Collaborative Research: The Geometric Dual Space of A Unipotent Group in Characteristic p

主要研究者在正特征域上继续对单能群的几何特征理论和几何表征理论进行研究。他们将这样一个群的“对偶空间”定义为一个代数几何对象，参数化了群上的l包的字符串。然后，主要研究人员发展了关于对偶空间的等变派生范畴的谱分解的推测几何理论，这类似于将表示分解为不可约的直接积分的经典分解。在轨道法适用的情况下，对这种谱分解提出了若干猜想，并讨论了张量范畴理论中相关的量化问题。这些量化问题与标准问题的区别在于，普遍包络代数的作用是由群代数来扮演的。该项目结合了当前数学和数学物理研究的几个活跃方向——几何表示理论、代数几何、张量范畴理论、量子群和共形场理论。几何谱分解的概念将加深我们对几何表示理论的一般模式的理解。该研究还将导致量子化哲学在张量范畴理论中的新应用。

The principal investigators continue their research in geometric character theory and geometric representation theory of unipotent groups over a field of positive characteristic. They define the "dual space" of such a group as an algebro-geometric object parameterizing L-packets of character sheaves on the group. The principal investigators then develop a conjectural geometric theory of the spectral decomposition of the equivariant derived category with respect to the dual space, which is similar to the classical decomposition of representations as direct integrals of irreducible ones. They also formulate several conjectures on this spectral decomposition in the setting where the orbit method is applicable and discuss certain related quantization problems in the theory of tensor categories. The difference between these quantization problems and the standard ones is that the role of the universal envelopingalgebras is played by group algebras.The proposed project combines several active directions of current research in mathematics and mathematical physics -- geometric representation theory, algebraic geometry, the theory of tensor categories, quantum groups, and conformal field theory. The notion of geometric spectral decomposition will deepen our understanding of the general patterns of geometric representation theory. The research will also lead to new applications of the quantization philosophy in the theory of tensor categories.