A proof of Casselman’s comparison theorem

Let G G be a real linear reductive group and K K be a maximal compact subgroup. Let P P be a minimal parabolic subgroup of G G with complexified Lie algebra p \mathfrak {p} , and n \mathfrak {n} be its nilradical. In this paper we show that: for any admissible finitely generated moderate growth smooth Fréchet representation V V of G G , the inclusion V K ⊂ V V_{K}\subset V induces isomorphisms H i ( n , V K ) ≅ H i ( n , V ) H_{i}(\mathfrak {n},V_{K})\cong H_{i}(\mathfrak {n},V) ( i ≥ 0 i\geq 0 ), where V K V_{K} denotes the ( g , K ) (\mathfrak {g},K) module of K K finite vectors in V V . This is called Casselman’s comparison theorem (see Henryk Hecht and Joseph L. Taylor [A remark on Casselman’s comparison theorem, Birkhäuser Boston, Boston, Ma, 1998, pp. 139–146]). As a consequence, we show that: for any k ≥ 1 k\geq 1 , n k V \mathfrak {n}^{k}V is a closed subspace of V V and the inclusion V K ⊂ V V_{K}\subset V induces an isomorphism V K / n k V K = V / n k V V_{K}/\mathfrak {n}^{k}V_{K}= V/\mathfrak {n}^{k}V . This strengthens Casselman’s automatic continuity theorem (see W. Casselman [Canad. J. Math. 41 (1989), pp. 385–438] and Nolan R. Wallach [Real reductive groups, Academic Press, Boston, MA, 1992]).