Irreducible components of affine Lusztig varieties

Abstract

Let $\breve{G}$ be a loop group and $\tilde W$ be its Iwahori-Weyl group. The affine Lusztig variety $Y_w(γ)$ describes the intersection of the Bruhat cell $\mathcal{I} \dot{w} \mathcal{I}$ for $w \in \tilde W$ with the conjugacy class of $γ\in \breve{G}$, 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 $Y_w(γ)$ to the numbers of orbits on top-dimensional components of $X_w(b)$ and the affine Springer fibers. For split groups and elements $γ$ 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 $Y_μ(γ)$ within the affine Grassmannian equals to the dimension of a specific weight space in a representation of the Langlands dual group.

Theorems & Definitions (31)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 2.1