LIMIT MULTIPLICITIES FOR PRINCIPAL CONGRUENCE SUBGROUPS OF AND

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$\text{GL}(n)$and$\text{SL}(n)$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 [Finis, Lapid, and Müller,Ann. of Math. (2)174(1) (2011), 173–195; Finis and Lapid,Ann. of Math. (2)174(1) (2011), 197–223], which allows us to show that for$\text{GL}(n)$and$\text{SL}(n)$the contribution of the continuous spectrum is negligible in the limit.