Deligne Lusztig Research Articles

Action of automorphisms on irreducible characters of symplectic groups

Assume G is a finite symplectic group Sp2n(q) over a finite field Fq of odd characteristic. We describe the action of the automorphism group Aut(G) on the set Irr(G) of ordinary irreducible characters of G. This description relies on the equivariance of Deligne–Lusztig induction with respect to automorphisms. We state a version of this equivariance which gives a precise way to compute the automorphism on the corresponding Levi subgroup; this may be of independent interest. As an application we prove that the global condition in Späth's criterion for the inductive McKay condition holds for the irreducible characters of Sp2n(q).

Euler characteristic of analogues of a Deligne–Lusztig variety for GLn

We give a combinatorial formula to calculate the Euler characteristic of an analogue of a Deligne–Lusztig variety, denoted Yw,g, which is attached to an element w in the Weyl group of GLn and g∈GLn. The main theorem of this paper states that the Euler characteristic of Yw,g only depends on the unipotent part of the Jordan decomposition of g and the conjugacy class of w. It generalizes the formula of the Euler characteristic of Springer fibers for type A.

On the Mackey formula for connected centre groups

Abstract Let 𝐆 {\mathbf{G}} be a connected reductive algebraic group over 𝔽 ¯ p {\overline{\mathbb{F}}_{p}} and let F : 𝐆 → 𝐆 {F:\mathbf{G}\to\mathbf{G}} be a Frobenius endomorphism endowing 𝐆 {\mathbf{G}} with an 𝔽 q {\mathbb{F}_{q}} -rational structure. Bonnafé–Michel have shown that the Mackey formula for Deligne–Lusztig induction and restriction holds for the pair ( 𝐆 , F ) {(\mathbf{G},F)} except in the case where q = 2 {q=2} and 𝐆 {\mathbf{G}} has a quasi-simple component of type 𝖤 6 {\mathsf{E}_{6}} , 𝖤 7 {\mathsf{E}_{7}} , or 𝖤 8 {\mathsf{E}_{8}} . Using their techniques, we show that if q = 2 {q=2} and Z ⁢ ( 𝐆 ) {Z(\mathbf{G})} is connected then the Mackey formula holds unless 𝐆 {\mathbf{G}} has a quasi-simple component of type 𝖤 8 {\mathsf{E}_{8}} . This establishes the Mackey formula, for instance, in the case where ( 𝐆 , F ) {(\mathbf{G},F)} is of type 𝖤 7 ⁢ ( 2 ) {\mathsf{E}_{7}(2)} . Using this, together with work of Bonnafé–Michel, we can conclude that the Mackey formula holds on the space of unipotently supported class functions if Z ⁢ ( 𝐆 ) {Z(\mathbf{G})} is connected.

Equivalences of tame blocks for p-adic linear groups

Let p and $$\ell $$ be two distinct primes, F a p-adic field and n an integer. We show that any level 0 block of the category of smooth $$\overline{{\mathbb {Z}}}_{\ell }$$ -valued representations of $$ \mathrm{GL}_{n}(F)$$ is equivalent to the unipotent block of an appropriate product of $$\mathrm{GL}_{n_{i}}(F_{i})$$ . More precisely, we show that level 0 blocks of p-adic general linear groups obey the “functoriality principle for blocks” introduced in Dat (Contemp Math 691:103–131, 2017). The overall strategy is to use the “Jordan decomposition of blocks” of finite general linear groups, and the theory of coefficients systems on the relevant buildings as a bridge between the p-adic and finite worlds. To overcome the main technical difficulty, we need fine properties of Deligne–Lusztig cohomology proved in Bonnafe et al. (Ann Math 185(2):609–670, 2017).

The cohomology of Deligne–Lusztig varieties for the general linear group

We propose two inductive approaches for determining the cohomology of Deligne-Lusztig varieties in the case of G = GLn.

Deligne–Lusztig duality and wonderful compactification

We use geometry of the wonderful compactification to obtain a new proof of the relation between Deligne–Lusztig (or Alvis–Curtis) duality for p-adic groups and homological duality. This provides a new way to introduce an involution on the set of irreducible representations of the group which has been defined by A. Zelevinsky for $$G=GL(n)$$ and by A.-M. Aubert in general (less direct geometric approaches to this duality have been developed earlier by Schneider-Stuhler and by the second author). As a byproduct, we describe the Serre functor for representations of a p-adic group.

Some Ree and Suzuki curves are not Galois covered by the Hermitian curve

The Deligne–Lusztig curves associated to the algebraic groups of type A22, B22, and G22 are classical examples of maximal curves over finite fields. The Hermitian curve Hq is maximal over Fq2, for any prime power q, the Suzuki curve Sq is maximal over Fq4, for q=22h+1, h≥1, and the Ree curve Rq is maximal over Fq6, for q=32h+1, h≥0. In this paper we show that S8 is not Galois covered by H64. We also prove an unpublished result due to Rains and Zieve stating that R3 is not Galois covered by H27. Furthermore, we determine the spectrum of genera of Galois subcovers of H27, and we point out that some Galois subcovers of R3 are not Galois subcovers of H27.

Ramified automorphic induction and zero-dimensional affine Deligne–Lusztig varieties

To any connected reductive group G over a non-archimedean local field F and to any maximal torus T of G, we attach a family of extended affine Deligne–Lusztig varieties (and families of torsors over them) over the residue field of F. This construction generalizes affine Deligne–Lusztig varieties of Rapoport, which are attached only to unramified tori of G. Via this construction, we can attach to any maximal torus T of G and any character of T a representation of G. This procedure should conjecturally realize the automorphic induction from T to G. For $$G = {{\mathrm{GL}}}_2$$ in the equal characteristic case, we prove that our construction indeed realizes the automorphic induction from at most tamely ramified tori. Moreover, if the torus is purely tamely ramified, then the varieties realizing this correspondence turn out to be (quite complicate) combinatorial objects: they are zero-dimensional and reduced, i.e., just disjoint unions of points.

Corrigendum: Connected components of affine Deligne–Lusztig varieties in mixed characteristic

We correct an error in our paper ‘Connected components of affine Deligne–Lusztig varieties in mixed characteristic’.

Stable basic sets for finite special linear and unitary groups

In this paper we show, using Deligne–Lusztig theory and Kawanaka's theory of generalised Gelfand–Graev representations, that the decomposition matrix of the special linear and unitary group in non-defining characteristic can be made unitriangular with respect to a basic set that is stable under the action of automorphisms.

L’espace symétrique de Drinfeld et correspondance de Langlands locale II

We study the geometry and the cohomology of the tamely ramified cover of Drinfeld’s p-adic symmetric space over a p-adic field K. For this tame level, we prove, in a purely local way, most of a conjecture of Harris on the form of the $$\ell $$ -adic cohomologies, as well as the torsion freeness of the integral cohomology. In this paper, we also compute the $$\ell $$ -adic cohomology of Coxeter Deligne–Lusztig variety associated to $$\mathrm{GL}_d,$$ and some results of independent interest on the coefficient systems over the Bruhat–Tits building associated to $$\mathrm{GL}_d(K)$$ have been established.

A note on perfect isometries between finite general linear and unitary groups at unitary primes

Let q be a power of a prime, ℓ a prime not dividing q, d a positive integer coprime to both ℓ and the multiplicative order of qmodℓ and n a positive integer. A. Watanabe proved that there is a perfect isometry between the principal ℓ-blocks of GLn(q) and GLn(qd) where the correspondence of characters is given by Shintani descent. In the same paper Watanabe also proved that if ℓ and q are odd and ℓ does not divide |GLn(q2)|/|Un(q)| then there is a perfect isometry between the principal ℓ-blocks of Un(q) and GLn(q2) with the correspondence of characters also given by Shintani descent. R. Kessar extended this first result to all unipotent blocks of GLn(q) and GLn(qd). In this paper we extend this second result to all unipotent blocks of Un(q) and GLn(q2). In particular this proves that any two unipotent blocks of Un(q) at unitary primes (for possibly different n) with the same weight are perfectly isometric. We also prove that this perfect isometry commutes with Deligne–Lusztig induction at the level of characters.

Affine Deligne–Lusztig varieties of higher level and the local Langlands correspondence for GL2

In the present article we define coverings of affine Deligne–Lusztig varieties attached to a connected reductive group over a non-archimedean local field. In the case of GL2 and positive characteristic, the unramified cuspidal part of the local Langlands correspondence is realized in the ℓ-adic cohomology of these varieties. We show this by giving a detailed comparison with the realization of local Langlands via cuspidal types by Bushnell–Henniart. All proofs are purely local.

Deligne–Lusztig constructions for division algebras and the local Langlands correspondence

In 1979, Lusztig proposed a cohomological construction of supercuspidal representations of reductive p-adic groups, analogous to Deligne–Lusztig theory for finite reductive groups. In this paper we establish a new instance of Lusztig's program. Precisely, let D be the quaternion algebra over a local non-Archimedean field K of positive characteristic, and let X be the p-adic Deligne–Lusztig ind-scheme associated to D×. There is a natural correspondence between quasi-characters of the (multiplicative group of the) unramified quadratic extension of K and representations of D× given by θ↦Hi(X)[θ]. We show that this correspondence is a bijection (after a mild restriction of the domain and target), and matches the bijection given by local Langlands and Jacquet–Langlands.

Weil representations of finite symplectic groups and finite odd-dimensional orthogonal groups

In [8], Srinivasan decomposes the uniform projection of the character of Weil representation ωG,G′ of a finite reductive dual pair (G,G′) in terms of Deligne–Lusztig virtual characters when the dual pair is one of the following: (1) two general linear groups; (2) two unitary groups; (3) a symplectic group and an even-dimensional orthogonal group, assuming the order of the finite field is large enough. In this article we obtain the decomposition for the remaining case, i.e., the dual pair of a symplectic group and an odd-dimensional orthogonal group. In particular, we have the decomposition of the uniform projection of the character of the Weil representation of a finite symplectic group.

Class polynomials for some affine Hecke algebras

Class polynomials attached to affine Hecke algebras were first introduced by He in [13]. They play an important role in the study of affine Deligne–Lusztig varieties. Motivated by [14], we compute the class polynomials attached to an affine Hecke algebra of type (twisted) A˜2. Using these class polynomials, we prove a conjecture of Görtz–Haines–Kottwitz–Reuman for the general linear group, unitary group, and division algebra of semisimple rank 2. Furthermore, we discuss some interesting patterns of affine Deligne–Lusztig varieties.

Connected components of affine Deligne–Lusztig varieties in mixed characteristic

We determine the set of connected components of minuscule affine Deligne–Lusztig varieties for hyperspecial maximal compact subgroups of unramified connected reductive groups. Partial results are also obtained for non-minuscule closed affine Deligne–Lusztig varieties. We consider both the function field case and its analog in mixed characteristic. In particular, we determine the set of connected components of unramified Rapoport–Zink spaces.

Decomposition matrices for low-rank unitary groups

We study the decomposition matrices of the unipotent ℓ-blocks of finite special unitary groups SU n ( q ) for unitary primes ℓ larger than n. Up to few unknown entries, we give a complete solution for n = 2 , ... , 10 . We also prove a general result for two-column partitions when ℓ divides q + 1 . This is achieved using projective modules coming from the ℓ-adic cohomology of Deligne–Lusztig varieties.

$$P$$ P -alcoves, parabolic subalgebras and cocenters of affine Hecke algebras

The cocenter of an affine Hecke algebra plays an important role in the study of representations of the affine Hecke algebra and the geometry of affine Deligne–Lusztig varieties (see for example, He and Nie in Compos Math 150(11):1903–1927, 2014; He in Ann Math 179:367–404, 2014; Ciubotaru and He in Cocenter and representations of affine Hecke algebras, 2014). In this paper, we give a Bernstein–Lusztig type presentation of the cocenter. We also obtain a comparison theorem between the class polynomials of the affine Hecke algebra and those of its parabolic subalgebras, which is an algebraic analog of the Hodge–Newton decomposition theorem for affine Deligne–Lusztig varieties. As a consequence, we present a new proof of the emptiness pattern of affine Deligne–Lusztig varieties (Gortz et al. in Compos Math 146(5):1339–1382, 2010; Gortz et al. in Ann Sci Ecole Norm Sup, 2012).

Fundamental elements of an affine Weyl group

Fundamental elements are certain special elements of affine Weyl groups introduced by Gortz, Haines, Kottwitz and Reuman. They play an important role in the study of affine Deligne–Lusztig varieties. In this paper, we obtain characterizations of the fundamental elements and their natural generalizations. We also derive an inverse to a version of “Newton-Hodge decomposition” in affine flag varieties. As an application, we obtain a group-theoretic generalization of Oort’s results on minimal \(p\)-divisible groups, and we show that, in certain good reduction of PEL Shimura datum, each Newton stratum contains a minimal Ekedahl–Oort stratum. This generalizes a result of Viehmann and Wedhorn.