• Corpus ID: 231839552

A proof of Casselman's comparison theorem for standard minimal parabolic subalgebra

@inproceedings{Li2021APO,
  title={A proof of Casselman's comparison theorem for standard minimal parabolic subalgebra},
  author={Ning Li and Gang Liu and Jun Yu},
  year={2021}
}

• N. Li, Gang Liu, Jun Yu
• Published 5 February 2021
• Mathematics

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 p, and n 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(n, VK) ∼= Hi(n, V ) (i ≥ 0), where VK denotes the (g,K) module of K finite vectors in V . This is called Casselman’s comparison theorem ([12]). As a consequence, we show that: for any k ≥ 1, nV is a closed subspace of V and the inclusion VK ⊂ V induces an isomorphism VK/n kVK = V/n V . This strengthens Casselman’s automatic continuity theorem ([7],[21]).