Affine Deligne–Lusztig varieties at infinite level

Abstract

In this paper, we study the emptiness/nonemptiness and the dimension formulas of affine Deligne-Lusztig varieties for Sp 4 (L). We mainly calculate the degree of class polynomials for the Iwahori-Hecke algebra of type C 2 . Then, give an explicit description on the emptiness/nonemptiness and dimension formulas of affine Deligne-Lusztig varieties for the group Sp 4 (L).

The Drinfeld stratification for ${\rm GL}_n$

“…In earlier work [CI18], we proved that for inner forms of GL n , Lusztig's loop Deligne-Lusztig set [Lus79] is closely related to a finite-ring analogue of the Drinfeld upper half space. This allowed us to endow this set with a scheme structure (a statement which is still conjectural for any group outside GL n ) and define its cohomology.…”

“…This allowed us to endow this set with a scheme structure (a statement which is still conjectural for any group outside GL n ) and define its cohomology. Under a regularity condition, we prove in [CI18] that the cohomology of loop Deligne-Lusztig varieties for inner forms of GL n realize certain irreducible supercuspidal representation and describe these within the context of the local Langlands and Jacquet-Langlands correspondences. In [CI19b], we are able to relax this regularity condition to something quite general by using highly nontrivial input obtained by studying the cohomology of a stratification-the Drinfeld stratification-which comes from the aforementioned stratification of the Drinfeld upper half space.…”

“…The first equality holds by (4.4). The second equality is an explicit computation: in the division algebra setting, see [Lus79, Equation (2.2)], [Boy12, Lemma 4.4], [Cha19, Section 2.1]; in the present setting of arbitrary inner forms of GL n , see [CI18,Section 6]. We give an exposition of these works here.…”

Cohomological representations of parahoric subgroups

“…q n has trivial Gal(F q n /F q )stabilizer. This is not equivalent to (though is implied by) the condition that the image of x in F × q n is a generator although this last condition is sometimes also associated to the same terminology [Hen92,BW13,CI18]. ♦ Note that if s ∈ Px is unramified very regular, then we may consider the W x (T )-homogeneous space W x (T, Z • (s)) (see Section 2.8).…”

“…When G is any inner form of GL n over k and T is an unramified maximal elliptic torus, we prove in [CI18] that the semi-infinite Deligne-Lusztig set of Lusztig [Lus79] is a scheme and its cohomology realizes the compact induction to G(k) of (an extension of) the P (O k )representations R θ T,U . Already in this setting, it is not enough to study R θ T,U for reductive P ; for example, when G is an anisotropic modulo center inner form of GL n , the relevant parahoric is an Iwahori subgroup.…”