Cohomological representations for real reductive groups

TL;DR

This paper investigates cohomological A-parameters for real reductive groups, establishing a structure theorem, analyzing their behavior under morphisms, and providing explicit calculations of cohomology ranks across packets.

Contribution

It introduces a structure theorem for cohomological A-parameters, explores their behavior under L-group morphisms, and computes cohomology ranks for all classical groups.

Findings

A structure theorem for cohomological A-parameters is proved.

Morphisms of L-groups preserving regular unipotent elements respect cohomological A-parameters.

The sum of cohomology ranks in a packet is independent of the packet and explicitly computable.

Abstract

For a connected reductive group $G$ over ${\mathbb R}$, we study cohomological $A$-parameters, which are Arthur parameters with the infinitesimal character of a finite-dimensional representation of $G({\mathbb C})$. We prove a structure theorem for such $A$-parameters, and deduce from it that a morphism of $L$-groups which takes a regular unipotent element to a regular unipotent element respects cohomological $A$-parameters. This is used to give complete understanding of cohomological $A$-parameters for all classical groups. We review the parametrization of Adams-Johnson packets of cohomological representations of $G({\mathbb R})$ by cohomological $A$-parameters and discuss various examples. We prove that the sum of the ranks of cohomology groups in a packet on any real group (and with any infinitesimal character) is independent of the packet under consideration, and can be explicitly calculated. This result has a particularly nice form when summed over all pure inner forms.