Mod p Bernstein centres of p-adic groups
The centre of the category of smooth mod p representations of a p-adic reductive group does not distinguish the blocks of finite length representations, in contrast with Bernstein's theory in characteristic zero. Motivated by this observaton and the known connections between the Bernstein centre and the local Langlands correspondence in families, we consider the case of GL_2(Q_p) and we prove that its category of representations extends to a stack on the Zariski site of a simple geometric object: a chain X of projective lines, whose points are in bijection with Paskunas's blocks. Taking the centre over each open subset we obtain a sheaf of rings on X, and we expect the resulting space to be closely related to the Emerton--Gee stack for 2-dimensional representations of the absolute Galois group of Q_p. Joint work in progress with Matthew Emerton and Toby Gee.