  Anna Melnikov provided a parametrization of Borel orbits in the affine
variety of square-zero $n \times n$ matrices by the set of involutions in the
symmetric group. A related combinatorics leads to a construction a
Bott-Samelson type resolution of the orbit closures. This allows to compute
cohomological and K-theoretic invariants of the orbits: fundamental classes,
Chern-Schwartz-MacPherson classes and motivic Chern classes in
torus-equivariant theories. The formulas are given in terms of Demazure-Lusztig
operations. The case of square-zero upper-triangular matrices is reach enough
to include information about cohomological and K-theoretic classes of the
double Borel orbits in $Hom(\mathbb C^k,\mathbb C^m)$ for $k+m=n$. We recall
the relation with double Schubert polynomials and show analogous interpretation
of Rim\'anyi-Tarasov-Varchenko trigonometric weight function.

Algebraic Geometry;Algebraic Topology;Representation Theory