Harish-Chandra Characters and the Langlands Conjectures (March 13, 2025)

In this 54-minute lecture, explore how Harish-Chandra characters connect to the Langlands Conjectures as presented by Tasho Kaletha at the 2025 Simons Collaboration on Perfection in Algebra, Geometry and Topology Annual Meeting. Discover the fundamental theorem of Harish-Chandra that assigns a conjugation-invariant function (character) to irreducible complex-valued representations of reductive real or p-adic groups, uniquely determining the representation. Learn how these characters are reflected in sheaf theory on the space of G-bundles on the Fargues-Fontaine curve for p-adic groups. Examine the explicit formulas for discrete series representations—long established for real groups by Harish-Chandra but only recently developed for p-adic groups—and understand the remarkable parallels between real and p-adic cases. See how these developments enable both characterization and construction of the refined local Langlands correspondence under specific base field assumptions.