Cohomological representations for real reductive groups

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.

Considerably streamlined version. To appear in the Journal of the London Math Society

References in corpus (1)

• Multiplicity upon restriction to the derived subgroup

Cited by in corpus (2)

• Computing Galois cohomology of a real linear algebraic group
• Limit Multiplicity For Unitary Groups and The Stable Trace Formula