Ramanujan Duals and Automorphic Spectrum

We introduce the notion of the automorphic dual of a matrix algebraic group defined over Q . This is the part of the unitary dual that corresponds to arithmetic spectrum. Basic funcional properties of this set are derived and used both to deduce arithmetic vanishing theorems of "Ramanujan" type as well as to give a new construction of automorphic forms. Let G be a semisimple linear algebraic group defined over Q. In the arithmetic theory of automorphic forms the lattice Y — G(Z) and its congrunce subgroups T(7V) = {y e G(Z): y = I(N)}, N e N play a central role. A basic problem is to understand the decomposition into irreducibles of the regular representation of G(R) on L2(Y(N)\G(R)). In general this representation will not be a direct sum of irreducibles, and for our purposes of defining the spectrum, it is best to use the notions of weak containment and Fell topology on the unitary dual G(R) of the Lie group G(R). (See [D, 18.1].) For any closed subgroup H of G(R) we define the spectrum cr(H\G(R)) to be the subset of G(R) consisting of all n e G(R) that are weakly contained in L2(H\G(R)). Furthermore, if G{{R) is the set of irreducible spherical representations, we set ol(H\G(R)) := a(H\G(R)) n Gl(R). When H = Y(N), cr(r(N)\G(R)) consists of all n e G(R) occurring as subrepresentations of L2(Y(N)\G(R)) as well as those n 's that are in wave packets of unitary Eisenstein series [La]. The latter occur only when r\C7(R) is not compact. We now introduce the central object of this note. Definition. The automorphic (resp. Ramanujan) dual of G is defined by (i) cAut= (J^rwwO), N=l ^Raman = LrAut H G (R). Here closure is taken in the topological space G(R). Thus, ¿rAut is the smallest closed set containing all the congruence spectrum. Here is an alternative description of C7Raman. Let C7(R) = KAN be an Iwasawa decomposition of G(R) ; then the theory of spherical functions identifies C71 (R) with a subset of VLç/W, where 21 = Lie^4, W = Weyl(C7, A). Moreover, the Fell topology on Gl(R) coincides with the topology of Gl(R) viewed as a subset of 2t£/ W. Let D be the ring of invariant differential operators on the associated symmetric space X. Then the duality theorem [GPS] shows that the spectrum Received by the editors April 16, 1991 and, in revised form, July 1991. 1991 Mathematics Subject Classification. Primary 20G30. © 1992 American Mathematical Society 0273-0979/92 $1.00 + $.25 per page 253 254 M. BURGER, J. S. LI, AND P. SARNAK of D in L2(Y\X), say Spr(D) c OJ./W is the image of al(T\G(K)) in 21* ¡W under the above identification. In particular, (jRaman is identified with (JSpr(JV)(D)c2l*:/^. N=\ That there should be restrictions on (/Raman and GAut nas its roots in the representation theoretic reinterpretation of the classical Ramanujan conjectures due to Satake [Sa]. Identifying the above sets may be viewed as the general Ramanujan conjectures. For example, Selberg's 1/4-conjecture may be stated as follows: For G = SL2, (2) GRaman = {1} U G (R)temp , where, in general, G(M)temp := c(G(R)) is the set of tempered representations, and Gl(R)temP = G(R)temp n G'(R). (See [CHH] for equivalent definitions of temperedness.) While the individual sets a(Y(N)\G(R)) are intractable, the set GAut (and ORaman) enjoy certain functorial properties. Theorem 1. Let G be a connected semisimple linear algebraic group defined over Q and H < G a Q-subgroup (Í) Indtf(R) #Aut c GAut. (ii) Assume that H is semisimple; then Restf(R) GAut c HAui. (iii) GAut <8> GAu, C GAut. A word about the meaning of these inclusions. Firstly, Ind denotes unitary induction and Res stands for restriction. By the inclusion, say in (i), we mean that if n' € //Aut and n is weakly contained in Indulgí n' then n € GAut. (i) produces (after a local calculation) elements in GAut from ones in 77Aut and yields a new method for constructing automorphic representations. Observe also that if n e G(R) is an isolated point then n e GAut implies that n occurs as a subrepresentation in L2(Y(N)\G(R)) for some N. This fact will be used below to construct certain automorphic cohomological representations, (ii) allow one to transfer setwise upper bounds on HAut to C7Aut and for many G's gives nontrivial approximations to the Ramanujan conjectures, (iii) exhibits a certain internal structure of the set GAut. We illustrate the use of Theorem 1 with some examples. Example A. If H = {e} then (i) implies that (3) GAut D G(R)temp U {1}. When G(Z) is cocompact this follows also from de George-Wallach [GW]. In comparison with (2) one might hope that (3) is an equality. However, using other H's and (i) one finds typically that GAut contains nontrivial, nontempered spectrum. For G = Sp(4) the failure of the naive Ramanujan conjecture has been observed by Howe and Piatetski-Shapiro [HP-S] using theta liftings.