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

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 $H_i(\mathfrak{n},V_K)\cong H_i(\mathfrak{n},V)$ ($i\ge 0$), where $V_K$ denotes the $(\mathfrak{g},K)$ module of $K$ finite vectors in $V$. This is called Casselman's comparison theorem. As a consequence, we show that: for any $k\ge 1$, ${\mathfrak{n}}^{k}V$ 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.

Fill in details of proofs of several lemmas in Section 3