In this article we study the perverse sheaf on Drinfeld's compactification obtained by applying the geometric Jacquet functor (alias nearby cycles) to a nondegenerate Whittaker sheaf. Namely, we describe its restrictions along the defect stratification in terms of the Langlands dual Lie algebra, in particular showing that this nearby cycles sheaf is tilting. We also describe the subquotients of the monodromy filtration using the Picard-Lefschetz oscillators introduced by S. Schieder, including a proof that the subquotients are semisimple without reference to any theory of weights. This argument instead relies on the action of the Langlands dual Lie algebra on compactified Eisenstein series (or what is essentially the same, cohomology of quasimaps into the flag variety) constructed by Feigin, Finkelberg, Kuznetsov, and Mirkovic.
We prove that cuspidal automorphic D-modules have non-vanishing Whittaker coefficients, generalizing known results in the geometric Langlands program from GL_n to general reductive groups. The key tool is a microlocal interpretation of Whittaker coefficients. We establish various exactness properties in the geometric Langlands context that may be of independent interest. Specifically, we show Hecke functors are t-exact on the category of tempered D-modules, strengthening a classical result of Gaitsgory (with different hypotheses) for GL_n. We also show that Whittaker coefficient functors are t…
Whittaker Patterns in the Geometry of Moduli Spaces of Bundles on Curves
设G是有限域F_q上的可裂连通约化群，N是G的极大幂幺子群. V. Drinfeld在F_q上光滑射影曲线X上引入了N-丛的模叠加的一个显著的部分紧化。本文研究Drinfeld的模空间和它上面的一类反常层，这类反常层的定义是由群G（K）上的Whittaker函数的研究所激发的，这里K=F_q（（t））.我们证明了我们的范畴是半单的，并且这个范畴的不可约对象是“干净的”，即，它们是由0个局部系统在地层上支持的扩展。作为这些结果的应用，我们得到了Whittaker函数的Casselman-Shalika公式的一个纯几何证明。
Let G be an algebraic reductive group over a an algebraically closed field of positive characteristic. Choose a parabolic subgroup $P$ in $G$ and denote by $U$ its unipotent radical. Let $X$ be a $G$-variety. The purpose of this paper is to give two examples of a situation in which the functor of averaging of l-adic sheaves on $X$ with respect to a generic character of $U$ commutes with Verdier duality. In the first example we take $χ$ to be an arbitrary $G$-variety and we prove the above property for all $\overline{P}$-equivariant sheaves on $X$ where $\overline{P}$ is an opposite parabolic s…
Model category structures arising from Drinfeld vector bundles
给出了半分离概型$X$上拟相干层的无界链复形体范畴$\mathbb{C}(\mathfrak{Qco}(X))$上模型范畴结构的一般构造.该结构是基于兼容的过滤的各个模块的部分在开放的仿射子集的$X$。它不像以前的方法那样需要在直接限制下闭合。我们应用它来描述导出范畴$\mathbb D (\mathfrak{Qco}(X))$通过各种模型结构上的$\mathbb{C}(\mathgrak{Qco}(X))$。作为特殊的例子，我们恢复最近的结果准相干层的平坦模型结构。我们的方法还包括（无限维）向量丛的情况下，和限制平坦的Mittag-Leffler准相干层，介绍了德林费尔德。最后，我们证明了在一般情况下，无限制的情况下不会导出上述的模型范畴结构。
We study solutions to a generalized version of the classical Yang-Baxter equation (CYBE) with values in a central simple Lie algebra over a field of characteristic 0 from an algebro-geometric perspective. In particular, we describe such solutions using cohomology free sheaves of Lie algebras on projective curves. This framework leads to a new proof of the Belavin-Drinfeld trichotomy, which asserts that over the ground field $\mathbb{C}$ any non-degenerate solution of the CYBE is either elliptic, trigonometric or rational. Furthermore, we obtain new structural results for solutions of the gener…
We apply Treumann's "Smith theory for sheaves" in the context of the Iwahori--Whittaker model of the Satake category. We deduce two results in the representation theory of reductive algebraic groups over fields of positive characteristic: (a) a geometric proof of the linkage principle; (b) a character formula for tilting modules in terms of the $\ell$-canonical basis, valid in all blocks and in all characteristics.
We give cases in which nearby cycles commutes with pushforward from sheaves on the moduli stack of shtukas to a product of curves over a finite field. The proof systematically uses the property that taking nearby cycles of Satake sheaves on the Beilinson-Drinfeld Grassmannian with parahoric reduction is a central functor together with a "Zorro's lemma" argument similar to that given by Xue to prove the smoothness of cohomology sheaves at unramified places. As an application, for automorphic forms at the parahoric level, we characterize the image of tame inertia under the Langlands corresponden…
In a previous paper, the first and third authors gave an explicit realization of the geometric Langlands correspondence for hypergeometric sheaves, considered as $\textrm{GL}_n$-local systems. Certain hypergeometric local systems admit a symplectic or orthogonal structure, which can be viewed as $\check{G}$-local systems, for a classical group $\check{G}$. This article aims to realize the geometric Langlands correspondence for these $\check{G}$-local systems. We study this problem from two aspects. In the first approach, we define the hypergeometric automorphic data for a classical group $G$ i…
The standard Lie bialgebra structure on an affine Kac-Moody algebra induces a Lie bialgebra structure on the underlying loop algebra and its parabolic subalgebras. In this paper we classify all classical twists of the induced Lie bialgebra structures in terms of Belavin-Drinfeld quadruples up to a natural notion of equivalence. To obtain this classification we first show that the induced bialgebra structures are defined by certain solutions of the classical Yang-Baxter equation (CYBE) with two parameters. Then, using the algebro-geometric theory of CYBE, based on torsion free coherent sheaves,…
