DECOMPOSITIONS OF BERNSTEIN–SATO POLYNOMIALS AND SLICES

Abstract

Let G be a linearly reductive group acting on a vector space V, and f a semi-invariant polynomial on V. In this paper we study systematically decompositions of the Bernstein–Sato polynomial of f in parallel with some representation-theoretic properties of the action of G on V. We provide a technique based on a multiplicity one property, that we use to compute the Bernstein–Sato polynomials of several classical invariants in an elementary fashion. Furthermore, we derive a “slice method” which shows that the decomposition of V as a representation of G can induce a decomposition of the Bernstein–Sato polynomial of f into a product of two Bernstein–Sato polynomials – that of an ideal and that of a semi-invariant of smaller degree. Using the slice method, we compute Bernstein–Sato polynomials for a large class of semi-invariants of quivers.