Cohomological representations for real reductive groups

For a connected reductive group G over R, we study cohomological A-parameters, which are Arthur parameters with the infinitesimal character of a finite-dimensional representation of G(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(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.

Automated summary

For a connected reductive group G over R, the study on cohomological A-parameters leads to a structure theorem and understanding in morphisms of L-groups. The parametrization of Adams-Johnson packets and discussion on various examples help to further interpret the concept. Additionally, the independence of the sum of ranks of cohomology groups within a packet on any real group is proven, especially when summed over all pure inner forms.