Affine Deligne-Lusztig varieties associated with generic Newton points

Abstract

This paper gives an explicit formula of the dimension of affine Deligne-Lusztig varieties associated with generic Newton point in terms of Demazure product of Iwahori-Weyl groups.

Affine Bruhat order and Demazure products

“…We also get an explicit description of Bruhat covers in Ă W (Proposition 4.5) and of the semi-infinite order on Ă W (Corollary 4.10). Finally, combining the aforementioned result of Viehmann [Vie14] with ideas of He [He21], we present a new description of generic Newton points (Theorem 5.29).…”

“…Due to this, both the Bruhat order and Demazure products of affine Weyl groups have been used and studied in the past. We mention the definition of admissible sets due to Kottwitz and Rapoport [KR00;Rap02], the description of generic Newton points in terms of the Bruhat order due to Viehmann [Vie14] and the recent works on generic Newton points and Demazure products due to He and Nie [He21;HN21].…”

“…While the Demazure product is a somewhat simple Coxeter-theoretic notion, it is connected to the question of generic Newton points of elements in Ă W . He [He21] shows how to compute generic Newton points in terms of iterated Demazure products, a method that we will review in Section 5.3. Conversely, He and Nie [HN21] use the Milićević's formula for generic Newton points [Mil21] to show new properties of the Demazure product.…”

“…Throughout this section, we fix an element x " wε µ P Ă W . Following He [He21], we consider twisted Demazure powers of x. Definition 5.27.…”

Affine Bruhat order and Demazure products

“…We also get an explicit description of Bruhat covers in 𝑊 (Proposition 4.5) and of the semi-infinite order on 𝑊 (Corollary 4.10). Finally, combining the aforementioned result of Viehmann [31] with ideas of He [11], we present a new description of generic Newton points (Theorem 5.29).…”