Some geometric facets of the Langlands correspondence for real groups

This note concerns geometric aspects of the local Langlands correspondence for real groups as extended from Langlands' original work by Adams-Barbasch-Vogan, and further (conjectural) formulations by W. Soergel. The main result concerns purity (in the sense of weights in Hodge theory) of equivariant extension groups between simple objects on the Adams-Barbasch-Vogan geometric parameter space (for trivial infinitesimal character).