The Beilinson-Bernstein correspondence for quantized enveloping algebras

Affine Beilinson-Bernstein localization at the critical level for $GL_2$

We prove the rank 1 case of a conjecture of Frenkel-Gaitsgory: critical level Kac-Moody representations with regular central characters localize onto the affine Grassmannian. The method uses an analogue in local geometric Langlands of the existence of Whittaker models for most representations of $GL_2$ over a non-Archimedean field.

Beilinson-Bernstein localization over the Harish-Chandra center

We present a simple proof of a strengthening of the derived Beilinson-Bernstein localization theorem using the formalism of descent in derived algebraic geometry. The arguments and results apply to arbitrary modules without the need to fix infinitesimal character. Roughly speaking, we demonstrate that all Ug-modules are the invariants, or equivalently coinvariants, of the action of intertwining functors (a refined form of Weyl group symmetry). This is a quantum version of descent for the Grothendieck-Springer simultaneous resolution. In an appendix we present an alternative perspective, which …

On the Bernstein-Gelfand-Gelfand resolution for Kac-Moody algebras and quantized enveloping algebras

A Bernstein-Gelfand-Gelfand resolution for arbitrary Kac-Moody algebras and arbitrary subsets of the set of simple roots is proven. Moreover, quantum group analogs of the Bernstein-Gelfand-Gelfand resolution for symmetrizable Kac-Moody algebras are established. For quantized enveloping algebras with fixed complex nonzero deformation parameter q exactness is proven for all q which are not a root of unity.

A Beilinson-Bernstein Theorem for analytic quantum groups

We introduce a $p$-adic analytic analogue of Backelin and Kremnizer's construction of the quantum flag variety of a semisimple algebraic group, when $q$ is not a root of unity and $| q-1|<1$. We then define a category of $λ$-twisted $D$-modules on this analytic quantum flag variety. We show that when $λ$ is regular and dominant and when the characteristic of the residue field does not divide the order of the Weyl group, the global section functor gives an equivalence of categories between the coherent $λ$-twisted $D$-modules and the category of finitely generated modules over $\widehat{U_q^λ}$…

The Beilinson Equivalence for Differential Operators and Lie Algebroids

Let D be the ring of differential operators on a smooth irreducible affine variety X over the complex numbers; or, more generally, the enveloping algebra of any locally free Lie algebroid on X. The category of finitely-generated graded modules of the Rees algebra D has a natural quotient category qgr(D) which imitates the category of modules on Proj of a graded commutative ring. We show that the derived category D^b(qgr(D)) is equivalent to the derived category of finitely-generated modules of a sheaf of algebras E on X which is coherent over X. This generalizes the usual Beilinson equivalence…

论极小抛物诱导

On the minimal parabolic induction

本文受 Beilinson-Bernstein 证明 Jantzen 猜想的启发，为 Kac-Moody 代数定义极小抛物诱导函子并建立其基本性质，进而研究单最高权模的扩张群与零化子。