# Spectral theory for non-unitary twists

## TL;DR

This paper develops a spectral theory framework for non-unitary twists of representations of Lie groups, establishing a trace formula applicable to a broad class of locally compact groups.

## Contribution

It introduces a complete filtration for the $G$-representation on $L^2( ext{quotient space}, ext{twist})$ with irreducible quotients, extending spectral analysis to non-unitary cases.

## Key findings

- Established a complete filtration with irreducible quotients.
- Proved a trace formula for non-unitary twists.
- Extended spectral theory to arbitrary locally compact groups.

## Abstract

Let $G$ be a Lie-group and $\Ga\subset G$ a cocompact lattice. For a finite-dimensional, not necessarily unitary representation $\om$ of $\Ga$ we show that the $G$-representation on $L^2(\Ga\bs G,\om)$ admits a complete filtration with irreducible quotients. As a consequence, we show the trace formula for non-unitary twists and arbitrary locally compact groups.