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Integral homology of loop groups via Langlands dual groups. (English) Zbl 1267.57041
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”, arXiv:alg-geom/9511007], 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 $${\mathfrak g}^{\vee}_{{\mathbb C}}$$.
The 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.
Theorem: 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 $$B^{\vee}$$.

##### MSC:
 57T10 Homology and cohomology of Lie groups 20G07 Structure theory for linear algebraic groups
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##### References:
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