The Mackey Analogy for SL(n,R)

Author:

George, Christopher Yousuf

Graduate Program:

Mathematics

Degree:

Doctor of Philosophy

Document Type:

Dissertation

Date of Defense:

December 18, 2008

Committee Members:

Nigel David Higson, Dissertation Advisor/Co-Advisor
Nigel David Higson, Committee Chair/Co-Chair
John Roe, Committee Member
Nathanial Patrick Brown, Committee Member
Donald P Schneider, Committee Member

Keywords:

representation theory
Lie groups

Abstract:

Let $G$ be a connected semisimple Lie group with finite center, and let K be a maximal compact subgroup. Mackey suggested that there should be an analogy between almost all the unitary representations of $G$ and almost all of the unitary representations of an associated semidirect product group $G_0 = K ltimes mathfrak{g/k}$, where $mathfrak g$ and $mathfrak k$ are the Lie algebras of $G$ and $K$, respectively. Perhaps because Mackey typically viewed the dual of a Lie group as a Borel space, the analogy he proposed was between almost all unitary representations of $G$ and $G_0$. In the case of complex semisimple Lie groups, Higson completed Mackey's analogy at the level of parameters, constructing a reasonably natural bijection between all irreducible tempered representations of $G$ and all irreducible unitary representations of $G_0$. In this dissertation, following Higson, and relying heavily on Vogan's classification of representations by minimal $K$-types, we conjecture that there is a very natural bijection between $widehat G$ and $widehat G_0$ for any connected semisimple Lie group with finite center, and prove this conjecture for $G=slnr$.