Excursion functions on $p$-adic $\mathrm{SL}_2$

The Bernstein center of a $p$-adic group is a commutative ring of certain distributions on the group, and it interacts closely with the group’s representation theory. Fargues and Scholze provide an abstract construction of a class of elements of the Bernstein center called excursion operators, which encode a candidate for the (semisimplified) local Langlands correspondence. In this talk, I will present an approach to understanding excursion operators concretely as distributions on the group, with a special emphasis on the case of $G = \mathrm{SL}_2$ where everything can be made quite explicit. In the pre-talk, I will provide a gentle introduction to the Bernstein center and the local Langlands correspondence for $\mathrm{SL}_2$.