Holonomic functions and prehomogeneous spaces

Abstract

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.