Searcharxiv⌕ Search

arXiv subjects

David Nadler

Publications and source records attributed to David Nadler.

59 records · Page 4Linked to original sources
arXiv2007

Loop Spaces and Langlands Parameters

We apply the technique of S^1-equivariant localization to sheaves on loop spaces in derived algebraic geometry, and obtain a fundamental link between two families of categories at the heart of geometric representation theory. Namely, we categorify the well known relationship between free loop spaces, cyclic homology and de Rham cohomology to recover the category of D-modules on a smooth stack X as a localization of the category of S^1-equivariant coherent sheaves on its loop space LX. The main observation is that this procedure connects categories of equivariant D-modules on flag varieties with categories of equivariant coherent sheaves on the Steinberg variety and its relatives. This provides a direct connection between the geometry of finite and affine Hecke algebras and braid groups, and a uniform geometric construction of all of the categorical parameters for representations of real and complex reductive groups. This paper forms the first step in a project to apply the geometric Langlands program to the complex and real local Langlands programs, which we describe.

math.RT↗
arXiv2006

Morse theory and tilting sheaves

We give a geometric construction of tilting perverse sheaves using stratified Morse theory, torus actions, and nearby cycles.

math.RT↗
arXiv2004

Perverse Sheaves on Real Loop Grassmannians

The aim of this paper is to identify a certain tensor category of perverse sheaves on the real loop Grassmannian of a real form $G_{\mathbb R}$ of a connected reductive complex algebraic group $G$ with the category of finite-dimensional representations of a connected reductive complex algebraic subgroup $\check H$ of the dual group $\check G$. The root system of $\check H$ is closely related to the restricted root system of the real form $G_{\mathbb R}$. The fact that $\check H$ is reductive implies that an interesting family of real algebraic maps satisfies the conclusion of the Decomposition Theorem of Beilinson-Bernstein-Deligne.

math.AG↗
arXiv2004

Matsuki correspondence for the affine Grassmannian

We present a version of the Matsuki correspondence for the affine Grassmannian $Gr=G(C((t)))/G(C[[t]])$ of a connected reductive complex algebraic group $G$. The main statement is an anti-isomorphism between the orbit posets of two subgroups of $G(C((t)))$ acting on $Gr$. The first is the polynomial loop group $LG_R$ of a real form $G_R$ of $G$; the second is the loop group $K(C((t)))$ of the complexification $K$ of a maximal compact subgroup $K_c$ of $G_R$. The orbit poset itself turns out to be simple to describe.

math.AG↗