorphism xing the common boundary. We also consider a larger compacti cation of V , the extended L-parabolic compacti cation. This is a manifold with corners and is obtained by the (radial) blow up of Z. eLV =  LV ; Z  :

Lemma 3. There is a canonical isomorphism eL V =  LV ; Z  ! V ; @ L ; N @ V 

with the space obtained from V by parabolic blow up, with respect to the boundary conormal direction, of @ L . The group of linear transformations of V preserving L lifts to a smooth action on eL V .

The smooth lifting of these linear transformations means that these notions can be extended directly to the case of a vector bundle V with a sub-bundle L. We use the same notation for the bre-by- bre compacti cations; V , LV and eL V : Ther

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U+ L

W’ ~ Z W

U_

Figure 2. Extended L-parabolic compacti cation.

are natural smooth blow-down giving a commutative diagramme eLV BB z zz zz z zz

BB BB BB

O

V EE }

L

b

!

EE EE EE

{ {{ { {{ {{ =

V

V consistent with the identi cations over the interior. 



.

?

Q

1

2. Riemann-Weyl quantization Let X be a compact Riemann manifold with metric g. Let U  X 2 be a small geodesically convex neighborhood of the diagonal in X 2 ; so if (x; y) 2 U there is a unique short geodesic from y to x lying in U. Denote the midpoint of the geodesic by m(x; y) and by (x; y) the tangent vector at the midpoint with respect to arclength parameterization. Shrinking U if necessary, (3) M : U 3 (x; y) 7?! (m(x; y); (x; y)) 2 TX is a di eomorphism onto a neighborhood of the zero section. Proposition 1. If gs is a smooth family of metrics on X and Ms are the corresponding di eomorphisms (3) then the vector eld Vs on TX , near the zero section, generating the di eomorphisms Ms  M0?1 = exp(Vs ) by integration is of the form X d M  ' = M  (V ') ; Vs = i j Wij ; ds s s s where i are (linear) functions vanishing at 0 in TX and the Wij are smooth vector elds which are tangent to the zero section.

This proposition represents the well-known `high invariance' properties of the Riemann-Weyl bration of W given by (3). The standard pseudodi erential operators on X can then be represented directly globally by choosing ' 2 C 1 (U) with '  1 near the diagonal and considering the

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kernels obtained by `Riemann-Weyl quantization' (4) ?k C 1 (T X) 3 a 7?! A = (2)?n '

Z

 Tm X (x;y )

ei (x;y) a(m(x; y); ) d  dgR :

Here d is the Riemann density on the bres and dgR is the density on the right factor in X 2 . The integral is best interpreted as a push-forward from the pull-back to U of T  X. Replacing the radial compacti cation by the two parabolic compacti cations L T X and eLT X gives the two classes of operators of interest here.

De nition 1. The spaces kL (X) and k;l eL (X) of L-parabolic pseudodi erential operators and extended L-parabolic pseudodi erential operators are de ned respectively as the sums of C 1 (X 2 ; R) and the ranges of the Riemann-Weyl quantization map (4) acting on ?k C 1 (L T X) and ?L k ?S l C 1 (eL T X). Notice here that L T X is a compact manifold with boundary,  2 C 1 (L T X) being a de ning function for the boundary and eLT X is a manifold with corners, L 2 C 1 (eLT X) and S 2 C 1 (eL T X) being de ning functions for the two

boundary hypersurfaces, the rst `arising from L' and the second, S arising from the second blow up in (1). It is implicit in this de nition that the class of kernels obtained is independent of the metric used in the de nition. This follows for Proposition 1 and a deformation argument. The space kL (X) is the same as that considered by Taylor [6] and by Beals and Greiner in [1] (except that they take L to be a line bundle); they prove coordinateinvariance and composition properties; see also the earlier work of Dynin. Similar results hold for the extended spaces. Proposition 2. The spaces kL (X) and k;l eL (X) form graded algebras of operators on C 1 (X) 0 kL (X)  kL0 (X)  kL+k (X) (5) k0 ;l0 k+k0 ;l+l0 k;l (X) eL (X)  eL (X)  eL and

2k;k k (6) kL (X)  k;k eL (X) ; (X)  eL (X) : It is straightforward to extend these de nitions and properties to operators acting from sections of one vector bundle to another, simply by taking tensor products with bundle homomorphisms. We are especially interested in the case that X is a contact manifold with contact line bundle L. Then we call L (X) and eL; (X) the Heisenberg and extended Heisenberg algebras respectively and denote them H (X) and eH; (X): From this point on we restrict attention to these contact cases. Taylor in [6] de nes two spaces similar to eH(X): These are respectively somewhat smaller and somewhat larger that the space considered here. Taylor's smaller space is an asymptotic completion of the sums of products of the Heisenberg and standard spaces. His larger space has non-polyhomogeneous symbols and is therefore less precise.

Z

ZZ Z

ZZ

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Example 3. One of the results shown by Taylor in [6] is that in the case of strictly pseudoconvex domains the operator +b on the boundary has inverse in the smaller of his calculi. See also the comments in section 4 below. 3. Symbols For a manifold with corners, X; let C_1 (X)  C 1 (X) be the space of functions vanishing to in nite order at the boundary. The quotient T (X) = C 1 (X)=C_1 (X) is the algebra of Taylor series at the boundary. More generally if i ; for i = 1; : : :; N; are de ning functions for the boundary hypersurfaces then k1    NkN T (X) = 1k    kNN C 1 (X)=C_1 (X) is the space of Laurent series with singularities of given order. We denote the union over orders by ?1    ?N T (X): Riemann-Weyl quantization (4) de nes full symbol maps FS : A (X) = (X)= ?1 (X) ! ? T (TX ) HFS : AH(X) = H(X)= ?1 (X) ! ? T (L TX ) (7) eHFS : AeH(X) = eH (X)= ?1 (X) ! ?H S? T (eLTX ): Since ?1 (X) is an ideal in each algebra these quotients are algebras and RiemannWeyl quantization, with inverse the appropriate full symbol map in (7), induces associative products on the Laurent series spaces. In the standard case this is the `?-product' 1

Z Z Z Z Z Z Z Z

(8)

a?b=

1

Z Z ZZ

1 X l=0

Pl (a; b)

where Pl is a bilinear di erential operator on T (T X) of the special form Pl (a; b) = l P~l (a; b) where P~l is generated by vector elds tangent to to the boundary of T X: The rst two terms are P0 (a; b) = ab; P1(a; b) = 2i1 fa; bg (9) in terms of the Poisson bracket. Thus the principal symbol, which is independent of the choice of metric, k : k (X)= k?1 (X) ! C 1 (SX; R?k ) is multiplicative. Here R is the conormal bundle to the boundary SX of T X: There are similar principal symbol maps, independent of the choice of metric, for the Heisenberg algebras k H k : kH (X)= kH?1 (X) ! C 1 (L SX; R? H ) (10) k?1;l?1 ?k;?l): (X) ! C 1 (eLSX; ReH eHk : keH (X)= eH In the rst case RH is the conormal bundle to LSX as the boundary of L T X: In the second, eL SX is not a manifold. Rather it is the union of the two boundary

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hypersurfaces of eH T X with ReH being the exterior product of the two conor?k;?l ) includes the necessary compatibility mal bundles; the notation C 1 (eLSX; ReH condition at the common boundary hypersurface of the two components. Let XL be the double cover of X given by the orientation sheaf of L: Then LSX n Z is a vector bundle, U; over XL : The choice of a non-vanishing (local) section, ; of L reduces the bundle locally to T  X=L over XL and the di erential of induces a symplectic structure on the bres of U; this structure is independent of the choice of : Thus U is a symplectic vector bundle over XL : On any symplectic vector space U with symplectic form !; there is a natural associative product on Schwartz' space S (U) : # : S (U)  S (U) ?! S (U); Z (11) u#v() = ei!(;) u( + )v( + )dd: As an algebra this is isomorphic to the composition product on S (R2n) considered as the kernels of operators on S (Rn); in such a way that the continuous extension gives the algebra of isotropic pseudodi erential operators on Rn : ?k C 1 (W )  kiso (Rn): Except that we have the parabolic compacti cation of U and we need to take into account the e ect of order, this gives the product on the Heisenberg principal symbol; some further comments on the necessary normalizations are made below. Proposition 3. Let iso(U) be the bundle of istropic algebras, with symbols having half-integral step expansions, on the bres of the bundle U over XL then k (U)= 1 H k : kH(X)= kH?1 (X) ?! ?2iso

Z 1 2

1 2

is an isomorphism of algebras, where the equivalence relation 1 requires that under the orientation isomorphism of U+ and U? the operator is invariant modulo

?1 iso : That is, the leading symbol product in the Heisenberg calculus is H k+k0 (AA0 ) = Hk (A)#Hk0 (A0 ): Thus the principal symbol composition for the Heisenberg algebra can be reduced to the global composition of isotropic pseudodi erential operators. The product for the extended Heisenberg algebra is essentially the same, except that the `link' between the two parts of L ST X n Z given by 1 is broken. Namely the induced product ?k;?l )  C 1 (L SX; R?k0 ;?l0 ) ?! C 1 (L SX; R?k?k0;?l?l0 ) C 1 (L SX; ReH eH eH is given by #; the isotropic product on the Heisenberg boundary hypersurface, and by pointwise product on the radial hypersurface S: The compatibility condition between these products is just the multiplicativity of the principal symbol in the isotropic calculus. 4. Kernels The notion of parabolic blow up (or iterated radial blow up) already alluded to allows the kernels of Heisenberg, and extended Heisenberg, pseudodi erential operators to be reduced to polyhomogeneous conormal distributions.

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For a contact manifold X the di erence of the lift of the contact line bundle from the two factors de nes a subbundle L~  N  Diag of the conormal bundle to the diagonal Diag  X 2 : The parabolic blow up of the diagonal with respect to this subbundle, as discussed in [4], gives a manifold with boundary XH2 = [X 2 ; Diag; L~ ] ?! X 2 : The kernels of Heisenberg pseudodi erential operators of negative order are locally integrable functions (really right densities) on X 2 ; smooth away from the diagonal and lifting to polyhomogeneous conormal distributions, with respect to the boundary, on XH2 : The kernels can be precisely characterized in terms of their expansions (with logarithmic terms occuring only for operators of integral order). This description of the kernels is completely analogous to the description of pseudodi erential operators as singular integral operators in terms of their kernels. The two cases can be combined to give a similar description of the kernels of extended Heisenberg operators. The appropriate space can be obtained from XH2 by the blow up of a submanifold, Z 0 ; of the boundary 2 = [X 2 ; Z 0] ?! X 2 ?! X 2 : (12) XeH H H 0 Here Z is de ned as the boundary of the lift of a submanifold through the diagonal with normal L~ :

Theorem 1. The kernels of extended Heisenberg pseudodi erential operators with 2 : both orders negative lift to be polyhomogeneous conormal on HeH

Once again the distributions which occur can be precisely characterized in terms of their orders and the coecients of logarithmic terms. 5. The @ -Neumann problem In joint work with Rafe Mazzeo this explicit description of the kernels of extended Heisenberg pseudodi erential operators is exploited to give a similarly explicit description of the kernel of the solution operator to the @-Neumann problem of J.J. Kohn, on a strictly pseudoconvex domain. This is done by construction a class of operators which combines the features of the Poisson operator for an elliptic boundary problem with the parabolic operators discussed in [4]. This construction is quite analogous to the combination above of standard and Heisenberg pseudodi erential operators to give the extended Heisenberg operators. In particular appropriate composition theorems show the Poisson operator for the @-Neumann operator to be in this class. 6. Boundedness, compactness and closure The symbolic properties of the Heisenberg algebras allow Hormander's elegant proof of L2 boundedness to be invoked in this setting. Proposition 4. The elements of H (X) and eH; (X) which extend to bounded operators on L2 (X) are precisely those in 0H (X) and 0eH;0(X) respectively. The elements extending to be compact operators on L2 (X) are those in ?H1 (X) and ?1;?1(X); those giving Fredholm operators ar distinguished by the invertibility of eH

Z

their symbols.

ZZ

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As usual these results extend directly to operators acting on sections of vector bundles. The Heisenberg algebra has a natural ideal. In terms of the de nition through Riemann-Weyl quantization it consists of the sums of elements of the smoothing ideal together with operators given by (4) where a 2 ?k C 1 (L T X); k a  0 at Z; i.e. the symbol vanishes to in nite order at Z: In fact this is easily understood in terms of the extended Heisenberg algebra since it is simply eH;?1(X)  eH; (X) in which it is also an ideal. Let AH(X); JH (X) and K(X) be the C  algebras on L2 (X) obtained by closure of 0H (X); 0eH;?1 (X) and ?1 (X); similarly let AeH (X) be the closure of 0eH;0 (X): Thus K(X) is just the Kalkin algebra.

Z

ZZ

Proposition 5. There are short exact sequences of algebras K(X) ?! JH (X) ?! C 0 (X; K(Rn)) and JH (X) ?! AH (X) ?! C 0 (SH)

where H = T  X=L is the contact hyperplane bundle.

These ideal decompositions can be used to compute the K-theory of the completed algebras.

Z

7. Hochschild homology The (continuous) Hochschild homology of an algebra such as A = H(X) is determined by the Hochschild di erential acting on the completed tensor products k) bk : A ^ (k+1) ?! A ^ k ; HHk (A) = null(b : bk+1A ^ (k+2) On A ^ A; b1 gives the commutator and on higher chain spaces gives `higher commutators'. The dual Hochschild cohomology therefore represents higher traces. In particular HH0(A) is the space of trace functionals on A: For the algebra of smoothing operators HH0 ( ?1 (X)) = C with all other cohomology vanishing (it is homologous to a matrix algebra). The same is true for the residual calculus of the isotropic algebra on any vector space. The full symbol algebra of the isotropic algebra has Hochschild cohomology 2n?k?S(V  V 0 )  S1: (13) HHk ( iso (V )= ?1 iso (V )) ' H Here S(V  V 0) is the sphere of the cotangent space V  V 0 ; S1 is the circle and n = dimV: This identi cation can be seen from the work of Brylinski and Getzler [2]. Using these ideas the homology of subquotients can be identi ed ; ?1 2n?k?SL  S1 (14) HHk ( ?1 eH (X)= (X)) ' H where now dimX = 2n ? 1 is a contact manifold with contact bundle L of which SL is the associated `circle' (i.e. orientation) bundle. Similarly HHk ( ?eH ;1(X)= ?1 (X)) ' H 2n?k(ST X  S1; @L  S1) (15) is the cohomology relative to the boundary of L  ST X:

Z

Z

Z

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8. Index formul Using the properties of the calculus and these homological computations the index of extended Heisenberg operators, acting on bundles, can be computed. There are two constituent index theorems. First, the Atiyah-Singer index theorem gives the index for standard pseudodifferential operators. Secondly, for elements of the Heisenberg calculus of the type Id +A where A 2 ?1 H (X); E) and the non-commutative symbol of Id +A is invertible at each point, the index can be deduced from the index formula of Boutet de Monvel [3]. The combination of these to formul to give a general index formula uses a deformation argument which allows the operator to be retracted into the standard calculus, modulo a term for which the index can be computed using the Toeplitz index theorem of Boutet de Monvel. This deformation is possible because the ellipticity of the operator, in the Heisenberg sense, implies the vanishing of its index as a family of isotropic operators { see also the family index theorem of Fedosov; see [5]. References [1] Richard Beals and Peter Greiner, Calculus on Heisenberg manifolds, Annals of Mathematics Studies, vol. 119, Princeton University Press, Princeton, NJ, 1988. [2] J.-L. Brylinski and E. Getzler, The homology of algebras of pseudo-di erential symbols and the noncommutative residue, K-theory 1 (1987), 385{403. [3] L. Boutet de Monvel, On the index of toeplitz operators of several complex variables, Invent. Math. (1979), no. 50, 249{272. [4] C.L. Epstein, R.B. Melrose, and G. Mendoza, Resolvent of the laplacian on strictly pseudoconvex domains, Acta Math 167 (1991), 1{106. [5] B.V. Fedosov, Index Theorems, Partial Di erential Equations, VIII, Encyclopedia of Mathematics, Springer Verlag, 1996, pp. 155{258. [6] M.E. Taylor, Noncommutative microlocal analysis. I, Mem. Amer. Math. Soc., vol. 313, AMS, 1984. Department of Mathematics, MIT

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