Speculations on the mod <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math> representation theory of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math>-adic groups

Harris, Michael

doi:10.5802/afst.1499

Cited by 11 publications

(10 citation statements)

References 25 publications

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“…[41, Chapter 1, Appendix 1, Corollary]). Further, the theorem above partially answers a question posed by Harris in [21,Question 4.5]. We also note that there is related work of Sorensen relating Kohlhaase's smooth duals to a duality operation on the category D(dgMod−H • ), assuming the group G is compact.…”

Section: Introductionsupporting

confidence: 60%

Functorial properties of pro‐p‐Iwahori cohomology

Koziol

2021

J. London Math. Soc.

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Suppose F is a finite extension of Qp, G is the group of F -points of a connected reductive F -group, and I1 is a pro-p-Iwahori subgroup of G. We construct two spectral sequences relating derived functors on mod-p representations of G to the analogous functors on Hecke modules coming from pro-p-Iwahori cohomology. More specifically: (1) using results of Ollivier-Vignéras, we provide a link between the right adjoint of parabolic induction on pro-p-Iwahori cohomology and Emerton's functors of derived ordinary parts; and (2) we establish a 'Poincaré duality spectral sequence' relating duality on pro-p-Iwahori cohomology to Kohlhaase's functors of higher smooth duals. As applications, we calculate various examples of the Hecke modules H i (I1, π).

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Section: Introductionsupporting

confidence: 60%

Functorial properties of pro‐p‐Iwahori cohomology

Koziol

2021

J. London Math. Soc.

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“…Although the main arguments remain unchanged we now, by appealing to a general theorem of Keller, have arranged them in a way which makes the reasoning more transparent. In the context of the search for a p-adic local Langlands program, there is increasing interest in studying derived situations; see [Harris 2015]. We also have now [Ollivier and Schneider 2015] the first examples of explicit computations of the cohomology groups H i (I, ind G I (1)).…”

Section: Background and Motivationmentioning

confidence: 99%

Smooth representations and Hecke modules in characteristicp

Schneider

2015

Pacific J. Math.

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Let G be a p-adic Lie group and I ⊆ G be a compact open subgroup which is a torsionfree prop-group. Working over a coefficient field k of characteristic p we introduce a differential graded Hecke algebra for the pair (G, I) and show that the derived category of smooth representations of G in k-vector spaces is naturally equivalent to the derived category of differential graded modules over this Hecke DGA.

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“…From this point of view, to understand S on the whole derived category of G we should first understand the higher smooth duals of the generator S i (ind G I 1). In fact this ties to a question posed by Harris in [Har16,Q. 4.5] as to whether there is any relation between Kohlhaase's S i and E-linear duality on D(H • I ).…”

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confidence: 74%

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

Cited by 11 publications

References 25 publications

Functorial properties of pro‐p‐Iwahori cohomology

Functorial properties of pro‐p‐Iwahori cohomology

Smooth representations and Hecke modules in characteristicp

A vanishing result for higher smooth duals

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