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 () denote an irreducible representation of G₀: = G (OF) with coefficients in a sufficiently large field of characteristic p. Set G: = G (F), etc. Assuming 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-ind₆䃐^G (L () ) ) 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 ₈ ₙExt₆^{i (c-ind₆䃐^G (L () ), ~c-ind₆䃐^G (L () ) ) ₈ ₙExtₓ^i (c-indₓ䃐^T (Lⁿ (U₀, L () ) ), ~c-indₓ䃐^T (Lⁿ (U₀, L () ) ) ) } indexed by n=-F: Qₚ (U), , 0, which we refer to as derived Satake morphisms. For =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.