Modular generalized Springer correspondence I: the general linear group

Achar, Pramod N.; Henderson, Anthony; Juteau, Daniel; Riche, Simon

doi:10.4171/jems/618

Cited by 15 publications

(63 citation statements)

References 42 publications

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“…With characteristic-0 coefficients, the Decomposition Theorem implies that the induction fuctor I G L⊂P takes semisimple perverse sheaves to semisimple perverse sheaves, so in that case, the notions of 'cuspidal' and 'supercuspidal' coincide, as do the notions of 'induction series' and 'induction superseries.' These properties no longer hold in positive characteristic; see [AHJR2,Remark 3.2].…”

Section: 2mentioning

confidence: 99%

“…Let (O, E) ∈ N 0-cusp G,k , and let E K be the equivariant K-local system on O such that θ G (O, E K ) = (O, E). Then IC(O, E) occurs in the modular reduction of IC(O, E K ), so it is cuspidal by [AHJR2,Proposition 2.22].…”

Section: 2mentioning

confidence: 99%

“…Since induction commutes with modular reduction (see [AHJR2,Remark 2.23]), IC(O, E) occurs as a composition factor in the perverse sheaf…”

Section: 2mentioning

confidence: 99%

“…Here Y (L,OL) is the Lusztig stratum associated to (L, O L ); see [AHJR2,§2.6] or [AHJR3, §2.1]. (When necessary, below we will add a superscript 'G' to the notation.)…”

Section: 4mentioning

confidence: 99%

“…In this paper, which follows the series [AHJR2, AHJR3, AHJR4] on the modular generalized Springer correspondence, we consider G-equivariant constructible sheaves on the nilpotent cone N G of G with coefficients in a field k of rather good characteristic. We will study not only the category of G-equivariant perverse sheaves Perv G (N G , k), as in [AHJR2,AHJR3,AHJR4], but also the G-equivariant bounded derived category D b G (N G , k). The main idea is that in rather good characteristic, it is possible to make meaningful comparisons with characteristic-0 generalized Springer theory, a subject that has been extensively studied for over thirty years since the work of Lusztig [Lu1].…”

Section: Introductionmentioning

confidence: 99%

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Constructible sheaves on nilpotent cones in rather good characteristic

Achar¹,

Henderson²,

Juteau³

et al. 2016

Sel. Math. New Ser.

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We study some aspects of modular generalized Springer theory for a complex reductive group $G$ with coefficients in a field $\mathbb k$ under the assumption that the characteristic $\ell$ of $\mathbb k$ is rather good for $G$, i.e., $\ell$ is good and does not divide the order of the component group of the centre of $G$. We prove a comparison theorem relating the characteristic-$\ell$ generalized Springer correspondence to the characteristic-$0$ version. We also consider Mautner's characteristic-$\ell$ `cleanness conjecture'; we prove it in some cases; and we deduce several consequences, including a classification of supercuspidal sheaves and an orthogonal decomposition of the equivariant derived category of the nilpotent cone.Comment: 26 page

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Section: 2mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

“…Since induction commutes with modular reduction (see [AHJR2,Remark 2.23]), IC(O, E) occurs as a composition factor in the perverse sheaf…”

Section: 2mentioning

confidence: 99%

“…Here Y (L,OL) is the Lusztig stratum associated to (L, O L ); see [AHJR2,§2.6] or [AHJR3, §2.1]. (When necessary, below we will add a superscript 'G' to the notation.)…”

Section: 4mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Constructible sheaves on nilpotent cones in rather good characteristic

Achar¹,

Henderson²,

Juteau³

et al. 2016

Sel. Math. New Ser.

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Local Langlands and Springer Correspondences

Aubert

2019

Progress in Mathematics

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This notes are the written version of a course given by the author at the workshop "Representation theory of p-adic groups" that was held at IISER Pune, India, in July 2017 (to appear in Representations of p-adic groups: Contributions from Pune, Progress in Math., Birkhäuser). They give notably an overview of results obtained jointly with Ahmed Moussaoui and Maarten Solleveld on the local Langlands correspondence, focusing on the links of the latter with both the generalized Springer correspondence and the geometric conjecture, the so-called ABPS Conjecture, introduced in collaboration with Paul Baum, Roger Plymen and Maarten Solleveld. Langlands parameters correspond by the LLC to the irreducible supercuspidal representations of G. The validity of this conjecture is proved for representations with unipotent reduction of the group G of the F-rational points of any connected reductive algebraic group which splits over an unramified extension of F in [FOS, Theorem 2] (when G is simple of adjoint type it is a special case of [Lus4], [Lus5]), for the Deligne-Lusztig depth-zero supercuspidal representations (as a consequence of [DeRe]), and also for general linear groups and split classical p-adic groups (any representation) (see [Mou1]), for inner forms of linear groups and of special linear groups, and for quasi-split unitary p-adic groups (any representation) (see [AMS1, § 6]).

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A geometric Schur functor

Mautner

2014

Sel. Math. New Ser.

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Abstract. We give geometric descriptions of the category C k (n, d) of rational polynomial representations of GLn over a field k of degree d for d ≤ n, the Schur functor and Schur-Weyl duality. The descriptions and proofs use a modular version of Springer theory and relationships between the equivariant geometry of the affine Grassmannian and the nilpotent cone for the general linear groups. Motivated by this description, we propose generalizations for an arbitrary connected complex reductive group of the category C k (n, d) and the Schur functor.

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Modular generalized Springer correspondence I: the general linear group

Cited by 15 publications

References 42 publications

Constructible sheaves on nilpotent cones in rather good characteristic

Constructible sheaves on nilpotent cones in rather good characteristic

Local Langlands and Springer Correspondences

A geometric Schur functor

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