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(qm), 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(qm). For a unitary representation Π, denote by nΠ 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.KeywordsUnitary RepresentationCohomology ClassParabolic SubgroupEisenstein SeriesCongruence SubgroupThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.