Stable combinations of special unipotent representations

TL;DR

This paper constructs a canonical basis for the space of stable characters of special unipotent representations of semisimple Lie groups, extending Arthur packet characters and connecting to nilpotent cone structures.

Contribution

It provides a canonical basis for stable characters of special unipotent representations, extending Arthur packet characters and linking to nilpotent cone geometry.

Findings

Constructed a canonical basis for the space of stable characters.

Extended stable characters from Arthur packets to a larger space.

Connected representation theory with geometric structures of nilpotent cones.

Abstract

The set of special unipotent representations of a semisimple Lie group was defined by Barbasch and Vogan. According to predictions of Arthur (established by Adams, Barbasch, and Vogan), this set is an overlapping union of Arthur packets. Each packet gives rise to a canonical virtual representation whose character is stable in the sense of Langlands and Shelstad. But these stable characters seldom span the space of all stable characters of virtual special unipotent representations. Under natural hypotheses, we give a canonical basis of this latter space in terms of rational forms of special pieces of the nilpotent cone of the Langlands dual Lie algebra. (This basis extends the stable characters arising from set of Arthur packets.) Some consequences of our results have nothing to do with real groups; for example, we are able to define a partial section of the Springer-Steinberg map from the Weyl group to nilpotent adjoint orbits.