Irreducible components of affine Lusztig varieties

TL;DR

This paper investigates the irreducible components of affine Lusztig varieties, establishing relations with affine Deligne-Lusztig varieties and confirming conjectures about their top-dimensional components in certain cases.

Contribution

It relates the orbit counts of top-dimensional components of affine Lusztig and affine Deligne-Lusztig varieties, and verifies Chi's conjecture for specific split groups and elements.

Findings

Number of orbits on top components match for most w in split groups.

Confirmed Chi's conjecture relating components to Langlands dual group representations.

Established relations between geometric properties of affine Lusztig and affine Deligne-Lusztig varieties.

Abstract

Let $\breve{G}$ be a loop group and $\tilde W$ be its Iwahori-Weyl group. The affine Lusztig variety $Y_w(\gamma)$ describes the intersection of the Bruhat cell $\mathcal{I} \dot{w} \mathcal{I}$ for $w \in \tilde W$ with the conjugacy class of $\gamma \in \breve{G}$, while the affine Deligne-Lusztig variety $X_w(b)$ describes the intersection of the Bruhat cell $\mathcal{I} \dot{w} \mathcal{I}$ with the Frobenius-twisted conjugacy class of $b \in \breve{G}$. Although the geometric connections between these varieties are unknown, numerical relations exist in their geometric properties. This paper explores the irreducible components of affine Lusztig varieties. The centralizer of $\g$ acts on $Y_w(\g)$ and the Frobenius-twisted centralizer of $b$ acts on $X_w(b)$. We relate the number of orbits on the top-dimensional components of $Y_w(\gamma)$ to the numbers of orbits on top-dimensional components of $X_w(b)$ and the affine Springer fibers. For split groups and elements $\gamma$ with integral Newton points, we show that, for most $w$, the numbers of orbits for the affine Lusztig variety and the associated affine Deligne-Lusztig variety match. Moreover, for these $\g$, we verify Chi's conjecture that the number of top-dimensional components in $Y_\mu(\gamma)$ within the affine Grassmannian equals to the dimension of a specific weight space in a representation of the Langlands dual group.