A vanishing result for higher smooth duals

Abstract

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 ...

International Congress of Mathematicians

“…For complex s, the integral (184) converges when the real part <s > 0. Integration by parts gives .s C 1/ D s.s/; (185) and, using this formula, one can extend the definition of .s/ to all values of s, except s D 0; 1; 2; : : : From (185) and the base case .1/ D 1, we conclude…”

“…under tensor product, inflation and induction and proved that for V 2 Mod R .G/ a , the integer For G unramified, 31 K a hyperspecial subgroup of G, W 2 Mod F ac p .K/ and i > dim Q p U , we have S i .ind G K W / D 0 (Claus Sorensen [185]).…”