The geometric Langlands correspondence has been interpreted as the mirror symmetry of the Hitchin fibrations for two dual reductive groups. This mirror symmetry, in turn, reduces to T–duality on the generic Hitchin fibers, which are smooth tori. In this paper we study what happens when the Hitchin fibers on the B-model side develop orbifold singularities. These singularities correspond to local...

For a real semisimple group of matrices, every unitary principal series representation splits into inequivalent irreducible representations [5]. This multiplicity-one result was conjectured [7] in 1971 because one expected in general and knew in some special cases that all reducibility was accounted for by canonical geometric constructions, such as the imbedding of a space of analytic functions...

It is surprising that the following question has not been addressed in the literature: what is the contragredient in terms of Langlands parameters? Thus suppose G is a connected, reductive algebraic group defined over a local field F , and G(F ) is its F -points. According to the local Langlands conjecture, associated to a homomorphism φ from the Weil-Deligne group of F into the L-group of G(F ...

It is surprising that the following question has not been addressed in the literature: what is the contragredient in terms of Langlands parameters? Thus suppose G is a connected, reductive algebraic group defined over a local field F , and G(F ) is its F -points. According to the local Langlands conjecture, associated to an admissible homomorphism φ from the WeilDeligne group of F into the L-gr...

These notes are a rough write-up of lectures presented at the workshop on “Geometry and Integrability”, held at the University of Melbourne, 6-15 Feb, 2008. My interest in Geometric Representation Theory stems from recent developments in the Geometric Langlands program. These lectures are an introduction into material which are a prerequisite to anyone interested in studying the Geometric Langl...

We expose a new procedure of quantization of fields, based on the Geometric Langlands Correspondence. Starting from fields in the target space, we first reduce them to the case of fields on one-complex-variable target space, at the same time increasing the possible symmetry group G. Use the sigma model and momentum maps, we reduce the problem to a problem of quantization of trivial vector bundl...

This article describes ways in which to get an overview of some of the goals and methods of Langlands program, and it points to treatments of some examples that are worth keeping in mind.

The representation theory of reductive groups, such as the group GLn of invertible complex matrices, is an important topic, with applications to number theory, algebraic geometry, mathematical physics, and quantum topology. One way to study this representation theory is through the geometric Satake correspondence (also known as geometric Langlands duality). This correspondence relates the geome...

Suppose G(R) is a real reductive group, with L-group ∨ G, and φ : WR → ∨ G is an L-homomorphism. There is a close relationship between the Lpacket associated to φ and characters the component group of the centralizer of the image of φ. This was originally observed by Shelstad [10], and further refined by Arthur, Kottwitz, Langlands and Shelstad. This is reinterpreted in [3], to (among other thi...

The classical Gindikin-Karpelevich formula appears in Langlands’ calculation of the constant terms of Eisenstein series on reductive groups and in Macdonald’s work on p-adic groups and affine Hecke algebras. The formula has been generalized in the work of Garland to the affine Kac-Moody case, and the affine case has been geometrically constructed in a recent paper of Braverman, Finkelberg, and ...