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.