Derived Equivalences and Mixed Categories in Representation Theory

过去十五年中几何表示论的一系列重大进展都采取了“相干可构造等价”的形式：这些定理断言，与归约群相关的某个簇上的相干层范畴等价（或导出等价）到与朗兰兹对偶群相关的不同簇上的可构造（或反常）层范畴。 也许最著名的这类结果是“几何佐竹等价”，由于卢斯蒂格，金兹伯格，和Mirkovic-Vilonen。 它将群G的表示（等价地：点上的凝聚层）与对偶群的仿射格拉斯曼上的球面逆层联系起来。 另一个主要结果是Arkhipov-Bezrukavnikov-Ginzburg（ABG）等价，它指出仿射格拉斯曼群上的反常层和G的旗簇余切丛上的相干层之间存在一个导出等价。 这两个结果都可以看作是几何朗兰兹程序的一部分。 私家侦探希望通过以下两个项目为这一图景做出贡献：（I）与S.里奇那个私家侦探希望证明抛物型版本的几个已知的相干构造等价，这些结果将包括ABG和几何佐竹等价作为特殊情况。(II)私家侦探希望发展一个新的公理框架，期望在一个非常一般的设置下导出等价和更高的外消失定理，以期应用于幂零锥上的相干层和旗变量的余切丛上。“矩阵群”是一组可逆方阵，包含其成员的所有乘积和逆。 一个典型的例子是SU（2），2x2酉复矩阵群。 这样一个群的“表示”是一个规则，它为群的每个成员分配一个向量空间的线性变换，以这种方式将矩阵乘法转换为线性变换的合成。 SU（2）有一个自然的二维表示--这个规则分配给SU（2）本身的每个元素--但也有许多其他的表示：例如，SU（2）在二元多项式空间上有一个表示，由变量中的线性替换给出。 SU（2）也是一个拓扑空间：事实上，它在拓扑上等价于一个3-球面。 表示论中的许多现代结果涉及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，这两个等价都将矩阵群的表示与称为“仿射格拉斯曼空间”的无限维空间上的“D-模”（一种处理微分方程局部解空间的复杂方法）联系起来。“拟议的研究包括两个项目：（一）P.I.，与S合作。Riche，希望证明一个一般的“抛物线parahoric等价定理”，将包括几何佐竹和ABG等价作为特殊情况;和（II）P.I.希望以一种公理化的方式来研究在已知的等价定理中出现的某些“正性现象”，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.