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1 October 2001 Multiplets of representations and Kostant's Dirac operator for equal rank loop groups
Gregory D. Landweber
Duke Math. J. 110(1): 121-160 (1 October 2001). DOI: 10.1215/S0012-7094-01-11014-4


Let $\mathfrak {g}$ be a semisimple Lie algebra, and let $\mathfrak {h}$ be a reductive subalgebra of maximal rank in $\mathfrak {g}$. Given any irreducible representation of $\mathfrak {g}$, consider its tensor product with the spin representation associated to the orthogonal complement of $\mathfrak {h}$ in $\mathfrak {g}$. Recently, B. Gross, B. Kostant, P. Ramond, and S. Sternberg [2] proved a generalization of the Weyl character formula which decomposes the signed character of this product representation in terms of the characters of a set of irreducible representations of $\mathfrak {h}$, called a multiplet. Kostant [7] then constructed a formal $\mathfrak {h}$-equivariant Dirac operator on such product representations whose kernel is precisely the multiplet of $\mathfrak {h}$-representations corresponding to the given representation of $\mathfrak {g}$.

We reproduce these results in the Kac-Moody setting for the extended loop algebras $\tilde {L}\mathfrak {g}$ and $\tilde {L}\mathfrak {h}$. We prove a homogeneous generalization of the Weyl-Kac character formula, which now yields a multiplet of irreducible positive energy representations of $L\mathfrak {h}$ associated to any irreducible positive energy representation of $L\mathfrak {g}$. We construct an $L\mathfrak {h}$-equivariant operator, analogous to Kostant's Dirac operator, on the tensor product of a representation of $L\mathfrak {g}$ with the spin representation associated to the complement of $L\mathfrak {h}$ in $L\mathfrak {g}$. We then prove that the kernel of this operator gives the $L\mathfrak {h}$-multiplet corresponding to the original representation of $L\mathfrak {g}$.


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Gregory D. Landweber. "Multiplets of representations and Kostant's Dirac operator for equal rank loop groups." Duke Math. J. 110 (1) 121 - 160, 1 October 2001.


Published: 1 October 2001
First available in Project Euclid: 18 June 2004

zbMATH: 1018.17016
MathSciNet: MR1861090
Digital Object Identifier: 10.1215/S0012-7094-01-11014-4

Primary: 17B67
Secondary: 22E46 , 22E67 , 58J20 , 81R10

Rights: Copyright © 2001 Duke University Press


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Vol.110 • No. 1 • 1 October 2001
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