---

APPEARED IN BULLETIN OF THE

arXiv:math/9204229v1 [math.RT] 1 Apr 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 26, Number 2, April 1992, Pages 253-257RAMANUJAN DUALS AND AUTOMORPHIC SPECTRUMM. Burger, J. S. Li, and P. SarnakAbstract.

We introduce the notion of the automorphic dual of a matrix algebraicgroup defined over Q.This is the part of the unitary dual that corresponds toarithmetic spectrum.Basic functorial properties of this set are derived and usedboth to deduce arithmetic vanishing theorems of “Ramanujan” type as well as togive a new construction of automorphic forms.Let G be a semisimple linear algebraic group defined over Q. In the arithmetictheory of automorphic forms the lattice Γ = G(Z) and its congrunce subgroupsΓ(N) = {γ ∈G(Z): γ ≡I(N)},N ∈Nplay a central role.

A basic problem is to understand the decomposition into irre-ducibles of the regular representation of G(R) on L2(Γ(N)\G(R)). In general thisrepresentation will not be a direct sum of irreducibles, and for our purposes ofdefining the spectrum, it is best to use the notions of weak containment and Felltopology on the unitary dual bG(R) of the Lie group G(R).

(See [D, 18.1].) Forany closed subgroup H of G(R) we define the spectrum σ(H\G(R)) to be the sub-set of bG(R) consisting of all π ∈bG(R) that are weakly contained in L2(H\G(R)).Furthermore, if bG1(R) is the set of irreducible spherical representations, we setσ1(H\G(R)) := σ(H\G(R)) ∩bG1(R).

When H = Γ(N), σ(Γ(N)\G(R)) consists ofall π ∈bG(R) occurring as subrepresentations of L2(Γ(N)\G(R)) as well as thoseπ’s that are in wave packets of unitary Eisenstein series [La]. The latter occur onlywhen Γ\G(R) is not compact.

We now introduce the central object of this note.Definition. The automorphic (resp.

Ramanujan) dual of G is defined by(1)bGAut =∞[N=1σ(Γ(N)\G(R)),bGRaman = bGAut ∩bG1(R).Here closure is taken in the topological space ˆG(R).Thus, bGAut is the smallest closed set containing all the congruence spectrum.Here is an alternative description of bGRaman. Let G(R) = KAN be an Iwasawadecomposition of G(R); then the theory of spherical functions identifies bG1(R) with1991 Mathematics Subject Classification.

Primary 20G30.Received by the editors April 16, 1991 and, in revised form, July 1991c⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1

2M. BURGER, J. S. LI, AND P. SARNAKa subset of A∗C/W, where A = Lie A, W = Weyl(G, A).

Moreover, the Fell topologyon bG1(R) coincides with the topology of bG1(R) viewed as a subset of A∗C/W. LetD be the ring of invariant differential operators on the associated symmetric spaceX.

Then the duality theorem [GPS] shows that the spectrum of D in L2(Γ\X), saySpΓ(D) ⊂A∗C/W is the image of σ1(Γ\G(R)) in A∗C/W under the above identifica-tion. In particular, bGRaman is identified with∞[N=1SpΓ(N)(D) ⊂A∗C/W.That there should be restrictions on bGRaman and bGAut has its roots in the repre-sentation theoretic reinterpretation of the classical Ramanujan conjectures due toSatake [Sa].

Identifying the above sets may be viewed as the general Ramanujanconjectures. For example, Selberg’s 1/4-conjecture may be stated as follows: ForG = SL2,(2)bGRaman = {1} ∪bG1(R)temp,where, in general, bG(R)temp := σ(G(R)) is the set of tempered representations, andbG1(R)temp = bG(R)temp ∩bG1(R).

(See [CHH] for equivalent definitions of tempered-ness. )While the individual sets σ(Γ(N)\G(R)) are intractable, the set bGAut (andbGRaman) enjoy certain functorial properties.Theorem 1.

Let G be a connected semisimple linear algebraic group defined overQ and H < G a Q-subgroup(i) IndG(R)H(R) bHAut ⊂bGAut. (ii) Assume that H is semisimple; thenResH(R) bGAut ⊂bHAut.

(iii) bGAut ⊗bGAut ⊂bGAut.A word about the meaning of these inclusions.Firstly, Ind denotes unitaryinduction and Res stands for restriction. By the inclusion, say in (i), we mean that ifπ′ ∈bHAut and π is weakly contained in IndG(R)H(R) π′ then π ∈bGAut.

(i) produces (aftera local calculation) elements in bGAut from ones in bHAut and yields a new methodfor constructing automorphic representations. Observe also that if π ∈bG(R) isan isolated point then π ∈bGAut implies that π occurs as a subrepresentationin L2(Γ(N)\G(R)) for some N. This fact will be used below to construct certainautomorphic cohomological representations.

(ii) allow one to transfer setwise upperbounds on bHAut to bGAut and for many G’s gives nontrivial approximations to theRamanujan conjectures. (iii) exhibits a certain internal structure of the set bGAut.We illustrate the use of Theorem 1 with some examples.Example A.

If H = {e} then (i) implies that(3)bGAut ⊃[G(R)temp ∪{1}.

RAMANUJAN DUALS3When G(Z) is cocompact this follows also from de George-Wallach [GW]. In com-parison with (2) one might hope that (3) is an equality.

However, using other H’sand (i) one finds typically that bGAut contains nontrivial, nontempered spectrum.For G = Sp(4) the failure of the naive Ramanujan conjecture has been observed byHowe and Piatetski-Shapiro [HP-S] using theta liftings.Example B. Let k/Q be a totally real field, q a quadratic form over k suchthat q has signature (n, 1) over R, and all other conjugates are definite.

Let G =Resk/Q SO(q). Then G(R) is of R-rank one and the noncompact factor is SO(n, 1).We identify A∗with R by sending ρ to (n −1)/2.

With this normalization bG(R)1is identified with iR ∪[−ρ, ρ] ⊂C modulo{±1}. [K].

We parametrize bG1(R) bys ∈iR+ ∪[0, ρ] and denote the corresponding representation by πs. Let ϕ0, .

. .

, ϕnbe an orthogonal basis of q such that q(ϕi) > 0, 0 ≤i ≤n −1, and q(ϕn) < 0.Define H = Resk/Q{g ∈SO(q): g(ϕ1) = ϕ1}. Applying Theorem 1(i) to the trivialrepresentation 1 ∈bHAut we find thatσ(H(R)\G(R)) ⊂bGAut.Now σ1(H(R)\G(R)) has been computed ([F]), and we find(4)bGRaman ⊃{ρ, ρ −1, ρ −2, .

. . } ∪iR+.In particular, for n ≥4 there are nontrivial nontempered spherical automorphicrepresentations.To find upper bounds on bGRaman one uses Theorem 1(ii) andH = Resk/Q{g ∈SO(q): g(ϕi) = ϕi, 1 ≤i ≤n −4}.Combining the Jacquet-Langlands correspondence [JL] with the Gelbart-Jacquetlift [GJ] one concludes thatbHRaman ⊂iR+ ∪[0, 12] ∪{1}.Applying (ii) it follows that(5)bGRaman ⊂iR+ ∪[0, ρ −12] ∪{ρ}.In the special case k = Q, n ≥4 this result has also been obtained by [EGM] and[LP-SS] using Poincar´e series.

Assuming the Ramanujan conjecture at ∞for GL(2)one deduces(6)bGRaman ⊂iR+ ∪[0, ρ −1] ∪{ρ}. (Compare with (4).) The natural conjecture arising from (4) and (6) isbGRaman = iR+ ∪{ρ, ρ −1, .

. .

}.This is apparently consistent with Arthur’s conjectures [A].

4M. BURGER, J. S. LI, AND P. SARNAKExample C. Let F4(−20) be the R-rank one form of F4.

Using a method of Borel[B], one may find Q-groups H < G such that G(R), H(R) both have rank one,the noncompact simple factors being F4(−20) and Spin(8, 1) respectively.Withnotations similar to Example B, one may identify bG1(R) with iR+ ∪[0, 5] ∪{11},here ρ = 11. One may compute σ1(H(R)\G(R)) and using Theorem 1(i) find thatbGRaman ⊃iR+ ∪{3, 11}.Example D. Consider now F4(4), the split real form of F4.

The correspondingsymmetric space has dimension 28. For any cocompact lattice Γ ⊂F4(4) one knowsfrom Vogan-Zuckerman [VZ] that the Betti numbers βm(Γ) = 0 for 0 < m < 8 or20 < m < 28, m ̸= 4 or 24, in these latter dimensions all the cohomology comesfrom parallel forms of the symmetric space.

Nevertheless using Theorem 1(i) wehaveTheorem 2. For any cocompact lattice Γ in F4(4) and N ≥0 there exists Γ′ ⊆Γof finite index such that βm(Γ′)H ≥N for m = 8, 20.The proof of Theorem 2 makes use of Matsushima’s formula [BW] together witha recent result of Vogan ensuring that the unitary representation contributing tothe above Betti numbers is isolated in the unitary dual of F4(4).

The Q-subgroupthat we use in applying Theorem 1(i) has real points equal to Spin(5, 4) up tocompact factors. By the well-known result of Oshima-Matsuki [MO], we concludethat the discrete series of the symmetric space F4(4)/ Spin(5, 4) contain a unitaryrepresentation with nonzero cohomology in degrees 8 and 20, which is isolated inthe unitary dual.

The fact that we have dealt with every lattice in F4(4) follows fromMargulis’s arithmeticity theorem [M], together with the classification of algebraicgroups over number fields [T].This method of constructing cohomology is rather general. If π is isolated inbG(R) and is contained in the automorphic dual of G then it occurs discretely inL2(Γ\G(R)) for Γ a congruence subgroup of deep enough level.

David Vogan hasrecently obtained the necessary and sufficient conditions for a unitary representa-tion with nonzero cohomology to be isolated, which implies that most of them do.Theorem 1 then allows us to obtain nonvanishing of cohomology in a large numberof cases.To end, we remark that these ideas extend in a natural way to S-arithmeticgroups. The proof of Theorem 1(i) consists of approximating, in a suitable way,congruence subgroups of H(Z) by congruence subgroups of G(Z) and then applyingcriteria of weak containment.

For the proofs of Theorem 1(ii), (iii) we refer thereader to [BS].AcknowledgmentsWe would like to thank David Vogan for sharing his insights into unitary repre-sentations and F. Bien for interesting conversations.References[A]J. Arthur, On some problems suggested by the trace formula, Lecture Notes in Math.,vol. 1041, Springer-Verlag, New York, 1983, pp.

1–50.

RAMANUJAN DUALS5[B]B A. Borel, Compact Clifford-Klein forms of symmetric spaces, Topology 2 (1963),111–122.[BB]M. W. Baldoni-Silva and D. Barbasch, The unitary dual of real rank one groups, InventMath.

72 (1983), 27–55.[BS]M. Burger and P. Sarnak, Ramanujan Duals II, Invent.

Math. 106 (1991), 1–11.[BW]A.

Borel and N. Wallach, Continuous cohomology, discrete subgroups and representa-tions of reductive groups, Ann. of Math.

Stud. 94 (1980).[CHH]M.

Cowling, U. Haagerup and R. Howe, Almost L2-matrix coefficients, J. Reine Angew.Math. 387 (1988), 97–110.[D]J.

Dixmier, C∗-algebras, North-Holland, Amsterdam, 1977.[EGM]J. Elstrodt, F. Grunewald and T. Mennicke, Poincar´e series, Kloosterman sums, andeigenvalues of the Laplacian for congruence groups acting on hyperbolic spaces, Invent.Math.

101 (1990), 641–685.[F]J. Faraut, Distributions sph´eriques sur les espaces hyperboliques, J.

Math. Pures Appl.58 (1979), 369–444.[GJ]S.

S. Gelbart and H. Jacquet, A relation between automorphic representations on GL(2)and GL(3), Ann. Sci.

´Ecole Norm. Sup.

(4e) II (1978), 471–542. [GGP-S] I. Gelfand, M. Graev, and I. Piatetsky-Shapiro, Representation theory and automorphicfunctions, Saunders, San Francisco, 1969.

[GW]de George and N. Wallach, Limit formulas for multiplicities in L2(Γ\G), Ann. of Math.

(2) 107 (1978), 133–150.[HP-S]R. Howe and I. Piatetski-Shapiro, Some examples of automorphic forms on Sp(4), DukeMath.

J. 50 (1983), 55–106.[JL]H.

Jacquet and R. P. Langlands, Automorphic forms on GL(2), Lecture Notes in Math.,vol. 114, Springer-Verlag, New York, 1970.[K]B.

Kostant, On the existence and irreducibility of a certain series of representations,Bull. Amer.

Math. Soc.

75 (1969), 627–642.[LP-SS]J. S. Li, I. Piatetsky-Shapiro and P. Sarnak, Poincar´e series for SO(n, 1), Proc.

IndianAcad. Sci.

97 (1987), 231–237.[L]R. P. Langlands, On the functional equation satisfied by Eisenstein series, Lecture Notesin Math., vol.

544, Springer-Verlag, New York, 1976.[Li1]J. S. Li, Theta lifting for unitary representations with nonzero cohomology, Duke Math.J.

61 (1991), 913–937. [Li2], Nonvanishing theorems for the cohomology of certain arithmetic quotients,preprint, 1990.[M]G.

Margulis, Discrete subgroups of semi-simple Lie groups, Ergeb. Math.

Grenzeb. (3),vol.

17 Springer-Verlag, New York, 1989.[MO]T. Matsuki and T. Oshima, A description of the discrete series for semi-simple sym-metric spaces, Adv.

Stud. Pure Math.

4 (1984), 331–390.[MR]J. Millson and M. Raghunathan, Geometric construction of cohomology for arithmeticgroups I, Proc.

Indian Acad. Sci.

Math. Sci.

90 (1981), 103–123.[T]J. Tits, Classification of algebraic semisimple groups, Proc.

Sympos. Pure Math., vol.

IX,Amer. Math.

Soc., Providence, RI, 1966, pp. 33–62.Graduate Center, City University of New York, New York 10036Department of Mathematics, University of Maryland, College Park, Maryland20742Department of Mathematics, Princeton University, Princeton, New Jersey 08544and IBM Research Division, Almaden research center, 650 Harry Rd, San Jos´e,California 95120

---

출처: arXiv:9204.229 • 원문 보기