Mirković-Vilonen cycles and polytopes for a symmetric pair

Hong, Jiuzu

doi:10.1090/S1088-4165-09-00341-0

Mirković-Vilonen cycles and polytopes for a symmetric pair
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by Jiuzu Hong

Represent. Theory 13 (2009), 19-32

DOI: https://doi.org/10.1090/S1088-4165-09-00341-0

Published electronically: February 13, 2009

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Abstract:

Let $G$ be a connected, simply-connected, and almost simple algebraic group, and let $\sigma$ be a Dynkin automorphism on $G$. Then $(G,G^\sigma )$ is a symmetric pair. In this paper, we get a bijection between the set of $\sigma$-invariant MV cycles (polytopes) for $G$ and the set of MV cycles (polytopes) for $G^\sigma$, which is the fixed point subgroup of $G$; moreover, this bijection can be restricted to the set of MV cycles (polytopes) in irreducible representations. As an application, we obtain a new proof of the twining character formula.

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