A description of the integral depth-$r$ Bernstein center

Paper

Abstract

In this paper we give a description of the depth-

Bernstein center for non-negative integers

of a reductive simply connected group

over a non-archimedean local field as a limit of depth-

standard parahoric Hecke algebras. Using the description, we construct maps from the algebra of stable functions on the

-th Moy-Prasad filtration quotient of hyperspecial parahorics to the depth-

Bernstein center and use them to attach to each depth-

irreducible representation

an invariant

, called the depth-

Deligne-Lusztig parameter of

. We show that

is equal to the semi-simple part of minimal

-types of