On the Finkelberg-Ginzburg Mirabolic Monodromy Conjecture

TL;DR

This paper analyzes the monodromy of mirabolic Harish-Chandra D-modules across various parameters, revealing that the Finkelberg-Ginzburg conjecture fails at specific special values, with explicit computations in low and higher ranks.

Contribution

It provides explicit monodromy computations for all parameters in rank 1 and identifies the failure of the conjecture at special values in higher ranks.

Findings

Conjugacy of monodromy in rank 1 for all parameters.

Failure of the Finkelberg-Ginzburg conjecture at special parameter values.

Use of Opdam's shift operators and intertwiners to resolve resonances.

Abstract

We compute the monodromy of the mirabolic Harish-Chandra D-module for all values of the parameters (theta,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 (theta,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.