Quantum geometric Langlands correspondence in positive characteristic: the GL(N) case

We prove a version of quantum geometric Langlands conjecture in characteristic $p$. Namely, we construct an equivalence of certain localizations of derived categories of twisted crystalline $\mathcal D$-modules on the stack of rank $N$ vector bundles on an algebraic curve $C$ in characteristic $p$. The twisting parameters are related in the way predicted by the conjecture, and are assumed to be irrational (i.e., not in $\mathbb F_p$). We thus extend the results of arXiv:math/0602255 concerning the similar problem for the usual (non-quantum) geometric Langlands. In the course of the proof, we introduce a generalization of $p$-curvature for line bundles with non-flat connections, define quantum analogs of Hecke functors in characteristic $p$ and construct a Liouville vector field on the space of de Rham local systems on $C$.Comment: 57 pages, to appear in Duke Math Journa

Quantization of Hitchin integrable system via positive characteristic

In a celebrated unpublished manuscript Beilinson and Drinfeld quantize the Hitchin integrable system by showing that the global sections of critically twisted differential operators on the moduli stack of G-bundles on an algebraic curve is identified with the ring of regular functions on the space of G-opers; they deduce existence of an automorphic D-module corresponding to a local system carrying a structure of an oper. In this note we show for G=GL(n) that those results admit a short proof by reduction to positive characteristic, where the result is deduced from generic Langlands duality established earlier by the first author and A. Braverman. The appendix contains a proof of some properties of the p-curvature map restricted to the space of opers.Comment: paper by Roman Bezrukavnikov and Roman Travkin with an appendix by Roman Bezrukavnikov, Tsao-Hsien Chen and Xinwen Zhu. 13 page

Quantum geometric Langlands correspondence in positive characteristic: the GLN case

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012.In title on title page, "N" of GLN appears as subscript of upper case letter N. Cataloged from PDF version of thesis.Includes bibliographical references (p. 73).Let C be a smooth connected projective curve of genus > 1 over an algebraically closed field k of characteristic p > 0, and c [epsilon] k \ Fp. Let BunN be the stack of rank N vector bundles on C and Ldet the line bundle on BunN given by determinant of derived global sections. In this thesis, we construct an equivalence of derived categories of modules for certain localizations of the twisted crystalline differential operator algebras DBunNet and DBunN, L-1/cdet The first step of the argument is the same as that of [BB] for the non-quantum case: based on the Azumaya property of crystalline differential operators, the equivalence is constructed as a twisted version of Fourier-Mukai transform on the Hitchin fibration. However, there are some new ingredients. Along the way we introduce a generalization of p-curvature for line bundles with non-flat connections, and construct a Liouville vector field on the space of de Rham local systems on C.by Roman Travkin.Ph.D

Lagrangian subvarieties of hyperspherical varieties

Given a hyperspherical $G$-variety $\mathscr X$ we consider the zero moment level $\Lambda_{\mathscr X}\subset{\mathscr X}$ of the action of a Borel subgroup $B\subset G$. We conjecture that $\Lambda_{\mathscr X}$ is Lagrangian. For the dual $G^\vee$-variety ${\mathscr X}^\vee$, we conjecture that that there is a bijection between the sets of irreducible components $\mathrm{Irr}\Lambda_{\mathscr X}$ and $\mathrm{Irr}\Lambda_{{\mathscr X}^\vee}$. We check this conjecture for all the hyperspherical equivariant slices, and for all the basic classical Lie superalgebras

Orthosymplectic Satake equivalence

This is a companion paper of arXiv:1909.11492. We prove an equivalence relating representations of a degenerate orthosymplectic supergroup with the category of $SO(N-1,{\mathbb C}[\![t]\!])$-equivariant perverse sheaves on the affine Grassmannian of $SO_N$. We explain how this equivalence fits into a more general framework of conjectures due to Gaiotto and to Ben-Zvi, Sakellaridis and Venkatesh.Comment: v2: 31 pages, a few minor corrections; Conjecture 3.3.4 became Theorem 3.3.5. v3: 32 pages, proof of Lemma 2.3.3 correcte

Mirabolic affine Grassmannian and character sheaves

We compute the Frobenius trace functions of mirabolic character sheaves defined over a finite field. The answer is given in terms of the character values of general linear groups over the finite field, and the structure constants of multiplication in the mirabolic Hall-Littlewood basis of symmetric functions, introduced by Shoji.Comment: 22 pages. The final version to appear in Selecta Mat

Mirabolic Robinson-Shensted-Knuth correspondence, preprint math/0802.1651. Michael Finkelberg, Victor Ginzburg and Roman Travkin Michael Finkelberg IMU, IITP and State University Higher School of Economy Mathematics Department, rm. 517 20 Myasnitskaya st.

Abstract. The set of orbits of GL(V) in Fl(V) × Fl(V) × V is finite, and is parametrized by the set of certain decorated permutations in a work of Magyar, Weyman, Zelevinsky. We describe a Mirabolic RSK correspondence (bijective) between this set of decorated permutations and the set of triples: a pair of standard Young tableaux, and an extra partition. It gives rise to a partition of the set of orbits into combinatorial cells. We prove that the same partition is given by the type of a general conormal vector to an orbit. We conjecture that the same partition is given by the bimodule Kazhdan-Lusztig cells in the bimodule over the Iwahori-Hecke algebra of GL(V) arising from Fl(V) ×Fl(V) ×V. We also give conjectural applications to the classification of unipotent mirabolic character sheaves on GL(V) × V