Smooth representations and Hecke modules in characteristic<i>p</i>

Schneider, Peter

doi:10.2140/pjm.2015.279.447

Cited by 32 publications

(46 citation statements)

References 16 publications

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“…was shown in [21] to be an equivalence of categories for GL(2, F ) if F = Q p but not if F is of characteristic p nor if the residue field of F strictly contains F p ; it is not known what happens for more general G. However, Schneider has shown in [24] that, provided I(1) is torsion-free, the derived version of the functor (4.1) defines an equivalence of triangulated categories between the (unbounded) derived categories…”

Section: Derived Hecke Algebrasmentioning

confidence: 99%

“…Schneider's original paper [24] shows that H i (G, I (1)) vanishes for i not in [0, dim G]; this definitely fails if I(1) has torsion. The higher derived Hecke algebras H i (G, I (1)) are modules over H(G, I(1)) = H 0 (G, I(1)).…”

Section: Questionsmentioning

confidence: 99%

“…It is not hard to compute H i (G, I(1)) as an F-vector space for every i, but the algebra structure is practically unknown. Schneider has computed H dimG (G, I(1)) explicitly ( [24] Proposition 6) but even there the module structure is unclear. The only non-trivial results seem to be due to Ollivier and Schneider; they have shown, for instance, that, when G = GL(2, F ), H 1 (G, I (1)) contains a non-zero torsion submodule over H(G, I(1)) unless F = Q p , and that this torsion submodule consists of supersingular modules; this is related to the failure of the functor (4.1) to be an equivalence of categories.…”

Section: Questionsmentioning

confidence: 99%

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Speculations on the mod p representation theory of p-adic groups

Harris¹

2016

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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Section: Derived Hecke Algebrasmentioning

confidence: 99%

Section: Questionsmentioning

confidence: 99%

Section: Questionsmentioning

confidence: 99%

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Speculations on the mod p representation theory of p-adic groups

Harris¹

2016

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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“…Schneider proved in [Sch15,Thm. 9] that the unbounded derived category D Rep ∞ E (G) is equivalent to the derived category of differential graded modules over a certain DGA variant of the Hecke algebra H • I , defined relative to a torsionfree pro-p group I ⊂ G (and a choice of injective resolution).…”

mentioning

confidence: 94%

“…9] that the unbounded derived category D Rep ∞ E (G) is equivalent to the derived category of differential graded modules over a certain DGA variant of the Hecke algebra H • I , defined relative to a torsionfree pro-p group I ⊂ G (and a choice of injective resolution). A key ingredient was [Sch15,Prop. 6] which shows that ind G I 1 is a (compact) generator of D Rep ∞ E (G) .…”

mentioning

confidence: 99%

A vanishing result for higher smooth duals

Sorensen

2019

Alg. Number Th.

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In this paper we prove a general vanishing result for Kohlhaase's higher smooth duality functors S i . If G is any unramified connected reductive p-adic group, K is a hyperspecial subgroup, and V is a Serre weight, we show that S i (ind G K V ) = 0 for i > dim(G/B) where B is a Borel subgroup. (Here and throughout the paper dim refers to the dimension over Qp.) This is due to Kohlhaase for GL 2 (Qp) in which case it has applications to the calculation of S i for supersingular representations. Our proof avoids explicit matrix computations by making use of Lazard theory, and we deduce our result from an analogous statement for graded algebras via a spectral sequence argument. The graded case essentially follows from Koszul duality between symmetric and exterior algebras.whether the bound dim(G/B) is sharp for π = ind G I 1 in the sense that S dim(G/B) (ind G I 1) is nonzeroeven in the case of GL 2 (Q p ).We now state the main result of this paper which we alluded to above. Let F/Q p be a finite extension, and G /F a connected reductive group with a Borel subgroup B. We assume that G is unramified 2 and choose a hyperspecial maximal compact subgroup K ⊂ G along with a finite-dimensional smooth representation K → GL(V ) with coefficients in some algebraic extension E/F p . Let ind G K V be the compact induction. Then we have the following vanishing result for its higher smooth duals S i .). What we actually prove is a slightly stronger result on the vanishing of the transition maps. Namely, if N ⊳ K has an Iwahori factorization and acts trivially on V , then the restriction mapas long as m is greater than some constant depending only on i and V, N (and an auxiliary filtration on N ). Here (−) ∨ denotes Pontryagin duality, and Λ(N ) = E[[N ]] is the completed group algebra over E. Note that the set of p m -powers N p m is a group for m large enough for any p-valuable group N , cf. [Sch11, Rem. 26.9].We have been unable to show that S dim(G/B) (ind G K V ) = 0 but we believe this should be true under a suitable regularity condition on the weight V , cf. section 12. We hope to address this in future work, and to say more about the action of Hecke operators on S i (ind G K V ) for all i. Let us add that the bound dim(G/B) is not sharp for all V . Indeed S i (ind G K 1) = 0 for all i > 0, cf. Remark 11.2. For GL 2 (Q p ) Theorem 1.1 amounts to [Koh17, Thm. 5.11], which is one of the main results of that paper. There is a small difference coming from the center Z ≃ Q × p though. He assumes V is an irreducible representation of K = GL 2 (Z p ) which factors through GL 2 (F p ), extends the central character of V to Z by sending p → 1, and considers ind G KZ V instead. The latter carries a natural Hecke operator T = T V whose cokernel π V is an irreducible supersingular representation. As V varies this gives all supersingular representations of GL 2 (Q p ) (with p acting trivially), cf. [BL94, Prop. 4] and [Bre03, Thm. 1.1]. The shortgives rise to a long exact sequence of higher smooth dualsSince S i (ind G KZ V ) = 0 for i ...

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The left adjoint of derived parabolic induction

Heyer

2023

Math. Z.

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We prove that the derived parabolic induction functor, defined on the unbounded derived category of smooth mod p representations of a p-adic reductive group, admits a left adjoint $$\textrm{L}(U,-)$$ L ( U , - ) . We study the cohomology functors $$\textrm{H}^i\circ \textrm{L}(U,-)$$ H i ∘ L ( U , - ) in some detail and deduce that $$\textrm{L}(U,-)$$ L ( U , - ) preserves bounded complexes and global admissibility in the sense of Schneider–Sorensen. Using $$\textrm{L}(U,-)$$ L ( U , - ) we define a derived Satake homomorphism and prove that it encodes the mod p Satake homomorphisms defined explicitly by Herzig.

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Smooth representations and Hecke modules in characteristicp

Cited by 32 publications

References 16 publications

Speculations on the mod p representation theory of p-adic groups

Speculations on the mod p representation theory of p-adic groups

A vanishing result for higher smooth duals

The left adjoint of derived parabolic induction

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