Lattice associated to a Shi variety

Let $W$ be an irreducible Weyl group and Wa its affine Weyl group. In [4] the author defined an affine variety $\hat{X}_{W_a}$, called the Shi variety of $W_a$, whose integral points are in bijection with $W_a$. The set of irreducible components of $\hat{X}_{W_a}$, denoted $H^0 (\hat{X}_{W_a})$, is of some interest and we show in this article that $H^0 (\hat{X}_{W_a})$ has a structure of a semi-distributive lattice.

Keywords

affine Weyl groups, Shi variety, irreducible components

2010 Mathematics Subject Classification

Primary 06B99. Secondary 20F55.

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This work was partially supported by NSERC grants and by the LACIM at Université du Québec à Montréal.

Received 12 March 2021

Accepted 28 September 2021

Published 19 August 2022