A Classification of the Irreducible mod-$p$ Representations of $\textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$

Koziol, Karol

doi:10.5802/aif.3043

Cited by 8 publications

(6 citation statements)

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“…Of course if G has relative rank 0, all irreducible representations of G are finite dimensional and supercuspidal, and our classification theorem says nothing. If G has relative semisimple rank 1, the classification is rather simple (see also [BL1,BL2,Abd,Che,Ko,Ly2]). An irreducible admissible representation π of G falls into one (and only one) of the following cases: 1) π is supercuspidal (hence infinite dimensional), i.e.…”

Section: Vi5mentioning

confidence: 99%

“…I.2. In this paper we classify irreducible admissible representations of G in terms of parabolic induction and supercuspidal representations of Levi subgroups of G. Such a classification was obtained for G = GL 2 in the pioneering work of L. Barthel and R. Livné [BL1,BL2] -see also some recent work [Abd,Che,Ko,KX,Ly2] on situations where, mostly, G has relative semisimple rank 1. New ideas towards the general case were set forth by the third-named author [He1,He2], who gave the classification for G = GL n over a p-adic field F ; his ideas were further expanded by the first-named author [Abe] to treat the case of a split group G, still over a p-adic field F .…”

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A classification of irreducible admissible mod 𝑝 representations of 𝑝-adic reductive groups

Abe

Henniart

Herzig

et al. 2016

J. Amer. Math. Soc.

102

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Let F be a locally compact non-archimedean field, p its residue characteristic, and G a connected reductive group over F . Let C be an algebraically closed field of characteristic p. We give a complete classification of irreducible admissible Crepresentations of G = G(F ), in terms of supercuspidal C-representations of the Levi subgroups of G, and parabolic induction. Thus we push to their natural conclusion the ideas of the third-named author, who treated the case G = GLm, as further expanded by the first-named author, who treated split groups G. As in the split case, we first get a classification in terms of supersingular representations of Levi subgroups, and as a consequence show that supersingularity is the same as supercuspidality. ContentsStart with a parabolic subgroup P of G, with Levi quotient M , and a representation σ of M . Then there is a largest parabolic subgroup P (σ) of G, containing P , such that 4 N. ABE, G. HENNIART, F. HERZIG, AND M.-F. VIGNÉRAS B of G with Levi subgroup Z, the F -points of the centralizer of S, and we write U for the unipotent radical of B.Let V be an irreducible representation of K -it has finite dimension. If (π, W ) is an admissible representation of G, then Hom K (V, W ) is a finite-dimensional C-vector space; by Frobenius reciprocity Homand we focus on the corresponding characters of Z G (V ), which we call the (Hecke) eigenvalues of Z G (V ) in π.For any parabolic subgroup P of G containing B, with Levi component M containing Z and unipotent radical N , the space of coinvariants V N ∩K of N ∩ K in V provides an irreducible representation of M ∩ K and by [He1, He2, HV2] there is a natural injective algebra homomorphism. Both homomorphisms are localizations at a central element. A character χ : Z G (V ) → C is said to be supersingular if, in the above situation, it can be extended to a character of Z M (V N ∩K ) only when P = G (see Chapter III, part A) for details). A supersingular representation of G is an irreducible admissible representation (π, W ) such that for all weights V of π, all eigenvalues of Z G (V ) in π are supersingular 2 .A triple (P, σ, Q) as in I.3 is a B-triple if P contains B; it is said to be supersingular if it is a B-triple and σ is a supersingular representation of the Levi quotient of P .Theorems 1 to 3 are consequences of the following results.Theorem 4. For each supersingular triple (P, σ, Q), the representation I(P, σ, Q) is irreducible and admissible. If π is an irreducible admissible representation of G, there is a supersingular triple (P, σ, Q) such that π is isomorphic to I(P, σ, Q); moreover P and Q are unique and σ is unique up to isomorphism.Theorem 5. Let π be an irreducible admissible representation of G. Then π is supercuspidal if and only if it is supersingular.(For G = GL 2 this was discovered by Barthel and Livné.)Note that Theorem 5 implies, in particular, that the notion of supersingularity does not depend on the choices of S, K, B necessary for the definition -beware that in general two choices of K will not even be conjugat...

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Section: Vi5mentioning

confidence: 99%

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confidence: 99%

A classification of irreducible admissible mod 𝑝 representations of 𝑝-adic reductive groups

Abe

Henniart

Herzig

et al. 2016

J. Amer. Math. Soc.

102

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“…Supersingular representations of GL 2 (Q p ) was classified by Breuil and some mod-p Langlands correspondences appeared ( [4]). Up till now, except GL 2 (Q p ) and some related groups such as SL 2 (Q p ) ( [1], [6], [12]), supersingular representations for general groups (e.g. GL 3 (Q p ) or GL 2 (F ) when F = Q p ) remain mysteries.…”

Section: Introductionmentioning

confidence: 99%

A note on presentations of supersingular representations of $\text{GL}_2(F)$

2019

Preprint

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“…We do not know in which generality such resolutions exist 1 for a general group G. For groups of semisimple rank one, Kohlhaase has constructed a class of representations in [Koh18] to which our bound applies. We also do not know [Koz16] Koziol was able to bootstrap from this case and deal with the unramified unitary group U (1, 1) over Q p . As an application of our Theorem 1.1 there is forthcoming work of Jake Postema for his UC San Diego Ph.D. thesis in which he calculates all the S i for supersingular representations of both SL 2 (Q p ) and U (1, 1), and explores how S 1 flips the members of an L-packet.…”

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“…Similar descriptions of the supersingular representations exist for other rank one groups. In [Abd14] Abdellatif classifies the supersingulars of SL 2 (Q p ) by decomposing π V | SL 2 (Qp) into irreducibles, and in [Koz16] Koziol was able to bootstrap from this case and deal with the unramified unitary group U (1, 1) over Q p . As an application of our Theorem 1.1 there is forthcoming work of Jake Postema for his UC San Diego Ph.D. thesis in which he calculates all the S i for supersingular representations of both SL 2 (Q p ) and U (1, 1), and explores how S 1 flips the members of an L-packet.…”

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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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A Classification of the Irreducible mod-$p$ Representations of $\textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$

Cited by 8 publications

References 13 publications

A classification of irreducible admissible mod 𝑝 representations of 𝑝-adic reductive groups

A classification of irreducible admissible mod 𝑝 representations of 𝑝-adic reductive groups

A note on presentations of supersingular representations of $\text{GL}_2(F)$

A vanishing result for higher smooth duals

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