Reducible characteristic cycles of Harish-Chandra modules for $\mathrm{U}(p,q)$ and the Kashiwara-Saito singularity

Appearance of the Kashiwara-Saito singularity in the representation theory of $p$-adic $\mathop{GL}_{16}$

In 1993 David Vogan proposed a basis for the vector space of stable distributions on $p$-adic groups using the microlocal geometry of moduli spaces of Langlands parameters. In the case of general linear groups, distribution characters of irreducible admissible representations, taken up to equivalence, form a basis for the vector space of stable distributions. In this paper we show that these two bases, one putative, cannot be equal. Specifically, we use the Kashiwara-Saito singularity to find a non-Arthur type irreducible admissible representation of $p$-adic $\mathop{GL}_{16}$ whose ABV-packe…

多参数持续性上Kashiwara晶体的几何构造

Geometric construction of Kashiwara crystals on multiparameter persistence

本研究构建了多参数持续性模簇不可约分支上Kashiwara晶体的几何构造，以交换关系为核心区别于既有研究，并明确了1、2参数情形下的Kashiwara算子，为相关晶体结构提供具体认知。

Isospectral commuting variety and the Harish-Chandra D-module

Let g be a complex reductive Lie algebra with Cartan algebra h. Hotta and Kashiwara defined a holonomic D-module M, on g x h, called Harish-Chandra module. We give an explicit description of gr(M), the associated graded module with respect to a canonical Hodge filtration on M. The description involves the isospectral commuting variety, a subvariety X of g x g x h x h which is a finite extension of the variety of pairs of commuting elements of g. Our main result establishes an isomorphism of gr(M) with the structure sheaf of the normalization of X. It follows, thanks to the theory of polarized …

Harish-Chandra D-模用于双Whittaker约化

Harish-Chandra D-modules for bi-Whittaker reduction

本文定义双Whittaker量子哈密顿约化的可容许模和Harish-Chandra模，通过通用中心化子几何研究其相对于Kazhdan滤过的挠性质，并证明正则无穷小特征标下对应的D(G)-模是开Bruhat胞腔上非正则联络的最小扩张。

A reducible characteristic variety in type A

We show that simple highest weight modules for sl_12 may have reducible characteristic variety. This answers a question of Borho-Brylinski and Joseph from 1984. The relevant singularity under Beilinson-Bernstein localization is the (in)famous Kashiwara-Saito singularity. We sketch the rather indirect route via the p-canonical basis, W-graphs and decomposition numbers for perverse sheaves that led us to examine this singularity.

Isospectral commuting variety, the Harish-Chandra D-module, and principal nilpotent pairs

Let g be a complex reductive Lie algebra with Cartan algebra h. Hotta and Kashiwara defined a holonomic D-module M, on g x h, called Harish-Chandra module. We relate gr(M), an associated graded module with respect to a canonical Hodge filtration on M, to the isospectral commuting variety, a subvariety of g x g x h x h which is a ramified cover of the variety of pairs of commuting elements of g. Our main result establishes an isomorphism of gr(M) with the structure sheaf of X_norm, the normalization of the isospectral commuting variety. It follows, thanks to the theory of Hodge modules, that th…