Modelling Richardson orbits for SO_N via Delta-filtered modules

TL;DR

This paper extends the correspondence between parabolic orbits and Delta-filtered modules from SL_n to orthogonal groups, establishing a new algebraic framework for understanding Richardson orbits and their extensions.

Contribution

It introduces the Auslander algebra of k[T]/T^n times C_2 as a suitable model for orthogonal groups and constructs a map linking parabolic orbits to Delta-filtered modules, including the Richardson orbit.

Findings

The Auslander algebra effectively models orthogonal group orbits.

The constructed map correctly identifies the Richardson orbit with a module without self-extensions.

Extensions between certain Delta-filtered modules can be arbitrarily large.

Abstract

We study the 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 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 parabolic orbits to Delta-filtered modules and show that in the case of the Richardson orbit, the result has no self-extensions. One of the consequences of our work is that we are able to describe the extensions between special classes of Delta-filtered modules. In particular, we show that these extensions can grow arbitrarily large.