Character formul{\ae} and GKRS multiplets in equivariant K-theory

TL;DR

This paper explores the structure of equivariant K-theory modules for compact Lie groups, revealing bilinear pairings and multiplet structures analogous to representation theory, advancing understanding of equivariant topological invariants.

Contribution

It introduces new structural results about the $H$-equivariant K-ring as a module over the $G$-equivariant K-ring, including nonsingular pairings and representation-theoretic multiplets.

Findings

Existence of a nonsingular bilinear pairing on the module

Identification of multiplets analogous to GKRS multiplets

Structural insights into $K_H^*(X)$ as a $K_G^*(X)$-module

Abstract

Let $G$ be a compact Lie group, $H$ a closed subgroup of maximal rank and $X$ a topological $G$-space. We obtain a variety of results concerning the structure of the $H$-equivariant K-ring $K_H^*(X)$ viewed as a module over the $G$-equivariant K-ring $K_G^*(X)$. One result is that the module has a nonsingular bilinear pairing; another is that the module contains multiplets which are analogous to the Gross-Kostant-Ramond-Sternberg multiplets of representation theory.