# Schwartz homologies of representations of almost linear Nash groups

## TL;DR

This paper develops a Schwartz homology theory for smooth moderate growth representations of almost linear Nash groups, establishing key properties and applications in the context of representation theory.

## Contribution

It introduces a new Schwartz homology framework for Nash group representations and proves foundational results like Frobenius reciprocity and Shapiro's lemma within this setting.

## Key findings

- Established Schwartz homology for smooth moderate growth representations.
- Proved Frobenius reciprocity and Shapiro's lemma in this context.
- Provided criteria for automatic extensions of Schwartz homologies.

## Abstract

Let $G$ be an almost linear Nash group, namely, a Nash group which admits a Nash homomorphism with finite kernel to some $\GL_k(\mathbb R)$. A homology theory (the Schwartz homology) is established for the category of smooth \Fre representations of $G$ of moderate growth. Frobenius reciprocity and Shapiro's lemma are proved in this category. As an application, we give a criterion for automatic extensions of Schwartz homologies of Schwartz sections of a tempered $G$-vector bundle.