Let G be a split reductive algebraic group over a non-archimedean local field. We study the representation theory of a central extension G̃ of G by a cyclic group of order n, under some mild tameness assumptions on n. In particular, we focus our attention on the development of the theory of principal series representations for G̃ and its applications to the study of Hecke algebras via a Satake is...

We reprove the theorem of Feigin and Frenkel relating the center of the critical level enveloping algebra of the Kac-Moody algebra for a semisimple Lie algebra to opers (which are certain de Rham local systems with extra structure) for the Langlands dual group. Our proof incorporates a construction of Beilinson and Drinfeld relating the Feigin-Frenkel isomorphism to (more classical) Langlands d...

In this paper, we prove that there is at most one correspondence between parahoric-spherical representations and semisimple local Langlands parameters which satisfies certain natural properties. Our proof of uniqueness statement very formal. particular, the constructed by Fargues Scholze yield unique candidate for such representations. As a corollary, conjecture posits compatibility with paraho...

In this article, we distinguish Siegel cuspidal eigenforms of degree two on the full symplectic group from signs their Hecke eigenvalues. To establish our theorem, obtain a result towards simultaneous sign changes eigenvalues eigenforms. course proof, also prove that Satake p-parameters different are distinct for set primes p density 1. The main ingredient to latter is theory Galois representat...