The Bernstein–Sato $b$-function of the Space of Cyclic Pairs

Abstract

We compute the Bernstein-Sato polynomial of $f$, a function which given a pair $(M,v)$ in $X=M_n(\mathbb{C})\times {\mathbb{C}}^n$ tests whether $v$ is a cyclic vector for $M$. The proof includes a description of shift operators corresponding to the Calogero-Moser operator $L_k$ in the rational case.