Cordial elements and dimensions of affine Deligne–Lusztig varieties

Abstract

The affine Deligne–Lusztig variety $X_w(b)$ in the affine flag variety of a reductive group ${\mathbf G}$ depends on two parameters: the $\sigma $ -conjugacy class $[b]$ and the element w in the Iwahori–Weyl group $\tilde {W}$ of ${\mathbf G}$ . In this paper, for any given $\sigma $ -conjugacy class $[b]$ , we determine the nonemptiness pattern and the dimension formula of $X_w(b)$ for most $w \in \tilde {W}$ .

Generic Newton points and cordial elements

“…the generic Newton point of x, is an important open problem. Our first main result fully solves it, generalizing earlier partial results [Mil21;He21;Sad21;HN21].…”

“…The theory of cordial elements has been used by He [He21] to compute the dimensions of many affine Deligne-Lusztig varieties, even for non-cordial elements x P Ă W . In order to prove our main results, we introduce new methods and refine existing ones.…”

On affine Lusztig varieties

“…Based on the dimension formula for affine Springer fibers in [2], we may write the dimension of affine Lusztig varieties in the affine Grassmannian as dim Y µ (γ) = µ, ρ + We do not have a complete description of the nonemptiness pattern and dimension formula for affine Deligne-Lusztig varieties in the affine flag variety. However, [21] provides the answer for almost all cases. Combining Theorem 5.6 with [21, Theorem 6.1], we obtain the following result on affine Lusztig varieties in the affine flag variety.…”

Affine Deligne–Lusztig varieties and quantum Bruhat graph

“…We first discuss the proof of theorem 1.1[(i)]. The assumption ν − ν(b) ≥ 2ρ ∨ essentially enters the proof of [HY21][Theorem 6.1] in asserting that the KR stratum associated to certain element in W is top-dimensional in X G (µ, b) ∅ , and the dimension of this KR stratum is known only under such gap hypothesis from [He21b]. To circumvent this condition, we simply construct a different top dimensional element (one in the antidominant chamber) even under the weaker gap hypothesis, by appealing to the theory of cordial elements in affine Weyl group à la [MV20].…”