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

主要研究者继续研究正特征域上单位元群的几何特征标理论和几何表示理论。他们将这样一个群的“对偶空间”定义为一个代数几何对象，它对L群上的字符集包进行参数化处理。然后，主要研究者发展了关于对偶空间的等变派生范畴的谱分解的猜想几何理论，这类似于将表示作为不可约表示的直接积分的经典分解。他们还在轨道方法适用的情况下提出了关于这种谱分解的几个猜想，并讨论了张量范畴理论中的某些相关量子化问题。这些量子化问题与标准问题的不同之处在于，泛包络代数的作用是由群代数扮演的。该方案结合了当前数学和数学物理中几个活跃的研究方向--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。这一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.