Twisted representation rings and Dirac induction

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dc.creatorLandweber, Gregory D.
dc.date2004-03-30
dc.date2004-10-25
dc.date.accessioned2026-07-07T07:36:35Z
dc.date.available2026-07-07T07:36:35Z
dc.descriptionExtending 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 $Π(T^{*}G)$, 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 $Π(T^{*}H)$ to the twisted representation ring of $Π(T^{*}G)$, 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 $U$.
dc.description26 pages. Shortened the paper and cleaned up problems with cocycles vs. cohomology classes, Proposition 2, and other minor issues
dc.identifierhttps://arxiv.org/abs/math/0403524
dc.identifierhttp://arxiv.org/abs/math/0403524
dc.identifierJ. Pure Appl. Algebra 206 (2006), no. 1-2, 21-54
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/120476
dc.subjectRepresentation Theory
dc.subjectK-Theory and Homology
dc.subject19A22, 19L47 (primary), 17B10, 22E47 (secondary)
dc.titleTwisted representation rings and Dirac induction
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