$\Delta$-filtered modules and nilpotent
orbits of a parabolic subgroup in SO$_N$
We study certain Delta-filtered modules for the Auslander algebra of
k[T]/T^n times C_2 where C_2 is the cyclic group of order two.
The motivation for this is the bijection between parabolic orbits in the nilradical of a parabolic subgroup of SL_n and certain Delta-filtered modules for the Auslander algebra of k[T]/T^n as found by Hille and Roehrle and Bruestle 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.
In this paper, we establish the Auslander algebra of $k[T]/T^n\rtimes C_2$
as the right candidate for the
orthogonal groups. In particular, for
any parabolic subgroup of an
orthogonal group we construct a map
from the Richardson orbit to $\Delta$-filtered modules without
self-extensions.
One of the consequences of our work is that we are able to describe
the extensions between special classes $\Delta$-filtered modules.
In particular, we show that these extensions can grow arbitrarily
large.
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