Loop spaces and representations
type: Item
instance of: scholarly article
title: Loop spaces and representations
zbMATH Open document ID: 1271.22018
DOI: 10.1215/00127094-2266130
author: David Ben-Zvi
author: David Nadler
published in: Duke Mathematical Journal
publication date: 2013-07-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/1004.5120
full work available at URL: https://projecteuclid.org/euclid.dmj/1370955540
reviewed by: Pawel Sosna
Mathematics Subject Classification ID: 22E57
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 55P35
zbMATH DE Number: 6188779
zbMATH Keywords: differential graded categories
zbMATH Keywords: loop spaces
zbMATH Keywords: reductive groups
zbMATH Keywords: representations
zbMATH Keywords: Langlands parameters
MaRDI profile type: Publication
OpenAlex ID: W3102671783
arXiv ID: 1004.5120
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review text: The main purpose of the paper under review is to introduce loop spaces in the sense of derived algebraic geometry into the representation theory of reductive groups. For instance, the first main result can be roughly summarised as follows.NEWLINENEWLINELet \(G\) be a connected reductive complex algebraic group, \(\mathcal{B}\) be the flag variety parametrizing Borel subgroups \(B\) of \(G\) and consider the dg-category \(\mathcal{H}_G\) (by this the authors mean a small pre-triangulated \(\mathbb{C}\)-linear differential graded category considered as an object of some appropriate \(\infty\)-category) of \(B\)-equivariant \(\mathcal{D}\)-modules on the flag variety \(\mathcal{B}\cong G/B\). Furthermore, consider the unipotent Steinberg variety \(\mathcal{S}t^u\) consisting of elements \((g,B_1,B_2)\) of \(G^u\times \mathcal{B}\times \mathcal{B}\) (where \(G^u\) is the formal scheme given by the formal neighborhood of the unipotent elements of \(G\)) satisfying \(g\in B_1\cap B_2\). Note that there exists a canonical closed embedding \(\mathcal{B}\times\mathcal{B}\hookrightarrow \mathcal{S}t^u\). The dg-category of \(\mathbb{G}_m\)-equivariant coherent sheaves on \(\mathcal{S}t^u/G\) which are coherent along \(\mathcal{B}\times\mathcal{B}/G\) is denoted by \(\mathcal{H}_G^{\mathrm{aff}}\). Now, \(\mathcal{S}t^u/G\) is a loop space in the above sense and so carries an action of \(S^1\). In fact, there is an action of \(\mathbb{S}\), the semi-direct product of \(\mathbb{G}_m\) and the affinization of \(S^1\). Furthermore, the category \(\mathcal{C}\) of \(\mathbb{S}\)-equivariant objects is linear over \(\mathcal{O}(BS^1)\cong k[u]\), so it makes sense to localize it with respect to the action of the generator \(u\). The first main result of the paper states that there is a canonical equivalence between \(\mathcal{H}_G\) and the localized \(\mathbb{S}\)-invariants of \(\mathcal{H}_G^{\mathrm{aff}}\). Note that the result relates quasi-coherent sheaves and \(\mathcal{D}\…
description: scientific article; zbMATH DE number 6188779