Excursion functions on p-adic groups

I present material from my student Jacksyn Bakeberg's thesis. The Bernstein center of a p-adic group is its ring of conjugation-invariant distributions; this ring controls the representations of the group. Fargues-Scholze gives a geometric construction of a subring of the Bernstein center, consisting of excursion operators, which are labeled by elements of the Galois group. One can think of these operators as an encoding of the Langlands correspondence. It would be interesting to give a completely explicit description of these excursion operators. We do exactly this in the case of $ G=SL_{2}$, where we show that the excursion operator is represented by a function (an ”excursion function”) on a dense open locus.