On Hecke Algebra Isomorphisms and Types for Depth-zero Principal Series

These lectures describe Hecke algebra isomorphisms and types for depth-zero principal series blocks, a.k.a. Bernstein components Rs(G) for s = sχ = [T, e χ]G, where χ is a depth-zero character on T (O). (Here T is a split maximal torus in a p-adic group G.) We follow closely the treatment of A. Roche [Ro] with input from D. Goldstein [Gol] and L. Morris [Mor]. We give an elementary proof that (I, ρχ) is a type for sχ, in the sense of Bushnell-Kutzko [BK]. This is a very special case of a result of Roche [Ro]. Our method is to imitate Casselman’s proof of Borel’s theorem on unramified principal series (the case χ = 1 of the present theorem). In contrast to the situation for general principal series blocks (see [Ro]), in the depth-zero case there is no restriction on the residual characteristic of F .