2021/02/05 by Ning Li, Gang Liu, Li, Ning +3
Mathematics · #22E46 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:22E46
paper · pdf · doi:10.48550/arxiv.2102.03204
Fill in details of proofs of several lemmas in Section 3
openalex publication_date 2021/02/05 · openalex created_date 2021/02/15 · arxiv created 2021/08/25 · arxiv updated 2021/08/26 · openalex updated_date 2026/07/28
Let G be a real linear reductive group and K be a maximal compact subgroup. Let P be a minimal parabolic subgroup of G with complexified Lie algebra \mathfrakp, and \mathfrakn be its nilradical. In this paper we show that: for any admissible finitely generated moderate growth smooth Fréchet representation V of G, the inclusion VK⊂ V induces isomorphisms Hi(\mathfrakn,VK)≅ Hi(\mathfrakn,V) (i≥ 0), where VK denotes the (\mathfrakg,K) module of K finite vectors in V. This is called Casselman's comparison theorem. As a consequence, we show that: for any k≥ 1, \mathfraknkV is a closed subspace of V and the inclusion VK⊂ V induces an isomorphism VK/\mathfraknkVK= V/\mathfraknkV. This strengthens Casselman's automatic continuity theorem.