Derived Satake morphisms for p-small weights in characteristic p

Let F be a finite unramified extension of Qₚ with ring of integers OF, and let G denote a split, connected reductive group over OF. We fix a Borel subgroup B = TU with maximal torus T and unipotent radical U, and let L(λ) denote an irreducible representation of G₀ := G(OF) with coefficients in a sufficiently large field of characteristic p. Set G := G(F), etc. Assuming λ is a p-small and sufficiently regular character and that $p - 1$ is greater than the Coxeter number of G, we show that the complex L(U,c-indG₀G(L(λ))) 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 $${i ∈ ZExtGⁱ(c-indG_0G(L(λ)),~c-indG_0G(L(λ))) i ∈ ZExtTⁱ(c-indT_0T(L^n(U_0,L(λ))),~c-indT_0T(L^n(U_0,L(λ))))}$$ indexed by $n=-[F:Q_p](U), …, 0$, which we refer to as derived Satake morphisms. For $λ=0$ 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 $P = MN ⊂ G$.