Related papers: The Global Jacquet-Langlands Correspondence via Te…

In our paper arXiv:1701.03146 we established, for every simply-laced Lie algebra g, a canonical isomorphism between the spaces of deformed conformal blocks of the deformed W-algebra and the quantum affine algebra corresponding to g, which…

This is the final paper in the series of five, in which we prove the geometric Langlands conjecture (GLC). We conclude the proof of GLC by showing that there exists a unique (up to tensoring up by a vector space) Hecke eigensheaf…

In this paper, we give a method for characterizing the local Langlands conjectures in the vein of Scholze's alternate proof of the local Langlands conjecture for $\mathrm{GL}_n$. More specifically, we show that if a local Langlands…

Let F be a finite field and G=GL(6,F). In this paper, we explicitly describe a certain twisted Jacquet module of an irreducible cuspidal representation of G.

This communication is an introduction to the Langlands Program and to ($G$-) shtukas (over algebraic curves) over function fields. Modular curves and Drinfeld (elliptic) modules and shtukas are used in coding theory. From this point of view…

Lusztig \cite{L5,L6} gave a parametrization for $\rm{Irr}(G^F)$, where $G$ is a reductive algebraic group defined over $\mathbb{F}_q$, with Frobenius map $F$. This parametrization is known as Lusztig's Jordan decomposition or Lusztig…

We consider the groups G which arise from real semisimple Jordan algebras via the Tits-Koecher-Kantor construction. Such a G is characterized by the fact that it admits a parabolic subgroup P=LN which is conjugate to its opposite, and for…

Let F be a non-archimedean local field. We establish the local Langlands correspondence for all inner forms of the group $SL_n (F)$. It takes the form of a bijection between, on the one hand, conjugacy classes of Langlands parameters for…

In this paper, we obtain Atkin--Lehner decompositions for spaces of modular forms on definite quaternion algebras. Similar to Casselman's approach our methods are representation theoretic. Using Jacquet--Langlands correspondence we also…

The global geometric Langlands correspondence relates Hecke eigensheaves on the moduli stack of G-bundles on a smooth projective algebraic curve X and holomorphic G'-bundles with connection on X, where G' is the Langlands dual group of G.…

Motivated by some results in classical differential geometry, we give a constructive procedure for building up a connection over a (twisted) tensor product of two algebras, starting from connections defined on the factors. The curvature for…

We study the two-dimensional twisted (0,2) sigma-model on various smooth complex flag manifolds G/B, and explore its relevance to the geometric Langlands program. We find that an equivalence - at the level of the holomorphic chiral algebra…

We introduce graded Hecke algebras H based on a (possibly disconnected) complex reductive group G and a cuspidal local system L on a unipotent orbit of a Levi subgroup M of G. These generalize the graded Hecke algebras defined and…

We formulate a two-parameter generalization of the geometric Langlands correspondence, which we prove for all simply-laced Lie algebras. It identifies the q-conformal blocks of the quantum affine algebra and the deformed W-algebra…

We obtain a faithful representation of the twisted tensor product $B\otimes_{\chi} A$ of unital associative algebras, when $B$ is finite dimensional. This generalizes the representations of [C] where $B=K[X]/<X^2-X>$, [GGV] where…

We use a ${\mathcal B}$-adic completion and the $p$-adic local Langlands correspondence for ${\mathrm {GL}}_2({\mathbf Q}_p )$ to give a construction of Kisin's rings and the attached universal Galois representations (in dimension 2 and for…

We formulate a conjecture on local geometric Langlands for supercuspidal representations using Yu's data and Feigin-Frenkel isomorphism. We refine our conjecture for a large family of regular supercuspidal representations defined by…

When $G$ is a complex reductive algebraic group and $G/K$ is a reductive symmetric space, the decomposition of $\C[G/K]$ as a $K$-module was obtained (in a non-constructive way) by Richardson, generalizing the celebrated result of…

We study the tensor product decomposition of the split real quantum group $U_{q\tilde{q}}(sl(2,R))$ from the perspective of finite dimensional representation theory of compact quantum groups. It is known that the class of positive…

For a reductive group $G$ over a local non-archimedean field $K$ one can mimic the construction from the classical Deligne--Lusztig theory by using the loop space functor. We study this construction in special the case that $G$ is an inner…