Proc. Nat. Acad. Sci. USA Vol. 72, No. 4, pp. 1406-1410, April 1975 Intertwining Operators and Automorphic Forms for the Metaplectic Group (principal series representation/theta-functions/Eisenstein series/ noncuspidal automorphic forms ofhalf-integral weight) STEPHEN GELBART* AND PAUL SALLYt * Cornell University, Ithaca, New York 14850; and t University of Chicago, Chicago, Illinois 60637 Communicated by E. M. Stein, January 6, 1975 ABSTRACT The local results announced in this paper above. Let 0, denote the ring of integers of F, and set concern the analytic continuation of intertwining oper- ators for the metaplectic covering group of SL(2) over an K, equal to SL2(0,). Since a,, is chosen to be trivial on arbitrary local field and the decomposition of Weil's K, X K,, if the residual characteristic of F, is odd, we representation attached to the quadratic form q(x) = x2. can introduce the continuous two co-cycle a(g,g') = The global results concern Eisenstein series on the meta- plectic group and a Siegel-Weil type formula relating the fI av(g,,g,') on GA and use it to define a global meta- residues of these series to certain generalized theta func- V tions. plectic group GA. This group is a nontrivial topological central extension of GA by { A 1}. By quadratic reciproc- The purpose of this note is to announce some results ity, a is cohomologically trivial on GF, about intertwining operators for local metaplectic By B we denote the upper triangular subgroup of G; groups and their application to the global theory of A is the diagonal subgroup, and N the subgroup automorphic forms. The main results imply a kind of {[0 ] }. If H is any subgroup of Gv or GA, we denote its Siegel-Weil formula relating theta-functions on the complete inverse image in GO or GA by ft. If a is co- metaplectic group to residues of Eisenstein series. homologically trivial on H X H, then H is isomorphic to Many of these results were conjectured in a similar some subgroup of f which we again denote by H. In context in a preprint by the first named author.t A this case, ft = H X {I 1} . As already noted, important detailed treatment of the local theory presented here examples of such subgroups of G, are N, and K,(v odd); will appear in a joint paper (manuscript in preparation). A,, is not, since Hilbert's symbol, regarded as a two-co- The global theory pertains to theta-functions and Eisen- cycle on FZ, is nontrivial. stein series and extends earlier work of Kubota (refs. 1 The inverse image of { 1 } in Ov or GA will be denoted and 2), Shimura (refs. 3 and 4), and Siegel (refs. 5 and by Z2 and our interest will be exclusively in representa- 6). Proofs of this part of the theory will be given by tions of G, or GA which are nontrivial on Z2, i.e., repre- the first named author (manuscript in preparation). sentations which define genuine projective representa- Basic facts and notation tions of G,, or GA. We refer to such representations as genuine. Let G denote the algebraic group SL(2) and F a local field of characteristic zero. Weil's metaplectic group is 1. Intertwining operators for the metaplectic group (the then a topological central extension of GF by I1- }. p-adic case) This group is denoted by GF and is nontrivial as a group Since F, is fixed throughout the local theory, we shall extension iff F $ C. Actually, from ref. 7 it follows that drop the subscript v in F and the subscript F, in GF,. GF is the unique such extension. Also, elements of G will implicitly be viewed as elements To describe GF explicitly, we follow ref. 1 and use a of G via the lift g -- (g, 1). Finally, for simplicity of conveniently chosen two-cocycle a on GF. Thus, GF is exposition, we shall assume the residual characteristic realized as the set of pairs (g,~), with g E GF, r = + 1, of F is odd. Completely analogous results are valid in and multiplication is given by (g,¢) (g',~) = [gg',gst general. a(g,g') ]. If (a,b) denotes the Hilbert symbol of a and b in Let r denote a generator of the prime ideal P of 0 and FX, then at(g,g') = (aa') if g = [a-l] and g' = [0 (0)-1]. e a generator of the roots of unity in F. If x C Fx is arbi- Further details are given in the preprint mentioned trary, there exists an integer ord(x), completely deter- in footnote t. mined by x, with the property that x-ord(x) E ox. The Now suppose F is a global field. Let A(F) denote its genuine principal series representations of G are induced adele ring and v any place of F. For each v, the local from characters of A and these in turn are described by metaplectic group G, can be defined in terms of a,, as genuine characters of the double cover of Fx determined by Hilbert's symbol. t (1974) "Automorphic Forms on the Metaplectic Group," denote the set with x mimeographed notes, Cornell University Mathematics Depart- LEMMA 1. Let Px of pairs (x,¢) ment, Ithaca, N.Y. C FX, v = +1, and ?multiplication given by (x,¢) (x',V') 1406 Downloaded by guest on September 29, 2021 Proc. Nat. Acad. Sci. USA 72 (1975) Intertwining Operators and Automorphic Forms 1407 = (xx',(x,x')r'). Then each quasi-character g of PF non- p-adic gammafunction of ref. 9, and trivial on Z2 is of theform so(1) = 1, l p [(Xr) ] = r[sgn,(_-x) (TT) l/2][ -sgnf(x)1/2A(X) q - 1 _q-1/2(q - q8) for some quasi-character A of FZ. A q J(E) = Recall that sgn,(x) = (y,x) and IA(x) itself is of the - 1 + q'/2(q-8 - qs) form IxI.-8,*(x) for some s £ C and unitary character a* q of Ox (extended to Fx in the obvious way). We say that p is "unramified" iff , is; the correspondence ,i- = and (s,/. *) is one-to-one. Recall that g is nontrivial on Z2. = - Now set B equal to the inverse image in G of the upper ()(er) q28(1 q'++ q8-12 q'q-1+28/23\ triangular subgroup B of G. Since B = NA, we have q-1 q-l2-' B = NA, a semi-direct product. Thus, we can extend Let us write A (s,,u *) for A (,a). The operator A(s,,lu*) P(U[o °-j,]) = ,g(x,¢) to B and consider the induced then depends analytically on s if 0 < Re(s) < '/2 and representation defines a unitary operator when Re(s) = 0. Using Propo- sition 1 one can establish: pea) = Ind (B,G,p). THEOREM 1. The map s -- A(sju*), initially defined The natural inner product in the space of p(jz) is given for 0 < Re(s) < /2, extends to a meromorphic operator by an integration over K since 0 = NAK. From this it valued function in Re(s) > 0 whose only possible pole is at follows that p(g) is unitary iff 1 and hence p is. These S = 1/2; if /h* is nontrivial, i.e., if a is ramified, then are the genuine principal series representations of 0. A (s,. *) is actually entire for Re(s) > 0. [Recall A (s,,O*) intertwines representations of G nontrivial on Z2.] LEMMA 2. Let w denote the element [° I] of U and define This Theorem will be crucial for the global theory. 'aw(xXO ]r) = z((['OZ],¢)°w) = a([X'0,b). Then p(jiw) Analogous results for the real and complex fields are and p(Qu) are equivalent. already known (compare refs. 11 and 12) and will be By Bruhat's decomposition, G = B U BwN. Thus, recalled now for later use. each so: G -.o C in the space of p(p) is determined by its Suppose. first that F = R. Each genuine quasi-char- restriction +(x) = sp(w["f]). This means p(jz) may be acter g of FPisthenof theformg(x,¢) = Ixj8(ni)[1 -sgn(x) ]/2 realized in L2(F), with with s E C and v = 1. Furthermore, the correspon- dencep-uu(y) = 1- 1 onto the = 4(X + lxl8[sgn(x)][lI-gn(n)]/2is p(gffllb],)+) b) set of quasicharacters of Fx. So suppose p(s,,q) denotes and the obvious induced representation of 0F and Re(s) = O. Then p(s,77) and p(- s,7) are interwined by an operator p(jO)([ ])9(X) =|)(a2x). A(s,-q) which continues to a meromorphic function of s Using this noncompact realization of p(g) and the non- with poles at s = 1/2, 1 odd and positive. In particular, archimedean analysis of refs. 8 and 9, one can actually suppose f is in the space of p(s,7) and transforms under implement the equivalence of Lemma 2 explicitly as in S02(R) according to the character e"t '2, k odd. Then A (s,1)f has a pole at s = 1/2 if and only if k= 1(4) and Proposition 1 below. For simplicity, we assume g is unramified. k > 0; similarly, A (s,-1)f has a pole at s = 1/2 if and only if k = 3(4) and k < 0. In either case, if Api denotes PROPOSITION 1. Let x denote the canonical nontrivial the space of suchf, P('/2,1 1)/AA is equivalent to a additive character of F trivial on 0 but not on P'- (see so-called trash representation of SL2(R) (compare ref.