  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.
