Let F be a finite unramified extension of \(\mathbb {Q}_p\) with ring of integers \(\mathcal {O}_F\) , and let \(\textbf{G}\) denote a split, connected reductive group over \(\mathcal {O}_F\) . We fix a Borel subgroup \(\textbf{B} = \textbf{T}\textbf{U}\) with maximal torus \(\textbf{T}\) and unipotent radical \(\textbf{U}\) , and let \(L(\lambda )\) denote an irreducible representation of \(\textbf{G}(\mathcal {O}_F)\) with coefficients in a sufficiently large field of characteristic p. Under the assumption that \(\lambda \) is a p-small and sufficiently regular character and that p is greater than 1 plus the Coxeter number of \(\textbf{G}\) , we show that the complex \(L(\textbf{U}(F),\text {c-ind}_{\textbf{G}(\mathcal {O}_F)}^{\textbf{G}(F)}(L(\lambda )))\) splits as the orthogonal direct sum of its cohomology objects in the derived category of smooth \(\textbf{T}(F)\) -representations in characteristic p. (Here \(L(\textbf{U}(F), -)\) denotes Heyer’s left adjoint of parabolic induction, from the derived category of smooth \(\textbf{G}(F)\) -representations to the derived category of smooth \(\textbf{T}(F)\) -representations.) Consequently, this gives rise to a collection of morphisms of graded spherical Hecke algebras \(\begin{aligned} & {\bigoplus _{i \in \mathbb {Z}}\text {Ext}_{\textbf{G}(F)}^{i}\left( \text {c-ind}_{\textbf{G}(\mathcal {O}_F)}^{\textbf{G}(F)}(L(\lambda )),~\text {c-ind}_{\textbf{G}(\mathcal {O}_F)}^{\textbf{G}(F)}(L(\lambda ))\right) } \\ & \quad {\longrightarrow \bigoplus _{i \in \mathbb {Z}}\text {Ext}_{\textbf{T}(F)}^{i}\left( \text {c-ind}_{\textbf{T}(\mathcal {O}_F)}^{\textbf{T}(F)}(L^n(\textbf{U}(\mathcal {O}_F),L(\lambda ))),~\text {c-ind}_{\textbf{T}(\mathcal {O}_F)}^{\textbf{T}(F)}(L^n(\textbf{U}(\mathcal {O}_F),L(\lambda )))\right) } \end{aligned}\) indexed by \(n=-[F:\mathbb {Q}_p]\dim (\textbf{U}), \ldots , 0\) , which we refer to as derived Satake morphisms. For \(\lambda =0\) 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 \(\textbf{P} = \textbf{M}\textbf{N} \subset \textbf{G}\) .
