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].

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].

Reviewer: Olaf Teschke (Berlin)

##### 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 |