Derived Satake morphisms for $p$-small weights in characteristic $p$

Abstract

Let $F$ be a finite unramified extension of ${\mathbb{Q}}_p$ with ring of integers ${\mathcal{O}}_F$, and let $\mathbf{G}$ denote a split, connected reductive group over ${\mathcal{O}}_F$. We fix a Borel subgroup $\mathbf{B}=\mathbf{TU}$ with maximal torus $\mathbf{T}$ and unipotent radical $\mathbf{U}$, and let $L(\hat{I}»)$ denote an irreducible representation of $G_0:=\mathbf{G}({\mathcal{O}}_F)$ with coefficients in a sufficiently large field of characteristic $p$. Set $G:=\mathbf{G}(F)$, etc. Assuming $\hat{I}»$ is a $p$-small and sufficiently regular character and that $p-1$ is greater than the Coxeter number of $\mathbf{G}$, we show that the complex $L(U,{\text{c-ind}}_{G_0}^G(L(\hat{I}»)))$ splits as the orthogonal direct sum of its cohomology objects in the derived category of smooth $T$-representations in characteristic $p$. (Here $L(U,-)$ denotes Heyer's left adjoint of parabolic induction, from the derived category of smooth $G$-representations to the derived category of smooth $T$-representations.) Consequently, this gives rise to a collection of morphisms of graded spherical Hecke algebras $$\underset{i\in \mathbb{Z}}{⨁}{\text{Ext}}_G^i\left({\text{c-ind}}_{G_0}^G(L(\hat{I}»)),{\text{c-ind}}_{G_0}^G(L(\hat{I}»))\right)⟶\underset{i\in \mathbb{Z}}{⨁}{\text{Ext}}_T^i\left({\text{c-ind}}_{T_0}^T(L^n(U_0,L(\hat{I}»))),{\text{c-ind}}_{T_0}^T(L^n(U_0,L(\hat{I}»)))\right)$$ indexed by $n=-[F:{\mathbb{Q}}_p]\dim (\mathbf{U}),\dots ,0$, which we refer to as derived Satake morphisms. For $\hat{I}»=0$ and $n=0$, this recovers the graded mod $p$ Satake homomorphism constructed by Ronchetti. We also give some partial results for general standard parabolic subgroups $\mathbf{P}=\mathbf{MN}\subset \mathbf{G}$.

39 pages. v2: Removed Section 5, other minor changes following referee report. To appear in Math Annalen