This paper begins a series studying D-modules on the Feigin-Frenkel semi-infinite flag variety from the perspective of the Beilinson-Drinfeld factorization (or chiral) theory. Here we calculate Whittaker-twisted cohomology groups of Zastava spaces, which are certain finite-dimensional subvarieties of the affine Grassmannian. We show that such cohomology groups realize the nilradical of a Borel ...

We generalize a theorem of Tate and show that the second cohomology of the Weil group of a global or local field with coefficients in C∗ (or, more generally, with coefficients in the complex points of an algebraic torus over C) vanish, where the cohomology groups are defined using measurable cochains in the sense of Moore. We recover a theorem of Labesse stating that the admissible homomorphism...

Abstract Kottwitz’s conjecture describes the contribution of a supercuspidal representation to cohomology local Shimura variety in terms Langlands correspondence. A natural extension this concerns Scholze’s more general spaces shtukas. Using new Lefschetz–Verdier trace formula for v-stacks, we prove extended conjecture, disregarding action Weil group, and modulo virtual whose character vanishes...

This is an introduction to the work of Beilinson and Drinfeld [6] on the Langlands program. Mathematics Subject Classification (2000). Primary 11R39; Secondary 14D20.

We initiate the study of harmonic Cheeger-Simons characters, with applications to smooth versions of the Geometric Langlands program in the abelian case.

After explaining the concepts of Langlands dual and miniscule representations, we define an analog of the Gauss sum for any compact, simple Lie group with a simply laced Lie algebra. We then show a reciprocity property when a Lie group is exchanged with its Langlands dual. We also explore the relation with theta functions and modular transformations. In the non-simply laced case, we construct a...

In this paper we give an overview over the basic features of the continuous representation theory of p-adic Lie groups as it has emerged during the last five years. The main motivation for developing such a theory is a possible extension of the local Langlands program to p-adic Galois representations. This is still very much in its infancy. But in the last section we will describe a first appro...

The arguments of this article support the view that in TGD Universe number theoretic and geometric Langlands conjectures could be understood very naturally. The basic notions are following. 1. Zero energy ontology (ZEO) and the related notion of causal diamond CD (CD is short hand for the cartesian product of causal diamond of M and of CP2). ZEO leads to the notion of partonic 2-surfaces at the...

Two of the primary methods of constructing automorphic forms are the Langlands program and Howe's theory of dual pairs. The Langlands program concerns a reductive linear group G deened over a number eld. Associated to G is its dual group L G. The con-jectural principle of functoriality says that a homomorphism L H ! L G should provide a \transfer" of automorphic representations from H to those ...