EXPLICIT
REALIZATION
A METAPLECTIC
OF
REPRESENTATION
Y. FLICKER, t D. KAZHDAN tt AND G. SAVIN ~
0. Let F ~ C be a local field with char F + 2. In [W] Weil explicitly constructed a model of a genuine unitary representation 0 of the two-fold covering group Sp of the symplectic group Sp over F. In particular, the existence of the covering group Sp was first proven in [W]. It is now known (see, e.g., [M]) how to construct r-fold covering groups of split semi-simple groups over a field F ~ C containing a primitive rth root of unity. In particular, when r = 2, such F has char F ~ 2. In the case of GL(n) the analogous genuine unitarizable representation O of a covering group is defined in [KP 1] as a sub- or quotient of some induced representation. This O corresponds to the trivial representation of GL(n) by the metaplectic correspondence (see [KP2], [FK1 ]). The purpose of this paper is to construct an explicit model of the representation O = 03 of a two-fold covering group G of GL(3) over a local field F ~ C of characteristic ~ 2, analogous to the explicit model of the representation of Weil [W]. We also determine the unitary completion of the unitarizable 0 3. The unitary completion o f our model coincides with the model ofTorasso [T] when F = R. The existence o f our model has interesting applications in harmonic analysis. Some of these applications are discussed in detail in w In a sequel IF1] the techniques of this paper are generalized to construct an explicit model of On for any n >_ 3.
1. T h e representation To state our Theorem and its Corollaries, we begin by specifying the representation O to be studied. 1.1. Let F be a local field ~ C of characteristic ~ 2. For every integer n > 1 there exists (see [M]) a unique non-trivial topological central double covering group p : Sn --" SL(n, F). Choose a section s : SL(n, F) ~ Sn corresponding to a choice of a two-cocycle fl'~ : Sn • S~ ---"ker p which defines the group law on Sn. Embed Gn = GL(n, F) in SL(n + 1, F) by t Partially supported by an NSF grant and a Seed grant. This author wishes to express his deep gratitude to IHES for its hospitality when the paper was written. tt Partially supported by an NSF grant. Sloan Doctoral Fellow.
17 JOURNAL13'ANALYSEIdATH~MATIQUE,Vol. 55 (1990)
18
Y. FLICKER ET AL.
t : g---"
o)
det g -1 .
Denote by G" the preimage p-l(t(Gn)). Let (-, .)" F 2 • F2---- {1, - 1 } be the Hilbert symbol. Identify { 1 , - 1} with the kernel of p. Put f l ( g , g ' ) = fl'(g, g')(det g, det g') (g, g' in G,). Let s : G, ~ G~' be the restriction of the section used in the definition of Sn +,. Denote by G, the group which is equal to G~ as a set, whose product rule is given by s(g)~ 9s(g')~' = s(gg')~'fl(g, g'). Then G, is a non-trivial topological double covering group of 0,. Let d and B be the groups of diagonal and upper-triangular matrices in G,, and A and B their preimages in G,. Note that s is a homomorphism on the group N of upper-triangular unipotent matrices, and put N = s(N). Let 2 be the center of G, and Z the center of Gn. L e m m a 1. Let A z be the group o f squares in A, and put A 2 = p-l(d2). Then (i) the group ZA 2 is the center o f A , (ii) i f n is even then Z = A2 C~ p - ' ( Z ) , (iii) i f n is odd then Z = p - ~ ( 2 ) , and p defines an isomorphism p : Z I ( Z r A 2) ~ ZIZ, 2 -~ F X l F •
Proof.
See [KP 1], Prop. 0.1.1.
Define a map t = tn : A ~ A 2 by t(h) = s(h)2u(h), where
u(h) =
1I, ~ir ( a ) , where a is an involution o f G which we proceed to define. Let w. be the anti-diagonal matrix ( ( - 1)i+lJi,,+l_j) in G,. Consider w, as an element o f SL(n + 1, F ) v i a j . D e n o t e by # t h e involution #(g) = w [ ' 9tg-~. w, o f SL(n + 1, F). Since the Steinberg group St(n + 1, F) is generated by elementary matrices (see [M], p. 39), # maps e l e m e n t a r y matrices to e l e m e n t a r y matrices, and # preserves the relations which define St(n + 1, F), then # lifts to an involution o f St(n + 1, F), hence to an involution # o f G. Suppose that n is odd. T h e n both s and s ~ = # o s o # satisfy the conditions o f L e m m a 4. H e n c e there exists a character X : FX ~ { 1, - 1 } such that s ~ -- ~ o s. Define a = ~ o 0; it is an inyolution o f G. Since a o s = Z o # o s = s o # on ZA 2, we have
~(a(s(z)s(h2))) = ~(s(#z)s(#h2))
for all z C Z ,
h CA;
hence ~(a(x)) = J ( x ) for all x in ZA 2. By L e m m a 3(i) we have p~ o a m p t , where p~ is the unique extension o f ~ to A. H e n c e n6 ~ a ~ nr, and by L e m m a 3(ii) we have O o a ~ O. It follows that there exists a non-zero o p e r a t o r I : V ~ V such that O ( g ) l = I O ( a ( g ) ) for all g in G. Since O is irreducible, by Schur's l e m m a 12 is a scalar, which we normalize to be 1. This determines I uniquely up to a sign. T h e choice O ( a ) = I determines an extension o f O to the semi-direct product
G#=GM(a). Remark. j ~ i) that
(i) It is easy to check (consider first the case where hs = 1 for all
(r(s(diag(hi))) = s(diag(h~-+ll_i)) 9 IX i-,
hi, j-i+,
In particular ~(s(z)) = s(z-').
(z, - 1)"~"-w2
Consequently a(g) = ( - 1, det p ( g ) ) ( " - w2#(g)
and
for z C F x ~
Z.
Z(x) ----( -- 1, x ) ("- w2.
(ii) Since (det #(g), det #(g')) = (det g, det g') (g, g' C d ) , the formula in (i) for the involution a o n G defines also an involution a ' o n G ' which satisfies p * a ' = # o p on G ' and o r o s - - s o # o n ZA 2.
METAPLECTIC
REPRESENTATION
21
1.4. An explicit model for O 2 is easily obtained (see [F1], w Example, or [FM], and the proof of Proposition l, w below) from that of the even Weil representation (see [F], p. 145). Indeed, this Weil representation is a representation of $2, which extends to a representation of s(Z)S2 (by the character y = 7~ on s(Z)). The representation 02 is the GE-module induced from this extension to
s(z)&. In this paper we construct an explicit realization of the unitarizable G3-module 03. When F = R the unitary completion of 03, or at least its restriction to p-~(SL(3, R)), coincides with the unitary p-~(SL(3, R))-module constructed by Torasso [T]. 2. T h e realization The representation O = O3 will be realized in a space of functions on a two-fold covering space X of the punctured affine plane X = F • F - {(0, 0)}. Clearly -- F \ GL(2, F), where
It is easy to see that the restriction of s to F is a homomorphism. Hence we can define the double cover X of X to be s(F) \ G2. Then X is a homogeneous space under the action of G2. To be able to write explicit formulas for the action of G2 on X, recall the explicit construction of G2. Put X
=
d,
c = 0,
and
( x(gg') x(gg') ) fl(g' g') = \ x-~g) ' x(g')det g " Then G 2 is the group of pairs (g, () (g in GL(2, F), ( in ker p) with the multiplication law
(g, ()(g',
= (gg,, (('fl(g, g')).
Given ~ = (x, y) in .g', put x(~) = x ifx ~ 0 and x(~) = y i f x = 0. Identify Xwith .g • ker p by mapping the image in X of the element s(h)( of G, where
,)
Y
to the element (x, y; ((x(h), det h)) of .~" X ker p. Then the action of G2 on ,~ • ker p implied by this identification is given by
22
Y. FLICKER ET AL.
/ (~, ~)(g, ( ' ) = {~g, ~('
(,)
\
x(zg) \ x(2,) ' x(g) /
\ det g ) ] . /
R e m a r k . Replacing (., .) by the nth Hilbert symbol, (.) defines an n-fold covering of the punctured plane X as the homogeneous space s(F) \ G2. D e f i n i t i o n . A function f : X ~ C is called genuine iff(z() = (f(z) for ( in ker p, z in X. It has bounded support if there is a compact subset of F • F which contains all g in ,~ with f(g; () v~ 0. It is called homogeneous iff(t2x, t2y; ~) = It I-If(x, y; () (t in FX). Let LE(X) be the space of genuine, square-integrable, complex-valued functions on X. Let C(X) be the space of smooth functions f i n L2(X). Denote by Cb(X) the space of f i n C(X) with bounded support. Denote by Ch(X) the space of homogeneous f i n C(X). Let P (D B) be the standard maximal parabolic subgroup of type (2, l) of G, and consider the subgroup P = p-~(P) of G. Define the action of P on L2(X) as follows (we denote the action by O): (1)
(2)
(3)
[O(S(o~))f](z)=ldetglU2f(zs(g))
0
s
O
1
v
0
1
[[i s
a
iI
(z) =
o
,(ux +
(z) = ~7(a)f(z)
(g in GL(2, F));
(u, v in F);
(a in FX).
a
Under the action (1) the space Ch(X) is a G2-module; it has a unique proper non-zero G2-submodule Ch(X) ~ isomorphic to O2@ [det ITM (see [F], p. 145). Indeed, the space
,,s, = ~ l a/b Iu2+~(g), a E F x, b E F • ~ E ker p}
23
METAPLECTIC REPRESENTATION
is a GE-module under the action p(g)r [detg[1/4~(hg). At s = - ~ it is reducible, of length two. Its unique proper non-zero submodule is 02 @ [det [u4.
The map r f((0, 1)g)= Idet g l-'/E-s~(g), establishes a G:-module isomorphism from I(s) to the space J(s)
=
{ f: X ~ C; f(b(x, y); () = ( I b I-'-2~f(x, y; 1), b EFX2},
with the G2-action p(g)f(z) = Idet p(g)13/4+Sf(zg) (z ~ X ) . D e f i n i t i o n . Denote by Cb(X) ~ the space of f i n Cb(X) for which there exists fo in Ch(X) ~ and A I > 0 such that f ( z ) = f o ( z ) for all z = ( x , y ; ( ) with m a x ( I x l, lYl)_- 0 such that
f(t2x, t ~ y ; ( ) = [ t [ - t f ( x , y ; ( )
i f m a x ( [ x l , lYl) 0
METAPLECTICREPRESENTATION
33
and f ' : F x --- C x satisfying f'(xa 2) = [a [-1/2f'(x)(x, a in F x) with f ( x ) = f ' ( x ) for Ix [ --
0 , and c ( 0 < c < 8 9 such that Itlf(t2x, t2y; r , ) = f ( x , y ; ~) for m a x ( I x l , l y l ) _- 