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Modular categories, crossed S-matrices and Shintani descent

Let $\mathscr{C}$ be a modular tensor category over an algebraically closed field $k$ of characteristic 0. Then there is the ubiquitous notion of the S-matrix $S(\mathscr{C})$ associated with the modular category. The matrix $S(\mathscr{C})$ is a symmetric matrix, its entries are cyclotomic integers and the matrix $(\dim \mathscr{C})^{-\frac{1}{2}}\cdot S(\mathscr{C})$ is a unitary matrix. Here $\dim \mathscr{C}\in k$ denotes the categorical dimension of $\mathscr{C}$ and it is a totally positive cyclotomic integer. Now suppose that we also have a modular autoequivalence $F:\mathscr{C}\to \mathscr{C}$. In this paper, we will define and study the notion of a crossed S-matrix associated with the modular autoequivalence $F$. We will see that the crossed S-matrix occurs as a submatrix of the usual S-matrix of some "bigger" modular category and hence the entries of a crossed S-matrix are also cyclotomic integers. We will prove that the crossed S-matrix (normalized by the factor $(\dim \mathscr{C})^{-\frac{1}{2}}$) associated with any modular autoequivalence is a unitary matrix. We will also prove that the crossed S-matrix is essentially the "character table" of a certain semisimple commutative Frobenius $k$-algebra associated with the modular autoequivalence $F$. The motivation for most of our results comes from the theory of character sheaves on algebraic groups, where we expect that the transition matrices between irreducible characters and character sheaves can be obtained as certain crossed S-matrices. In the character theory of algebraic groups defined over finite fields, there is the notion of Shintani descent of Frobenius stable characters. We will define and study a categorical analogue of this notion of Shintani descent in the setting of modular categories.