Holonomic functions and prehomogeneous spaces - Lőrincz, András Cristian (Selecta mathematica (Basel, Switzerland), 2023)
A function that is analytic on a domain of 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 D -modules.