Informal Friday Seminar: Romanov -- The Lusztig-Vogan module of the Hecke algebra

Let G be a real reductive Lie group (think GL(n,R)).  When studying the representation
theory of such a group, one quickly encounters a well-behaved class of representations
called admissible representations.  The combinatorial behaviour of these representations
(e.g.  composition series multiplicities of standard representations) is captured
by a certain geometrically-defined module over the associated Hecke algebra, the
Lusztig-Vogan module.  In this talk, I will describe the construction of the
Lusztig-Vogan module, then we will see what it looks like explicitly in some SL2
examples.  If we are lucky, we might see a glimpse of a mysterious feature called Vogan
duality.  This talk is related to my previous IFS talks on unitary representation
theory, equivariant cohomology, and the admissible dual of SL(2,R), but I will assume
that the audience has no recollection of anything I have previously said.