# Note on Trace Class Groups

## TL;DR

This paper investigates conditions under which certain semidirect product Lie groups are trace class, establishing that non-compact semisimple algebraic subgroups lead to non-trace class groups, with initial exploration of Heisenberg-type groups.

## Contribution

It proves that semidirect products of R^n with non-compact semisimple algebraic groups are not trace class, and begins studying similar properties for Heisenberg-type groups.

## Key findings

- Semidirect products with non-compact G are not trace class.
- The converse for compact G is known from previous work.
- Initial analysis of semidirect products with Heisenberg groups.

## Abstract

A Lie group G is called a trace class group if for every irreducible unitary representation R of G and every C-infinity function f with compact support the operator R(f) is of trace class. In this note we prove that the semidirect product of R^n and a real semisimple algebraic subgroup G of GL(n;R) is a trace class group only if G is compact. The converse has been shown elsewhere. We also make a descent start with the study of semidirect products with Heisenberg-type groups.