On limit multiplicities of discrete series representations in spaces of automorphic forms
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 \textit{D. L. DeGeorge} and \textit{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 \textit{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 \textit{R. P. Langlands} [Base change for \(GL_ 2\) (Ann. Math. Stud. 96) (1980; Zbl 0444.22007), pp. 227-230].
dimensions of spaces of automorphic forms
scientific article; zbMATH DE number 3931307
scientific article