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Semi-infinite flags. I: Case of global curve $$\mathbb{P}^1$$. (English) Zbl 1076.14512
Astashkevich, Alexander (ed.) et al., Differential topology, infinite-dimensional Lie algebras, and applications. D. B. Fuchs’ 60th anniversary collection. Providence, RI: American Mathematical Society (ISBN 0-8218-2032-X/hbk). Transl., Ser. 2, Am. Math. Soc. 194(44), 81-112 (1999).
The paper under review is the first part of a series of two papers, the second one being [B. Feigin, M. Finkelberg, A. Kusnetsov and I. Mirkovic, in: Differential topology, infinite-dimensional Lie algebras, and applications. D. B. Fuchs’ 60th anniversary collection. AMS Transl., Ser. 2, Am. Math. Soc. 194(44), 113–148 (1999; Zbl 1076.14511)], where semi-infinite flag varieties are studied together with perverse sheaves on them. For a Borel subgroup $$B\subset G$$, the notion of the semi-infinite flag space $$\mathcal Z$$ was introduced by B. L. Feigin and E. V. Frenkel [Commun. Math. Phys. 128, No. 1, 161–189 (1990; Zbl 0722.17019)] as the quotient of $$G((z))$$ modulo the connected component of $$B((z))$$. The authors are interested in the space $$\mathcal{PS}$$ of perverse sheaves on $$\mathcal Z$$ equivariant with respect to the Iwahori subgroup $$I\subset G[[z]]$$.
The first part is devoted to the construction of $$\mathcal{PS}$$ as an ind-scheme, using spaces $$\mathcal Q^\alpha$$ of “quasi-maps” from $$\mathbb{P}^1$$ to the flag variety of $$G$$ (which are Drinfeld compactifications of maps of degree $$\alpha\in H^2(\mathcal B)$$. Then, a system of of subvarieties $$\mathcal Z^\alpha \subset \mathcal Q^\alpha$$ is considered, which consists of quasi-maps $$f: \mathbb{P}^1\to\mathcal B$$ defined at $$\infty$$ fulfilling $$f(\infty)=B_-$$. Then $$\mathcal{PS}$$ is constructed as collections of perverse sheaves on $$\mathcal Z^\alpha$$, together with so-called factorizations [cf. R. Bezrukavnikov, M. Finkelberg and V. Schechtman, “Factorizable sheaves and quantum groups”, Lect. Notes Math. 1691 (1998; Zbl 0938.17016)].
In the third chapter, a convolution functor from the category of perverse sheaves on $$G((z))/G[[z]]$$, constant along the orbits of $$I$$, to $$\mathcal{PS}$$ is constructed. This serves as a geometric counterpart of V. Ginzburg’s [Perverse sheaves on a Loop group and Langlands’ duality, preprint, http://arxiv.org/abs/alg-geom/9511007] restriction functor to the principal block in the category of modules over the quantum group at a root of unity, which has been conjectured to be equivalent to $$\mathcal{PS}$$.
For the entire collection see [Zbl 0921.00044].

##### MSC:
 14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) 14M15 Grassmannians, Schubert varieties, flag manifolds 20G10 Cohomology theory for linear algebraic groups 20G42 Quantum groups (quantized function algebras) and their representations