# Loop Spaces and Langlands Parameters

@article{BenZvi2007LoopSA, title={Loop Spaces and Langlands Parameters}, author={David Ben-Zvi and David Nadler}, journal={arXiv: Representation Theory}, year={2007} }

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…

## 13 Citations

### Loop spaces and representations

- Mathematics
- 2010

We introduce loop spaces (in the sense of derived algebraic geometry) into the representation theory of reductive groups. In particular, we apply the theory developed in our previous paper…

### Loop spaces and connections

- Mathematics
- 2012

We examine the geometry of loop spaces in derived algebraic geometry and extend in several directions the well‐known connection between rotation of loops and the de Rham differential. Our main…

### Chern Character, Loop Spaces and Derived Algebraic Geometry

- Mathematics
- 2008

In this note we present a work in progress whose main purpose is to establish a categorified version of sheaf theory. We present a notion of derived categorical sheaves, which is a categorified…

### Betti Geometric Langlands

- MathematicsAlgebraic Geometry: Salt Lake City 2015
- 2018

We introduce and survey a Betti form of the geometric Langlands conjecture, parallel to the de Rham form developed by Beilinson-Drinfeld and Arinkin-Gaitsgory, and the Dolbeault form of…

### GEOMETRICITY OF THE HODGE FILTRATION ON THE ∞-STACK OF PERFECT COMPLEXES OVER X

- Mathematics
- 2005

We construct a locally geometric ∞-stack MHod(X, Perf) of perfect complexes with λ-connection structure on a smooth projective variety X. This maps to A1/Gm, so it can be considered as the Hodge…

### Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry

- Mathematics
- 2008

We study the interaction between geometric operations on stacks and algebraic operations on their categories of sheaves. We work in the general setting of derived algebraic geometry: our basic…

### Equivariant Satake category and Kostant-Whittaker reduction

- Mathematics
- 2007

We explain (following V. Drinfeld) how the equivariant derived category of the affine Grassmannian can be described in terms of coherent sheaves on the Langlands dual Lie algebra equivariant with…

### Geometricity of the Hodge filtration on the $\infty$-stack of perfect complexes over $X_{DR}$

- Mathematics
- 2005

We construct a locally geometric $\infty$-stack $M_{Hod}(X,Perf)$ of perfect complexes with $\lambda$-connection structure on a smooth projective variety $X$. This maps to $A ^1 / G_m$, so it can be…

### emes HKR multiplicatifs

- Mathematics
- 2011

This work establishes a comparison between functions on derived loop spaces (Toen and Vezzosi, Chern character, loop spaces and derived algebraic geometry, in Algebraic topol- ogy: the Abel symposium…

### Workshop on the homotopy theory of homotopy theories

- Mathematics
- 2011

These notes are from a series of lectures given at the Workshop on the Homotopy Theory of Homotopy Theories which took place in Caesarea, Israel, in May 2010. The workshop was organized by David…

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We explain (following V. Drinfeld) how the equivariant derived category of the affine Grassmannian can be described in terms of coherent sheaves on the Langlands dual Lie algebra equivariant with…

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