Joint IAS/PU Arithmetic Geometry

Local Langlands correspondence (LLC) is a conjectural finite-to-one map from representations of a p-adic reductive group G to the set of L-parameters for G. Recently there have been two major advances in this area: Kaletha's characterization of the LLC in terms of explicit character formulas and harmonic analysis, and then a more geometric construction, due to Fargues-Scholze, involving excursion operators. This talk represents work in progress on what happens when you marry these approaches: you can write down an expected formula for the excursion operator, assuming Kaletha's formulas. Interestingly, the excursion operator is convolution with a function, the "excursion function", which comes entirely out of harmonic analysis.