In joint work with Karin Baur (ETH, Zurich) and Karin Erdmann (Oxford),
we study certain Delta-filtered modules for the Auslander
algebra of k[T]/T^n\rtimes C_2 where C_2 is the cyclic group
of order two.
The motivation of this lies in the problem of describing the $P$-orbit
structure for the action of a parabolic subgroup $P$ of a linear algebraic
group on its nilradical \mathfrak{n}. In general, there are
infinitely P-orbits in \mathfrak{n} and it is a ``wild'' problem to describe them.
However, in the case of a parabolic subgroup of SL_N, there
exists a bijection between P-orbits in the nilradical and
certain (Delta-filtered) modules for the Auslander algebra of k[T]/T^n,
due to work of Hille and Rohrle and Brustle et al..
Under this bijection, the Richardson orbit (i.e. the
dense orbit) corresponds to the Delta-filtered module without
self-extensions.
It has remained an open problem to describe such
a correspondence for other classical groups.