Proof of the De Concini-Kac-Procesi conjecture

In this paper we prove a conjecture by De Concini, Kac and Procesi \cite{CP} (Corollary \ref{conj}): The dimension of any M\in U_q-\mood^\chi is divisible by $l^{codim_\mathcal{B}\mathcal{B}_\chi}$

A Localization Theorem for Finite W-algebras

Following the work of Beilinson-Bernstein and Kashiwara-Rouquier, we give a geometric interpretation of certain categories of modules over the finite W-algebra. As an application we reprove the Skryabin equivalence.Comment: Preliminary Version. Comments Welcom

Fr\'echet Modules and Descent

We study several aspects of the study of Ind-Banach modules over Banach rings thereby synthesizing some aspects of homological algebra and functional analysis. This includes a study of nuclear modules and of modules which are flat with respect to the projective tensor product. We also study metrizable and Fr\'{e}chet Ind-Banach modules. We give explicit descriptions of projective limits of Banach rings as ind-objects. We study exactness properties of projective tensor product with respect to kernels and countable products. As applications, we describe a theory of quasi-coherent modules in Banach algebraic geometry. We prove descent theorems for quasi-coherent modules in various analytic and arithmetic contexts.Comment: improved versio

Quantum flag varieties, equivariant quantum D-modules and localization of quantum groups

Let \Oq(G) be the algebra of quantized functions on an algebraic group $G$ and \Oq(B) its quotient algebra corresponding to a Borel subgroup $B$ of $G$. We define the category of sheaves on the "quantum flag variety of $G$" to be the \Oq(B)-equivariant \Oq(G)-modules and proves that this is a proj-category. We construct a category of equivariant quantum $\mathcal{D}$-modules on this quantized flag variety and prove the Beilinson-Bernsteins localization theorem for this category in the case when $q$ is not a root of unity

2-gerbes and 2-Tate spaces

We construct a central extension of the group of automorphisms of a 2-Tate vector space viewed as a discrete 2-group. This is done using an action of this 2-group on a 2-gerbe of gerbel theories. This central extension is used to define central extensions of double loop groups.Comment: Uses Paul Taylor`s diagram