This talk will be a continuation of last week’s IFS talk on the Lusztig-Vogan module of the Hecke algebra. More details can be found below. 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 get a glimpse of a mysterious feature called Vogan duality.