A comparison between pro-p Iwahori–Hecke modules and mod p representations

Abstract

We give an equivalence of categories between certain subcategories of modules of pro-p Iwahori-Hecke algebras and modulo p representations.2010 Mathematics Subject Classification. 22E50,20C08.Let J be a subset of ∆ and denote the corresponding standard parabolic subgroup by P J . Let L J be the Levi part of P J containing Z. Then K ∩L J is a special parahoric subgroup and I(1) J = I(1)∩L J a pro-p Iwahori subgroup. Attached to these, we have many objects. For such objects we add a suffix J, for example, the pro-p Iwahori-Hecke algebra attached to (L J , I(1) J ) is denoted by H J . There are two exceptions: base T w and E(w) for H J is denoted by T J w and E J (w), respectively. For each J ⊂ ∆, we have twoAn element E(λ) belongs toThe second one follows from the following fact and [Abe16, Lemma 2.6]: a basis of H − J ∩ A J is given by {E(λ)} where λ runs through as above [Abe, Lemma 4.2].) Since w containsLemma 3.12. We regard A w J as a subalgebra of A J via the above embedding. Then n w J M is uniquely extended to A J , namely there exists a unique A J -module M J such that supp M ) are both isomorphisms.Proof. The first one is isomorphism by the similar argument in the proof of Lemma 3.11. Take λ 0 ∈ Λ(1) such thatSince E J (λ 0 ) is invertible in A J , it is also invertible on n w J M J . (Note that n w J M J is an A J -module.) Hence the second homomorphism is an isomorphism.Therefore we get

EXTENSION BETWEEN SIMPLE MODULES OF PRO-p-IWAHORI HECKE ALGEBRAS

“…As far as the author knows, a calculation of extensions was done only when G = GL 2 . Our calculation is in general, namely we do not assume anything about G. We also remark a related result in [1]. If π 1 ,π 2 are modulo p irreducible subquotients of principal series of G, by the main theorem of [1], then we have an embedding Ext 1 H (π…”

“…• Ω acts on (W aff ,S aff ). (1). The parameter (q,c) is maps q : S aff → C and c : Ref(W aff (1)) → C[Z κ ] with the following conditions.…”

“…For each standard parabolic subgroup P, let H P be the prop-Iwahori Hecke algebra of the Levi subgroup of P. Then for a module σ of H P , we can consider: the parabolic induction I P (σ) which is an H-module, a certain parabolic subgroup P (σ) containing P, a generalized Steinberg module St P (σ) Q (σ), where Q is a parabolic subgroup between P and P (σ). By [3], each simple module is constructed by three steps: (1) starting with a supersingular module σ of H P , where P is a parabolic subgroup; (2) (σ 2 )) be two simple modules here σ 1 (resp. σ 2 ) is a simple supersingular module of H P1 (resp.…”

“…(5) Now let H be a generic algebra and π 1 ,π 2 be simple supersinglar modules. The algebra has the following decomposition as vector spaces: (1)]. Here, H aff ⊂ H is an algebra called 'the affine subalgebra', Ω(1) is a certain commutative group acting on H aff , Z κ is a normal subgroup of Ω(1) and we have an embedding C[Z κ ] → H aff which is compatible with the action of Ω(1) on H aff .…”

“…Therefore, it is sufficient to calculate two groups: H 1 (Ω, Hom H aff (π 1 ,π 2 )) and Ext 1 H aff (π 1 ,π 2 ) Ω . By the classification result of supersingular simple modules [14,18], the restriction of π 1 ,π 2 to H aff are the direct sum of characters of H aff .…”