Smooth representations and Hecke modules in characteristicp

Abstract

Let G be a p-adic Lie group and I ⊆ G be a compact open subgroup which is a torsionfree pro-p-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. 1 Background and motivation Let G be a d-dimensional p-adic Lie group, and let k be any field. We denote by Modk(G) the category of smooth G-representations in k-vector spaces. It obviously has arbitrary direct sums. We fix a compact open subgroup I ⊆ G. In Modk(G) we then have the representation indI (1) := all k-valued functions with finite support on G/I with G acting by left translations. Being generated by a single element, which is the characteristic function of the trivial coset, indI (1) is a compact object in Modk(G). It generates the full subcategory Modk(G) of all representations V in Modk(G) which are generated by their I-fixed vectors V I . In general Modk(G) is not an abelian category. The Hecke algebra of I by definition is the endomorphism ring HI := EndModk(G)(ind G I (1)) op . We let Mod(HI) denote the category of left unital HI -modules. There is the pair of adjoint functors H : Modk(G) −→ Mod(HI) V 7−→ V I = HomModk(G)(ind G I (1), V ) , and T0 : Mod(HI) −→ Modk(G) ⊆ Modk(G) M 7−→ indI (1)⊗HI M .