Derived Equivalences and Mixed Categories in Representation Theory

在过去的十五年里，几何表示理论中的一系列重大进展都采取了“凝聚-可构造等价”的形式：这些定理断言：与还原群相关的某些簇上的凝聚层范畴与与朗兰兹对偶群相关的不同簇上的可构造(或倒立)层的范畴等价(或导出等价)。也许这类结果中最著名的是Lusztig、Ginzburg和Mirkovic-Vilonen提出的“几何Satake等价”。它将群G的表示(等价于：一点上的凝聚层)与对偶群的仿射Grassmanian上的球面反层层联系起来。另一个主要结果是Arkhipov-Bezrukavnikov-Ginzburg(ABG)等价，它指出在仿射Grassmanian上的倒序层与旗簇的余切丛上的相干层之间存在导出等价。这两个结果可视为几何朗兰兹程序的一部分。P.I.希望通过以下两个项目为这一图景做出贡献：(I)P.I.与S.Riche合作，希望证明几个已知的相干可构造等价的抛物线形式；这些结果将ABG等价和几何Satake等价作为特例包含在内。(Ii)P.I.希望发展一种新的公理框架，它有望在非常一般的情况下得到关于导出等价和高次外消的定理，以期应用于幂零锥上的凝聚层和旗簇的余切丛上。“矩阵群”是一组包含其成员的所有乘积和逆的可逆方阵。一个典型的例子是SU(2)，它是由2x2酉复矩阵组成的群。这种群的“表示”是这样一种规则，它为群中的每个成员分配某个向量空间的线性变换，其方式是将矩阵乘法转换为线性变换的组合。SU(2)有一个自然的二维表示--给SU(2)本身的每个元素赋值的规则--但还有许多其他的表示：例如，SU(2)在两个变量的多项式空间上有一个表示，通过变量的线性替换给出。SU(2)也是一个拓扑空间：实际上，它在拓扑上等价于一个3-球面。表示论中的一些现代结果涉及到矩阵群和相关空间的几何。两个著名的结果是几何Satake等价和Arkhipov-Bezrukavnikov-Ginzburg(ABG)等价，两者都将矩阵群的表示与某个无限维空间上的“D-模”(一种处理微分方程局部解空间的复杂方式)联系在一起，称为“仿射Grassmanian”。建议的研究包括两个项目：(I)P.I.与S.Riche合作，希望证明一个一般的“抛物线-意指等价定理”，该定理将几何Satake和ABG等价作为特例包括在内；(Ii)P.I.希望以公理的方式研究出现在已知等价定理中的某些“积极现象”，以期将这些定理推广到其他环境。

A series of major advances in geometric representation theory in the past fifteen years have taken the form of "coherent-constructible equivalences": these theorems assert that the category of coherent sheaves on some variety associated to a reductive group is equivalent (or derived-equivalent) to the category of constructible (or perverse) sheaves on a different variety, associated to the Langlands dual group. Perhaps the best-known such result is the "geometric Satake equivalence," due to Lusztig, Ginzburg, and Mirkovic-Vilonen. It relates representations of a group G (equivalently: coherent sheaves on a point) to spherical perverse sheaves on the affine Grassmannian for the dual group. Another major result is the Arkhipov-Bezrukavnikov-Ginzburg (ABG) equivalence, which states that there is a derived equivalence between perverse sheaves on the affine Grassmannian and coherent sheaves on the cotangent bundle of the flag variety of G. Both these results may be regarded as part of the geometric Langlands program. The P.I. hopes to contribute to this picture with the following two projects: (I) In collaboration with S. Riche, the P.I. hopes to prove parabolic versions of several known coherent-constructible equivalences; these results would encompass the ABG and geometric Satake equivalences as special cases. (II) The P.I. hopes to develop a new axiomatic framework which is expected to lead to theorems on derived equivalences and higher Ext-vanishing in a very general setting, with a view to applications to coherent sheaves on the nilpotent cone and on the cotangent bundle of the flag variety.A "matrix group" is a set of invertible square matrices that contains all products and inverses of its members. A typical example is SU(2), the group of 2x2 unitary complex matrices. A "representation" of such a group is a rule that assigns to each member of the group a linear transformation of some vector space, in such a way that it transforms matrix multiplication into composition of linear transformations. SU(2) has a natural 2-dimensional representation--the rule assigning to each element of SU(2) itself--but there are many others as well: for instance, SU(2) has a representation on the space of polynomials in two variables, given by linear substitutions in the variables. SU(2) is also a topological space: in fact, it is topologically equivalent to a 3-sphere. A number of modern results in representation theory involve the geometry of matrix groups and related spaces. Two celebrated results are the geometric Satake equivalence and the Arkhipov-Bezrukavnikov-Ginzburg (ABG) equivalence, both of which relate representations of a matrix group to "D-modules" (a sophisticated way of working with spaces of local solutions of differential equations) on a certain infinite-dimensional space called the "affine Grassmannian." The proposed research includes two projects: (I) the P.I., in collaboration with S. Riche, hopes to prove a general "parabolic-parahoric equivalence theorem" that would include the geometric Satake and ABG equivalences as special cases; and (II) the P.I. hopes to study in an axiomatic way certain "positivity phenomena" that occur in the known equivalence theorems, with a view to generalizing those theorems to other settings.