Twisted representation rings and Dirac induction

Abstract

Extending ideas of twisted equivariant $K$-theory, we construct twisted versions of the representation rings for Lie superalgebras and Lie supergroups, built from projective $\Z_{2}$-graded representations with a given cocycle. We then investigate the pullback and pushforward maps on these representation rings (and their completions) associated to homomorphisms of Lie superalgebras and Lie supergroups. As an application, we consider the Lie supergroup $\Pi (T^{*}G)$, obtained by taking the cotangent bundle of a compact Lie group and reversing the parity of its fibers. An inclusion $H \hookrightarrow G$ induces a homomorphism from the twisted representation ring of $\Pi(T^{*}H)$ to the twisted representation ring of $\Pi(T^{*}G)$, which pulls back via an algebraic version of the Thom isomorphism to give an additive homomorphism from $K_{H}(\mathrm{pt})$ to $K_{G}(\mathrm{pt})$ (possibly with twistings). We then show that this homomorphism is in fact Dirac induction, which takes an $H$-module $U$ to the $G$-equivariant index of the Dirac operator $\dirac \otimes U$ on the homogeneous space $G/H$ with values in the homogeneous bundle induced by $U$.