A Satake isomorphism for representations modulo p of reductive groups over local fields

Abstract

Abstract Let F be a local field with finite residue field of characteristic p. Let G be a connected reductive group over F and B a minimal parabolic subgroup of G with Levi decomposition B = Z U $B=ZU$ . Let K be a special parahoric subgroup of G, in good position relative to (Z,U). Fix an absolutely irreducible smooth representation of K on a vector space V over some field C of characteristic p. Writing ℋ ( G , K , V ) $\mathcal {H}(G,K,V)$ for the intertwining Hecke algebra of V in G, we define a natural algebra homomorphism from ℋ ( G , K , V ) $\mathcal {H}(G,K,V)$ to ℋ ( Z , Z ∩ K , V U ∩ K ) $\mathcal {H}(Z,Z\cap K,V^{U\cap K})$ , we show it is injective and identify its image. We thus generalize work of F. Herzig, who assumed F of characteristic 0, G unramified and K hyperspecial, and took for C an algebraic closure of the prime field 𝔽 p . We show that in the general case ℋ ( G , K , V ) $\mathcal {H}(G,K,V)$ need not be commutative; that is in contrast with the cases Herzig considers and with the more classical situation where V is trivial and the field of coefficients is the field of complex numbers.