On loop Deligne--Lusztig varieties of Coxeter-type for inner forms of ${\rm GL}_n$

Abstract

For a reductive group G over a local non-archimedean field K one can mimic the construction from the classical Deligne-Lusztig theory by using the loop space functor. We study this construction in special the case that G is an inner form of GLn and the loop Deligne-Lusztig variety is of Coxeter type. After simplifying the proof of its representability, our main result is that its -adic cohomology realizes many irreducible supercuspidal representations of G, notably almost all among those whose L-parameter factors through an unramified elliptic maximal torus of G. This gives a purely local, purely geometric and -in a sense -quite explicit way to realize special cases of the local Langlands and Jacquet-Langlands correspondences.

Geometric L-packets of Howe-unramified toral supercuspidal representations

“…From this theorem, we get the irreducibility of c-Ind G SGx,0 (R Gr Sr ,Ur (θ)) (Equation ( 6)) as a free consequence, but this is already highly nontrivial. In the setting that G is an inner form of GL n , a geometric proof of (6) is in [CI21a] (see Theorems 9.1 and 12.5) and under the relaxation of the 0-torality condition to the torality discussed in Section 7.3, a geometric proof of (6) is the subject of [CI19], which further relies on [CI20].…”

“…On the other hand, it is expected (and known in certain cases) that Theorem 6.3 holds beyond the 0-toral case as long as S is elliptic. When G is an inner form of GL n , Theorem 6.3 is known without any constraints on θ (see Theorem B of [CI19]), and we will discuss this in more detail in Section 7.3. For G arbitrary and S Coxeter, establishing Theorem 6.3 is recent work of Dudas-Ivanov (see Theorem 3.2.3 of [DI20]) announced during the preparation of the present paper.…”

“…is subtle, and we will discuss it later in this subsection. Before we do that, we note that Theorem 7.8 in particular yields an algebraic alternative to the geometric approach of [CI19] to establish the irreducibility of the θ-eigenspace of loop Deligne-Lusztig varieties:…”

“…We remark that the assumption q > > 0 is innocuous at least for Coxeter tori-we will prove in subsequent work that it is no stronger than the assumptions needed for Yu's construction [Yu01] (see Remark 5.10). In particular, our approach specialized to GL n relaxes the p > n assumption in [CI19] to p > 2 (see Section 7.3).…”

“…We make some comments about this in Section 7.4. In Section 7.3, we focus on the setting G = GL n , explicitly calculate the twisting character ε ram [θ], and show that a stronger version of the geometrically-proved supercuspidality results of [CI19] follows as a special case (Corollary 7.9) of our comparison theorem (Theorem 7.2). For convenience, we include a diagram depicting the main structure of the results needed to prove our comparison theorems (Theorems 7.2, 7.5):…”