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

TL;DR

This paper proves Casselman's comparison theorem for minimal parabolic subalgebras in real reductive groups, establishing isomorphisms between homology groups of $K$-finite vectors and the entire representation, with implications for automatic continuity.

Contribution

It extends Casselman's comparison theorem to minimal parabolic subalgebras, showing isomorphisms in homology and closedness of certain subspaces, strengthening previous automatic continuity results.

Findings

Isomorphisms between homology groups of $K$-finite vectors and the whole representation.

Closedness of $ abla^k V$ subspaces in the representation.

Strengthening of Casselman's automatic continuity 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 Fr\'echet 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. 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.