arXiv 1611.044862016-11-14math.QAmath.RT

双模范畴的中心与归纳限制函子

On centers of bimodule categories and induction-restriction functors

Tanmay Deshpande

备注：: 18 pages

摘要：

在本文中，我们研究了一个玩具类别版本的Lusztig的归纳和限制函子的字符层，但在抽象的设置多融合范畴。设$\mathscr{C}$是不可分解的多融合范畴，$\mathscr{M}$是可逆的$\mathscr{C}$ -双模范畴.则$\mathscr{M}$关于$\mathscr{C}$的中心$\mathscr{Z}_{\mathscr{C}}(\mathscr{M})$是Drinfeld中心$\mathscr{Z}(\mathscr{C})$上的可逆模范畴，该Drinfeld中心是一个编织融合范畴。令$ζ_{\mathscr{M}}:\mathscr{Z}_{\mathscr{C}}(\mathscr{M})\longrightarrow\mathscr{M}$表示遗忘函子，令$χ_{\mathscr{M}}:\mathscr{M}\longrightarrow\mathscr{Z}_{\mathscr{C}}(\mathscr{M})$为其右伴随函子。这些函子可以被认为是Lusztig用来定义（可能不连通的）约化群上的特征层的限制函子和归纳函子的玩具类似物。本文研究了上述函子下简单对象的像的分解与某些Grothendieck环的特征标表之间的关系。在$\mathscr{C}$具有球形结构且$\mathscr{M}$具有$\mathscr{C}$ -双模迹的情况下，我们将其与关联于$\mathscr{Z}(\mathscr{C})$ -模范畴$\mathscr{Z}_{\mathscr{C}}(\mathscr{M})$的交叉S-矩阵的概念联系起来。

英文摘要：

In this paper we study a toy categorical version of Lusztig's induction and restriction functors for character sheaves, but in the abstract setting of multifusion categories. Let $\mathscr{C}$ be an indecomposable multifusion category and let $\mathscr{M}$ be an invertible $\mathscr{C}$-bimodule category. Then the center $\mathscr{Z}_{\mathscr{C}}(\mathscr{M})$ of $\mathscr{M}$ with respect to $\mathscr{C}$ is an invertible module category over the Drinfeld center $\mathscr{Z}(\mathscr{C})$ which is a braided fusion category. Let $ζ_{\mathscr{M}}:\mathscr{Z}_{\mathscr{C}}(\mathscr{M})\longrightarrow\mathscr{M}$ denote the forgetful functor and let $χ_{\mathscr{M}}:\mathscr{M}\longrightarrow\mathscr{Z}_{\mathscr{C}}(\mathscr{M})$ be its right adjoint functor. These functors can be considered as toy analogues of the restriction and induction functors used by Lusztig to define character sheaves on (possibly disconnected) reductive groups. In this paper we look at the relationship between the decomposition of the images of the simple objects under the above functors and the character tables of certain Grothendieck rings. In case $\mathscr{C}$ is equipped with a spherical structure and $\mathscr{M}$ is equipped with a $\mathscr{C}$-bimodule trace, we relate this to the notion of the crossed S-matrix associated with the $\mathscr{Z}(\mathscr{C})$-module category $\mathscr{Z}_{\mathscr{C}}(\mathscr{M})$.

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