Loop spaces and representations
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 \(\mathcal{D}\)-modules. The proof consists of two parts. First, the authors make the relation between \(\mathcal{S}t^u/G\) and the loop space of \(B\setminus G/B\) precise and then use their own general results on loop spaces and connections from [J. Topol. 5, No. 2, 377--430 (2012; Zbl 1246.14027)].NEWLINENEWLINESimilarly to the above, the authors also relate \(\mathcal{D}\)-modules on the nilpotent cone with coherent sheaves on the commuting variety. Furthermore, they prove that the categorical Langlands parameters for real groups arise naturally from the study of loop spaces of flag varieties and their Jordan decomposition.NEWLINENEWLINEThe paper is organized as follows. After a nice introduction giving a lot of background and motivation, several results on loop spaces from [loc. cit.] are recalled in Section 2. In the following section the authors explain how one can view equivariant Steinberg varieties as loop spaces, while Section 4 is devoted to Langlands parameters.
scientific article; zbMATH DE number 6188779