On Centers of Bimodule Categories and Induction–Restriction Functors. (29th June 2017)

• Record Type: Journal Article
• Authors: Deshpande, Tanmay
• Is Part Of: International mathematics research notices. Volume 2019:Number 2(2019)
• Page Start: 578
• Page End: 605
• Publication Date: 2017-06-29
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnx143
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: In this article 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 $\zeta_{{\mathscr{M}}}:{\mathscr{Z}}_{\mathscr{C}}({\mathscr{M}}){\stackrel{{}}{\longrightarrow}}{\mathscr{M}}$ denote the forgetful functor and let $\chi_{\mathscr{M}}:{\mathscr{M}}{\stackrel{{}}{\longrightarrow}}{\mathscr{Z}}_{\mathscr{C}}({\mathscr{M}})$ 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 article, 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 ${{\mathscr{Z}}_{{\mathscr{C}}}({\mathscr{M}})}$ .