  A. Joseph invented multidegrees in [Jo84] to study orbital varieties, which
are the components of an orbital scheme, itself constructed by intersecting a
nilpotent orbit with a Borel subalgebra. Their multidegrees, known as Joseph
polynomials, give a basis of a (Springer) representation of the Weyl group. In
the case of the nilpotent orbit $\{ M^2=0 \}$, the orbital varieties can be
indexed by noncrossing chord diagrams in the disc. In this paper we study the
normal cone to the orbital scheme inside this nilpotent orbit $\{ M^2 = 0 \}$.
This gives a better-motivated construction of the Brauer loop scheme we
introduced in [KZJ07], whose components are indexed by all chord diagrams (now
possibly with crossings) in the disc. The multidegrees of its components, the
Brauer loop varieties, were shown to reproduce the ground state of the Brauer
loop model in statistical mechanics [DFZJ06]. Here, we reformulate and slightly
generalize these multidegrees in order to express them as solutions of the
rational quantum Knizhnik--Zamolodchikov equation associated to the Brauer
algebra. In particular, the vector of the multidegrees satisfies two sets of
equations, corresponding to the $e_i$ and $f_i$ generators of the Brauer
algebra. We describe here the geometric meaning of both $e_i$ and $f_i$
equations in our slightly extended setting. We also describe the corresponding
actions at the level of orbital varieties: while only the $e_i$ equations make
sense directly on the Joseph polynomials, the $f_i$ equations also appear if
one introduces a broader class of varieties. We explain the connection of the
latter with matrix Schubert varieties.
