Mini-course: $C^\ast$-algebras of Reductive Lie Groups (5 Lectures)

A (linear) reductive Lie group is a closed subgroup $G$ of $GL(n)$ that is also closed under taking conjugate transposes of matrices. For example, $G = SL(n)$. A unitary representation of $G$ is a group homomorphism from $G$ to the unitary group of a Hilbert space. Such representations are useful in the study of differential equations with symmetries described by an action by $G$, for example in quantum mechanics. The theory of these representations is very rich and deep, especially when $G$ is noncompact. Much of the information about these representations is contained in the full and reduced $C^\ast$-algebras of $G$. This opens up the possibility to use operator algebraic techniques to study representations. A key result is the Connes-Kasparov conjecture, proved by Wassermann, Lafforgue, Chabert, Echterhoff and Nest. This is an explicit computation of the $K$-theory of the reduced $C^\ast$-algebra of $G$ in terms of equivariant indices of operators parametrised by representations of a compact subgroup of $G$. In this lecture series, we first discuss some background in representation theory, and then go into the structure of the reduced $C^\ast$-algebra of $G$ and its $K$-theory. This will be based on papers by Wassermann and by Clare, Crisp and Higson. We will also see some trace maps that can be used to extract explicit representation theoretic information from the $K$-theory of the $C^\ast$-algebra of $G$. Prerequisites for this lecture series are basic knowledge of $C^\ast$-algebras and $K$-theory. No knowledge of Lie theory or representation theory will be assumed.