On the Reflection Type Decomposition of the Adjoint Reduced Phase Space of a Compact Semisimple Lie group

TL;DR

This paper explores the decomposition of the reduced phase space of a compact Lie group by reflection types, relating it to orbit type decomposition, and discusses implications for quantum Hilbert space structures.

Contribution

It introduces a reflection type decomposition approach for the reduced phase space and connects it with quantum costratification, providing explicit results for classical groups.

Findings

Reflection type and orbit type decompositions coincide for classical groups like SU(n) and Sp(n).

Derived relations for reflection type subsets using irreducible characters.

Provided explicit examples for low-rank classical groups.

Abstract

We consider a system with symmetries whose configuration space is a compact Lie group, acted upon by inner automorphisms. The classical reduced phase space of this system decomposes into connected components of orbit type subsets. To investigate hypothetical quantum effects of this decomposition one has to construct the associated costratification of the Hilbert space of the quantum system in the sense of Huebschmann. In the present paper, instead of the decomposition by orbit types, we consider the related decomposition by reflection types (conjugacy classes of reflection subgroups). These two decompositions turn out to coincide e.g. for the classical groups SU(n) and Sp(n). We derive defining relations for reflection type subsets in terms of irreducible characters and discuss how to obtain from that the corresponding costratification of the Hilbert space of the system. To illustrate the method, we give explicit results for some low rank classical groups.