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The elliptic Weyl character formula

Part of: Topological $K$-theory Homology and cohomology theories Lie groups

Published online by Cambridge University Press:  12 May 2014

Nora Ganter
Nora Ganter*
Affiliation:
Department of Mathematics and Statistics, The University of Melbourne, Parkville VIC 3010,Australia email nganter@unimelb.edu.au
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Abstract

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We calculate equivariant elliptic cohomology of the partial flag variety $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}G/H$, where $H\subseteq G$ are compact connected Lie groups of equal rank. We identify the ${\rm RO}(G)$-graded coefficients ${\mathcal{E}} ll_G^*$ as powers of Looijenga’s line bundle and prove that transfer along the map

$$\begin{equation*}\pi \,{:}\,G/H\longrightarrow {\rm pt} \end{equation*}$$
is calculated by the Weyl–Kac character formula. Treating ordinary cohomology, $K$-theory and elliptic cohomology in parallel, this paper organizes the theoretical framework for the elliptic Schubert calculus of [N. Ganter and A. Ram, Elliptic Schubert calculus, in preparation].

Keywords

elliptic cohomologyloop group characters

MSC classification

Secondary: 55N34: Elliptic cohomology 55N91: Equivariant homology and cohomology 19L47: Equivariant $K$-theory 22E67: Loop groups and related constructions, group-theoretic treatment

Information

Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 7 , July 2014 , pp. 1196 - 1234
Copyright
© The Author 2014 

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