We present some fundamental aspects of the representation theory of reductive p-adic groups, mapping out some recent developments, including the ABPS conjecture and the description of the structure of the reduced C*-algebras of reductive groups. The exposition proceeds fairly rapidly, but is essentially self-contained.

We start by introducing C*-algebras associated with locally compact groups. Next, the theory of Hilbert modules, C*-correspondences, crossed-product algebras, and Morita equivalence are discussed. This is followed with applications to Mackey’s theory of induction of representations and Howe’s theory of theta correspondence. Next, K-theory and (equivariant) KK-theory are introduced. Connections between isolated points in the unitary dual and generators of the K-theory of C*-algebras of liminal groups are discussed.

We start by describing the fundamental theory of representation theory of a semisimple Lie group. This is followed by a classification of almost all irreducible tempered representations of a connected, semisimple, linear Lie group. We apply this classification to describe the structure of the reduced group C*-algebras of such groups.

This chapter is an introduction to the geometric perspective of tempered representations of connected reductive Lie groups via Dirac induction and the Connes–Kasparov isomorphism. We start with discussing Dirac operators in spin geometry. We then recall some classical Dirac operators on homogeneous spaces arising in representation theory. Next, we introduce the theory of higher index to be used as a language to formulate the Connes–Kasparov conjecture. In Section 3.4, we introduce Dirac induction, the Connes–Kasparov conjecture, and survey a couple of ways for proving it. The final section is devoted to the harmonic-analytic approach to the Connes–Kasparov isomorphism illustrated by the example of connected semisimple groups.