Integral homology of loop groups via Langlands dual groups (Q2889328)

Let \(K\) be a connected compact Lie group and \(G\) its complexification. The authors determine the cohomology of \(\Omega K\) with coefficients in \(\mathbb Z\) in a perspective pioneered by Ginzburg that we recall first. Let \({\mathcal Gr}_{G}\) be the complex affine Grassmannian, \(G^{\vee}\) the Langlands dual group scheme of \(G\), defined over \(\mathbb Z\), and \({\mathfrak g}^{\vee}_{\mathbb C}\) the Lie algebra of \(G^{\vee}_{\mathbb C}\). Let \(e\) be a regular nilpotent element in \({\mathfrak g}^{\vee}_{\mathbb C}\), associated to the first Chern class of the determinant line bundle on \({\mathcal Gr}_{G}\). In [``Perverse sheaves on a loop group and Langlands' duality'', \url{arXiv:alg-geom/9511007}], \textit{V. Ginzburg} proved the existence of an isomorphism of Hopf algebras, \(H^*({\mathcal Gr}_{G};{\mathbb C})\cong U({\mathfrak g}^{\vee}_{{\mathbb C},e})\), where \({\mathfrak g}^{\vee}_{{\mathbb C},e}\) is the centralizer of \(e\) in \({\mathfrak g}^{\vee}_{{\mathbb C}}\).NEWLINENEWLINEThe main result of the paper under review is an extension of Ginzburg's theorem to the cohomology of \({\mathcal Gr}_{G}\) with coefficients in \(\mathbb Z\) that can be stated as follows.NEWLINENEWLINETheorem: Let \(G\) be a reductive connected group over \(\mathbb C\) such that its derived group, \(G^{\text{der}}\), is almost simple. Let \(K\) be a maximal compact subgroup of \(G\). Let \(\ell_{G}\) be the square of the ratio of the lengths of long roots and the short roots of \(G\) (so \(\ell_{G}=1\),\,2 {or} 3). Then there is a canonical isomorphism of group schemes over \({\mathbb Z}[1/\ell_{G}]\), \({\text{Spec}} \,H_{*}(\Omega K,{\mathbb Z}[1/\ell_{G}])\cong B_{e}^{\vee}[1/\ell_{G}]\), where \(B^{\vee}\) is a fixed Borel subgroup of \(G^{\vee}\), \(e\in {\text Lie}\,B^{\vee}\) is a well determined regular nilpotent element and \(B_{e}^{\vee}\) is the centralizer of \(e\) in \(B^{\vee}\).