A geometric Steinberg formula

Achar, Pramod N.; Riche, Simon

doi:10.48550/arxiv.2109.11980

Cited by 1 publication

(26 citation statements)

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“…Let us also record the following property, proved in [AR3,Lemma 2.7], which shows in particular that lengths always add in a decomposition given by (2.3).…”

Section: Combinatorics Of the Affine Weyl Groupmentioning

confidence: 96%

“…By [AR3,Lemma 2.6], on the right-hand side we have α, λ ≤ 0 for any α ∈ R + . Moreover, if w(α) ∈ −R + , then at least one simple root γ appearing in the decomposition of α as a sum of simple roots must satisfy w(γ) ∈ −R + ; we therefore have α, λ ≤ −1 in this case.…”

Section: Combinatorics Of the Affine Weyl Groupmentioning

confidence: 99%

“…In anticipation of this conjecture, one might look for "representation-theoretic phenomena" in Perv Iu (Gr, k), which might hold with milder assumptions on ℓ. The present paper and its companion paper [AR3] (where we established a formula for convolution products of certain simple perverse sheaves, modeled on the Steinberg tensor product formula for group representations, without any restriction on ℓ) both pursue this idea.…”

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confidence: 89%

“…(The partial order appearing below is Lusztig's "periodic order" on W ext , whose definition is recalled in §2.5. The definition of restricted elements in W ext can be found in [AR3,§2.4]. )…”

mentioning

confidence: 99%

“…where Fr * is pullback along the Frobenius morphism Fr : q G → q G (1) , Sat is the geometric Satake equivalence, and sw is a certain autoequivalence of Perv L + G (Gr, k). We refer the reader to [AR3,§1.2] for precise definitions of the notation, and further discussion of this statement.…”

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confidence: 99%

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A geometric model for blocks of Frobenius kernels

Achar¹,

Riche²

2022

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Building on a geometric counterpart of Steinberg's tensor product formula for simple representations of a connected reductive algebraic group q G over a field of positive characteristic proved in [AR3], and following an idea of Arkhipov-Bezrukavnikov-Braverman-Gaitsgory-Mirković, we define and initiate the study of some categories of perverse sheaves on the affine Grassmannian of the Langlands dual group to q G that should provide geometric models for blocks of representations of the Frobenius kernel q G 1 of q G. In particular, we show that these categories admit enough projective and injective objects, which are closely related to some tilting perverse sheaves, and that they are highest weight categories in an appropriate generalized sense. Contents 1. Introduction 1 2. Combinatorics of the affine Weyl group 6 3. Perverse sheaves on affine Grassmannians 13 4. Background from representation theory 26 5. Modules over the regular perverse sheaf 33 6. Averaging and wall-crossing functors 43 7. Projectives and injectives 51 8. Ungraded R-modules 60 9. Baby Verma and co-Verma modules 67 References 87

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“…Let us also record the following property, proved in [AR3,Lemma 2.7], which shows in particular that lengths always add in a decomposition given by (2.3).…”

Section: Combinatorics Of the Affine Weyl Groupmentioning

confidence: 96%

Section: Combinatorics Of the Affine Weyl Groupmentioning

confidence: 99%

mentioning

confidence: 89%

mentioning

confidence: 99%

mentioning

confidence: 99%

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