Invariant Distributions and the Bernstein Center

Let $G$ be a $p$-adic group. I shall introduce the Bernstein center, review its construction and properties and explain how to use it in order to study distributions on spherical spaces.

More specifically, we shall discuss the wave front of distributions and how to control it in case the distribution is ”finite“ with respect to the Bernstein center. Then we discuss whether this condition of ”finiteness“ is reasonable assumption.

The tools used will involve very little analysis, but rather facts from commutative algebra and connections of the representation theory of $p$-adic groups to commutative algebra.