  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 Rim\'anyi, and for all cases by Zhang.
