We study the completed cohomology Ĥ of a definite unitary group G in two variables associated with a CM-extension K/F . When the prime p splits, we prove that (under technical asumptions) the p-adic local Langlands correspondence for GL2(Qp) occurs in Ĥ. As an application, we obtain a result towards the Fontaine-Mazur conjecture over K: if x is a point on the eigenvariety such that ρx is geomet...

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...

Characteristic classes of Schubert varieties can be used to study the geometry and combinatorics homogeneous spaces. We prove a relation between elliptic on generalized full flag variety those its Langlands dual. This new symmetry is motivated by 3D mirror symmetry, it only revealed if calculus elevated from cohomology or K theory level.

By Langlands duality one usually understands a correspondence between automorphic representations of a reductive group G over the ring of adels of a field F , and homomorphisms from the Galois group Gal(F/F ) to the Langlands dual group GL. It was originally introduced in the case when F is a number field or the field of rational functions on a curve over a finite field [1]. Recently A. Beilins...