15 March 2005 LS galleries, the path model, and MV cycles
S. Gaussent, P. Littelmann
Duke Math. J. 127(1): 35-88 (15 March 2005). DOI: 10.1215/S0012-7094-04-12712-5


We give an interpretation of the path model of a representation (see [18]) of a complex semisimple algebraic group G in terms of the geometry of its affine Grassmannian. In this setting, the paths are replaced by Lakshmibai-Seshadri (LS) galleries in the affine Coxeter complex associated to the Weyl group of G. To explain the connection with geometry, consider a Demazure-Hansen-Bott-Samelson desingularization $\hat\Sigma(\lambda)$ of the closure of an orbit $G(\mathbb{C}[[t]])\cdot\lambda$ in the affine Grassmannian. The homology of $\hat\Sigma(\lambda)$ has a basis given by Białynicki-Birula (BB) cells, which are indexed by the T-fixed points in $\hat\Sigma(\lambda)$. Now the points of $\hat\Sigma(\lambda)$ can be identified with galleries of a fixed type in the affine Tits building associated to G, and the T-fixed points correspond in this language to combinatorial galleries of a fixed type in the affine Coxeter complex. We determine those galleries such that the associated cell has a nonempty intersection with $G(\mathbb{C}[[t]])\cdot\lambda$ (identified with an open subset of $\hat\Sigma(\lambda)$), and we show that the closures of the strata associated to LS galleries are exactly the Mirković-Vilonen (MV) cycles (see [25]) which form a basis of the representation V(λ) for the Langlands dual group $G^\vee$.


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S. Gaussent. P. Littelmann. "LS galleries, the path model, and MV cycles." Duke Math. J. 127 (1) 35 - 88, 15 March 2005. https://doi.org/10.1215/S0012-7094-04-12712-5


Published: 15 March 2005
First available in Project Euclid: 4 March 2005

zbMATH: 1078.22007
MathSciNet: MR2126496
Digital Object Identifier: 10.1215/S0012-7094-04-12712-5

Primary: 22E46
Secondary: 14M15

Rights: Copyright © 2005 Duke University Press


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Vol.127 • No. 1 • 15 March 2005
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