Limit Multiplicities for Principal Congruence Subgroups of And

Abstract

Abstract. We study the limiting behavior of the discrete spectra associated to the principal congruence subgroups of a reductive group over a number field. While this problem is well understood in the cocompact case (i.e., when the group is anisotropic modulo the center), we treat groups of unbounded rank. For the groups GL(n) and SL(n) we show that the suitably normalized spectra converge to the Plancherel measure (the limit multiplicity property). For general reductive groups we obtain a substantial reduction of the problem. Our main tool is the recent refinement of the spectral side of Arthur's trace formula obtained in [FLM11,FL11], which allows us to show that for GL(n) and SL(n) the contribution of the continuous spectrum is negligible in the limit.

Analytic torsion and $L^2$-torsion of compact locally symmetric manifolds

“…Denote by z 1 , ..., z m ∈ C \ S 1 be the poles of A v (z). Then (z − z 1 ) · · · (z − z m )A v (z) is a polynomial of degree n with coefficients in End(H Kv πv ) and by [FLM15,Corollary 5.18] we get…”

“…For GL(n) the pertinent estimations exist for the logarithmic derivatives itself [MS04, Proposition 0.2]. In general, the key ingredient for the estimation of the integrals is a generalization of the classical Bernstein inequality due to Borwein and Erdelélyi [FLM15,Corollary 5.18]. The application of this result involves the estimation of the order at ∞ of matrix coefficients of local normalized intertwining operators R P ,P (π v , s).…”

Fields of rationality of cusp forms

“…The primary difference is that we follow [Shi12] and [ST12] by fixing a discrete series representation at ∞ and examine the limit multiplicities only at finite places, whereas they look at limit multiplicities at infinite places. In [FLM14] they solved the Limit Multiplicity Problem for a large class of groups (specifically those satisfying properties (BD) and (TWN) as given in Section 5 of that paper. These groups include GL n and SL n ).…”

Refined Limit Multiplicity for Varying Conductor

“…In recent work, Abert, Bergeron, et al, Finis, Lapid, and Mueller, and Shin and Templier have solved the limit multiplicity problem for a large class of sequences of compact open subgroups in reductive groups G (see, for instance, [ABB+12], [FL15,FLM15], [Shi12,ST12]). In this paper, it is our goal to eliminate a pesky 'multiplicity' term from the statement of the Limit Multiplicity problem (at least for forms of GL n ) and to isolate representations of a given conductor, simply counting each with multiplicity 1.…”