On the Finkelberg–Ginzburg Mirabolic Monodromy Conjecture

Abstract

Abstract We compute the monodromy of the mirabolic $\mathcal{D}$-module for all values of the parameters $(\vartheta ,c)$ in rank 1 and outside an explicit codimension 2 set of values in ranks 2 and higher. This shows in particular that the Finkelberg–Ginzburg conjecture, which is known to hold for generic values of $(\vartheta ,c)$, fails at special values even in rank 1. Our main tools are Opdam’s shift operators and intertwiners for the extended affine Weyl group, which allow for the resolution of resonances outside the codimension two set.