A proof of Casselman’s comparison theorem

Abstract:

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 $\mathfrak {p}$, and $\mathfrak {n}$ be its nilradical. In this paper we show that: for any admissible finitely generated moderate growth smooth Fréchet representation $V$ of $G$, the inclusion $V_{K}\subset V$ induces isomorphisms $H_{i}(\mathfrak {n},V_{K})\cong H_{i}(\mathfrak {n},V)$ ($i\geq 0$), where $V_{K}$ denotes the $(\mathfrak {g},K)$ module of $K$ finite vectors in $V$. 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 $k\geq 1$, $\mathfrak {n}^{k}V$ is a closed subspace of $V$ and the inclusion $V_{K}\subset V$ induces an isomorphism $V_{K}/\mathfrak {n}^{k}V_{K}= V/\mathfrak {n}^{k}V$. 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]).