Unitary representations attached to parabolic subgroups: the case of abelian unipotent radical
Dan Ciubotaru
TL;DR
The paper classifies unitary representations with integral infinitesimal character within Lusztig's unipotent framework for parabolic subgroups with abelian unipotent radicals, organizing them into microlocal Arthur packets. By exploiting the abelian nilradical, it proves that all such representations are unitary when the infinitesimal character is correctly symmetric under the Weyl group, and none are hermitian otherwise. It then provides explicit geometric and representation-theoretic data for the abelian case across types A, C, Spin, D, E6, and E7, including orbit structures, isotropy, and D-module quivers, thereby validating the conjectural integral-unitarity description in this uniform setting. The results offer a concrete, testable parametric picture that connects Langlands–Arthur theory with ABV microlocal packets in a tractable geometric context, and set the stage for broader generalizations to maximal parabolics with abelian or spherical spaces.
Abstract
We classify the unitary representations with integral infinitesimal character in Lusztig's category of unipotent representations in the case when the geometric parameter space comes from the action of a Levi subgroup on the abelian nilradical of a (maximal) parabolic subalgebra. We organise the unitary representations into microlocal Arthur packets. This is a test case for investigating a conjectural description of unitary representations with integral infinitesimal character.