Abstract
By assuming the endoscopic classification of automorphic representations on inner forms of unitary groups, which is currently work in progress by Kaletha, Minguez, Shin, and White, we bound the growth of cohomology in congruence towers of locally symmetric spaces associated to$U(n,1)$. In the case of lattices arising from Hermitian forms, we expect that the growth exponents we obtain are sharp in all degrees.
Highlights
This paper studies the limit multiplicity problem for cohomological automorphic forms on arithmetic quotients of U (N − 1, 1)
We assume that G has signature (N − 1, 1) at one real place and compact factors at all other real places
We show in Proposition 3.1 that the representations π of U (N − 1, 1) contributing to hd(2)(Y (n)) all have p(π ) 2(N − 1)/d
Summary
This paper studies the limit multiplicity problem for cohomological automorphic forms on arithmetic quotients of U (N − 1, 1). The main part of the proof involves using the structure of the packets Πψ (G) to bound the right hand side of (1) in terms of global multiplicities on smaller quasisplit unitary groups, which we bound using a theorem of Savin. Because the parameters φi are simple generic, the packet Πφi (U (mi )) is stable, so all representations in it occur discretely on U (mi ) This implies that dimU(mi )(Ki (n), φi ). We would need a sharper uniform bound than dim π K(n) qd(d−1)n/2 when π runs over nongeneric representations of GLd
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