Geometric Langlands and mathematical physics

该建议有四个部分：1。Hecke eigen-functions over non-archimedian local fields（with D. Kazhdan）.设X是有限域k上的光滑射影曲线，G是可裂约化群.群G和X上有理函数整体域k（X）的自同态形式的经典理论的非分歧部分研究了所谓的Hecke算子的特征函数，这些算子作用于X上G-丛的模空间BunG_G（X）的k-点上的函数。朗兰兹猜想描述了X上l-adic局部系统的特征值，其结构群为G ^-G的朗兰兹对偶群。在我和D. Kazhdan的一个老工作中，如果用局部域替换k，并且如果用Bun_G（X）上的K点的半形式工作，则可以再次定义Hecke算子。对于阿基米德K Etingof，Frenkel和Kazhdan证明了这些算子作用在L^2半型空间上，并且具有离散的公共谱。他们用G ^来描述光谱（D. Gaiotto用量子场论得到了类似的结果）。我们建议研究非阿基米德K的类似问题。我们已经得到了G = SL（2）的部分结果。2. 3d和4d量子场论中的某些范畴等价（与M. Finkelberg和R. Travkin合著）。我们想证明D.Gaiotto的一个猜想（由物理学论证激发），该猜想描述了各种量子超群（例如量子化GL（N| M））在某个代数群的仿射格拉斯曼群上的某些层范畴上的性质。我们还解释了这个问题的潜在应用，局部几何朗兰兹对应。我们还讨论了一些其他的等价的范畴也动机的物理考虑。3.渐近Hecke代数的范畴化（与D. Kazhdan和S. Dawydiak合著）附着在约化群G上的仿射Hecke代数H是几何表示论的中心对象之一。它是一个代数环上的洛朗多项式在一些变量q（让我们称之为A）。在Kazhdan和Lusztig的著名工作之后，Bezrukavnikov定义了H的两个等价范畴（即K-环同构于H的monoidal范畴），并证明了它们的等价性。一种分类是根据G的仿射旗簇上的某些可构造层进行的。另一个是关于G ^的所谓导出Steinberg簇上的相干层。另一方面，Lusztig定义了一个整数上的代数J，它与H有着非常有趣的关系。我们建议定义J的类似的概念，这将有助于理解它与H的关系。

The proposal has 4 parts: 1. Hecke eigen-functions over non-archimedian local fields (with D.Kazhdan). Let X be a smooth projective curve curve over a finite field k and let G be a split reductive group. The unramified part of the classical theory of autormorphic forms for the group G and the global field k(X) of rational functions on X studies eigen-functions of the so called Hecke operators which act on functions on k-points of the moduli space BunG_G(X) of G-bundles on X. The (now proved in this case) Langlands conjecture describes the eigen-values in terms of l-adic local systems on X with structure group G^ - the Langlands dual group of G. It was shown in an old work of myself and D.Kazhdan that if one replaces k by a local field, and if one works with half-forms on K-points on Bun_G(X), one can again define Hecke operators. For archimedian K Etingof, Frenkel and Kazhdan conjectured that these operators act on the space of L^2 half-forms and have discrete common spectrum. They conjecturally describe the spectrum in terms of G^ (similar conjectures were produced by D.Gaiotto using quantum field theory). We propose to study similar problem for non-archimedian K. We have already obtained some partial results for G=SL(2). 2. Some category equivalences arising from 3d and 4d quantum field theories (with M.Finkelberg and R.Travkin). We would like to prove a conjecture of D.Gaiotto (motivated by physics arguments) describing categories of representations of various quantum super-groups (e.g. quantized GL(N|M)) in terms of certain categories of sheaves on the affine Grassmannian of some algebraic group. We also explain potential applications of this problem to local geometric Langlands correspondence. We discuss some other equivalences of categories also motivated by physical considerations. 3. Categorification of asymptotic Hecke algebra (with D.Kazhdan and S.Dawydiak) The affine Hecke algebra H attached to a reductive group G is one of the central objects of geometric representation theory. It is an algebra over the ring of Laurent polynomials in some variable q (let us call it A). Following a famous work of Kazhdan and Lusztig, Bezrukavnikov defined two categorifications of H (i.e. monoidal categories whose K-ring is isomorphic to H) and proved their equivalence. One categorification is in terms of certan constructible sheaves on the affine flag variety of G. The other is in terms of coherent sheaves on the so called derived Steinberg variety of G^. On the other hand, Lusztig has defined certain algebra J over the integers which has some very intriguing relation with H. We propose to define similar categorifications of J. This should help understand its relation with H.