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On limit multiplicities of discrete series representations in spaces of automorphic forms. (English) Zbl 0582.22012

Let G be a connected semi-simple real Lie group, \(\Gamma\) an arithmetic subgroup of G and let \(L^ 2(\Gamma \setminus G)\) be the space of square integrable functions on \(\Gamma\) \(\setminus G\) viewed as unitary G-module acted upon by right translations. Given an irreducible unitary representation \(\pi\) of G one would like to determine the multiplicity m(\(\pi\),\(\Gamma)\) with which \(\pi\) occurs in the right regular representation space \(L^ 2(\Gamma \setminus G)\). This, in turn, roughly amounts to compute dimensions of spaces of automorphic forms with respect to \(\Gamma\). In view of various applications in the theory of arithmetic varieties and the connections with the theory of automorphic forms also non-vanishing results are of interest.
If G admits discrete series representations there is a conjecture saying that for a given discrete series representation \(\pi\) of G and a tower \((\Gamma_ i)\) of arithmetic subgroups of \(\Gamma\) one has the limit formula \[ \lim_{i\to \infty}\frac{m(\pi,\Gamma_ i)}{v(\Gamma_ i\setminus G)}=d(\pi) \] where d(\(\pi)\) denotes the formal degree of \(\pi\) with respect to a chosen Haar measure dg on G and \(v(\Gamma_ i\setminus G)\) denotes the volume of \(\Gamma_ i\setminus G\) with respect to the induced measure. If \(\Gamma\) \(\setminus G\) is compact the conjecture has been proved even in a slightly more general setting by D. L. DeGeorge and N. R. Wallach [Ann. Math., II. Ser. 107, 133-150 (1978; Zbl 0397.22007)], and there are some specific results in real rank one cases.
In the paper under review a weaker version of the conjecture above is proved in an adelic setting for congruence subgroups and a special class of towers. However, as an application one still gets non-vanishing results for \(m(\pi,\Gamma_ i)\) for sufficiently small congruence subgroups.
The proof uses the Selberg trace formula and Arthur’s results towards it. The basic idea is to plug in so-called pseudo-coefficients [cf. the author and P. Delorme, C. R. Acad. Sci., Paris, Sér. I 300, 385- 387 (1985)] isolating in this way a finite set of representations. This kind of arguments is already used in the case \(GL_ 2\) by R. P. Langlands [Base change for \(GL_ 2\) (Ann. Math. Stud. 96) (1980; Zbl 0444.22007), pp. 227-230].
Reviewer: J.Schwermer

MSC:

22E46 Semisimple Lie groups and their representations
22E40 Discrete subgroups of Lie groups
11F70 Representation-theoretic methods; automorphic representations over local and global fields
43A85 Harmonic analysis on homogeneous spaces
14K15 Arithmetic ground fields for abelian varieties

References:

[1] Arthur, J.: A Trace Formula for Reductive groups I. Duke Math. J.45, 911-954 (1978) · Zbl 0499.10032 · doi:10.1215/S0012-7094-78-04542-8
[2] Arthur, J.: A Trace Formula for Reductive groups II. Compos. Math.40, 87-121 (1980) · Zbl 0499.10033
[3] Arthur, J.: Eisenstein series and the Trace Formula. Proc. Symp. Pure Math.33, 1, 253-276 (1979) · Zbl 0431.22016
[4] Arthur, J.: The Trace formula in invariant form. Ann. Math.114, 1-74 (1981) · Zbl 0495.22006 · doi:10.2307/1971376
[5] Arthur, J.: A measure on the unipotent variety (Preprint) · Zbl 0589.22016
[6] Arthur, J.: The local behavior of weighted orbital integrals (In preparation) · Zbl 0649.10020
[7] Arthur, J.: A theorem on the Schwartz space of a reductive Lie group. Proc. Natl. Acad. Sci. USA72, 4718-19 (1975) · Zbl 0311.43010 · doi:10.1073/pnas.72.12.4718
[8] Barbasch, D., Moscovici, H.:L 2-index and the Selberg Trace Formula. J. Funct. Anal.53, 151-201 (1983) · Zbl 0537.58039 · doi:10.1016/0022-1236(83)90050-2
[9] Bernstein, J., Deligne, B., Kazhadan, D.: Trace Palew-Wiener Theorem for Reductivep-adic groups. (Preprint)
[10] Bernstein, I.N., Zelevinski, A.V.: Induced representations of reductivep-adic groups I. Ann. Sci ENS10, 441-472 (1977) · Zbl 0412.22015
[11] Borel, A.: Regularization theorems in Lie algebra cohomology. Applications. Duke Math. J.50, 605-623 (1983) · Zbl 0528.22010 · doi:10.1215/S0012-7094-83-05028-7
[12] Borel, A., Casselman, W.:L 2-cohomology of locally symmetric manifolds of finite volume. Duke Math. J.50, 625-647 (1983) · Zbl 0528.22012 · doi:10.1215/S0012-7094-83-05029-9
[13] Borel, A., Jacquet, H.: Automorphic forms and automorphic representations. Proc. Symp. Pure Math.33, 1, 189-202 (1979) · Zbl 0414.22020
[14] Borel, A., Wallach, N.: Continuous cohomology, discrete subgroups, and representations of reductive groups. Princeton: Princeton University Press 1980 · Zbl 0443.22010
[15] Casselman, W.: Introduction to the theory of admissible representations ofp-adic groups. (Mimeographed notes)
[16] Clozel, L., Delorme, P.: Sur le théorème de Paley-Wiener invariant pour les groupes réductifs réels. C.R.A.S. Paris (To appear) · Zbl 0584.22005
[17] Clozel, L., Delorme, P.: Pseudo-coefficients et cohomologie des groupes réductifs reels. C.R.A.S. Paris (To appear) · Zbl 0593.22010
[18] Clozel, L., Labesse, J.-P., Langlands, R.P.: Morning seminar on the Trace Formula (mimeographed notes). I.A.S., Princeton 1983-84
[19] DeGeorge, D.: On a Theorem of Osborne and Warner. J. Funct. Anal.48, 81-94 (1982) · Zbl 0503.22008 · doi:10.1016/0022-1236(82)90062-3
[20] DeGeorge, D., Wallach, N.: Limit formulas for multiplicities inL 2(?/G). Ann. Math.107, 133-150 (1978) · Zbl 0397.22007 · doi:10.2307/1971140
[21] Gèrardin, P.: Construction de Séries discrètesp-adiques. Lect. Notes462. Berlin Heidelberg New York: Springer 1975
[22] Harish-Chandra: Harmonic Analysis in Reductivep-adic groups. Proc. Symp. Pure Math.26, 167-192 (1974)
[23] Harish-Chandra: The Plancherel formula for reductivep-adic groups. In: Collected Papers, vol. 4. Berlin-Heidelberg-New York-Tokyo: Springer 1984
[24] Harish-Chandra: Harmonic Analysis on Reductivep-adic Groups. Lect. Notes in Mathematics 162. Berlin Heidelberg New York: Springer 1984
[25] Henniart, G.: La Conjecture de Langlands locale pourGL (3). Mém. Soc. Math. Fr. (To appear) · Zbl 0497.12008
[26] Kazhdan, D.: Arithmetic varieties and their fields of quasi-definition. Actes Cong. Intern. Math.2, 321-325 (1970)
[27] Kazhdan, D.: On Arithmetic varieties II. Isr. J. Math.44, 139-159 (1983) · Zbl 0543.14030 · doi:10.1007/BF02760617
[28] Labesse, J.P.: La formule des traces d’Arthur-Selberg. Seminaire Bourbaki, no. 636 (1984-85) · Zbl 0592.22011
[29] Mischenko, P.: Invariant tempered distributions on the reductive groupGL(n, F p). Thesis, Toronto 1982 · Zbl 0491.22008
[30] Osborne, M.S., Warner, G.: The theory of Eisenstein systems. New York: Academic Press 1981 · Zbl 0489.43009
[31] Rogawski, J.: Representations ofGL(n) and division algebras over ap-adic field. Duke Math. J.50, 161, 196 (1983) · Zbl 0523.22015 · doi:10.1215/S0012-7094-83-05006-8
[32] Silberger, A.J.: The Langlands Quotient Theorem forp-adic Groups. Math. Ann.236, 95-104 (1978) · doi:10.1007/BF01351383
[33] Varadarajan, V.S.: Harmonic Analysis on Real Reductive Groups. Lecture Notes in Mathematics 576. Berlin Heidelberg New York: Springer 1977 · Zbl 0354.43001
[34] Wallach, N.R.: On the Constant term of a square-integrable automorphic form. Proceedings of the Neptun Conference on Operator algebras and Group representations 1980 · Zbl 0554.22004
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