  A function that is analytic on a domain of $\mathbb{C}^n$ is holonomic if it
is the solution to a holonomic system of linear homogeneous differential
equations with polynomial coefficients. We define and study the Bernstein-Sato
polynomial of a holonomic function on a smooth algebraic variety. We analyze
the structure of certain sheaves of holonomic functions, such as the algebraic
functions along a hypersurface, determining their direct sum decompositions
into indecomposables, that further respect decompositions of Bernstein-Sato
polynomials. When the space is endowed with the action of a linear algebraic
group $G$, we study the class of $G$-finite analytic functions, i.e. functions
that under the action of the Lie algebra of $G$ generate a finite dimensional
rational $G$-module. These are automatically algebraic functions on a variety
with a dense orbit. When $G$ is reductive, we give several
representation-theoretic techniques toward the determination of Bernstein-Sato
polynomials of $G$-finite functions. We classify the $G$-finite functions on
all but one of the irreducible reduced prehomogeneous vector spaces, and
compute the Bernstein-Sato polynomials for distinguished $G$-finite functions.
The results can be used to construct explicitly equivariant
$\mathcal{D}$-modules.
