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

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.

💡 Research Summary

The paper investigates the relationship between the equivariant K‑theory of a compact Lie group G and that of a closed subgroup H of maximal rank, focusing on the module structure of the H‑equivariant K‑group K_H^(X) over the G‑equivariant K‑group K_G^(X) for an arbitrary G‑space X. The authors present two principal results, both of which generalize classical theorems from representation theory and index theory.

First, they establish a relative duality theorem (Theorem 4.1.5). By introducing a “twisted Spin^c‑induction” construction—built from central U(1)‑extensions σ of G and τ of H, together with a suitable orientation twist—they define a K_G^*(X)‑bilinear, non‑degenerate pairing
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