Local Euler obstructions for determinantal varieties

Abstract

The goal of this note is to explain a derivation of the formulas for the local Euler obstructions of determinantal varieties of general, symmetric and skew-symmetric matrices, by studying the invariant de Rham complex and using character formulas for simple equivariant D-modules. These calculations are then combined with standard arguments involving Kashiwara's local index formula and the description of characteristic cycles of simple equivariant D-modules. The formulas are implicit in the work of Boe and Fu, and in the case of general matrices they have also been obtained recently by Gaffney–Grulha–Ruas, for skew-symmetric matrices by Promtapan and Rimányi, and for all cases by Zhang.