Unitary representations of SL(n,ℝ) and the cohomology of congruence subgroups

Abstract

Let G=SL_±(n,ℝ) with G=g and K=O(n), and let Γ(q^m), m ε N\O, q an odd prime, be a congruence subgroup of SL(n,ℝ). We prove that for m large enough all unitary representations Π with H*(g,K,Π) ≠ 0 are automorphic representations of G/Γ(q^m). For a unitary representation Π, denote by n_Π the smallest integer with H^*(g,K,Π) ≠ 0 if such an integer exists. Representing cohomology classes by Eisenstein series we prove

$$H^{n_\Pi } (\Gamma (q^m ),\mathbb{C}) \ne 0$$

for m large.

This research was supported by NSF Grant MCS80-01854