Representation theory of noncompact real reductive groups

This page and its companion stop at the finite-dimensional real structure theory: complexification and conjugations, compact and split forms, Cartan involutions and Cartan decompositions, restricted roots, the Iwasawa decomposition, $\theta$-stable Cartan subalgebras and Cayley transforms, and the Vogan and Satake classification of real forms. Everything on this page is a statement about the real Lie algebra ${\mathfrak{g}}_0$ or about the global decomposition of a semisimple Lie group with finite center.

The representation theory of a noncompact real reductive group $G$ lies beyond this block and is not used anywhere on this page. Its subject matter is: the real reductive class and the Harish-Chandra class governed by axioms on the Lie algebra, the maximal compact subgroup $K$ and the center (Knapp, Chapter VII, §2); admissible representations and the Harish-Chandra module of $(\mathfrak{g},K)$-finite vectors, which forgets the topology of the representation in exchange for a purely algebraic module over the pair $(\mathfrak{g},K)$; the unitary dual and the problem of which irreducible admissible representations carry a positive definite invariant Hermitian form; the Plancherel theorem for $L^2(G)$ with its discrete and continuous parts; and the Langlands classification of irreducible admissible representations as quotients of standard parabolically induced representations.

Two features of the finite-dimensional theory that this page does build are the inputs that the analytic theory takes for granted: the Cartan decomposition and the existence of a maximal compact subgroup $K$, and the Iwasawa decomposition $G=KAN$ together with the restricted-root data of $\mathfrak{a}$ and the nilpotent algebra $\mathfrak{n}$. The further steps — admissibility, the Harish-Chandra homomorphism and its infinitesimal character, the asymptotics of matrix coefficients, and the harmonic analysis of $L^2(G)$ — use elliptic regularity, Fourier analysis on groups and invariant integral geometry, none of which is developed in this library track. For the same reason the compact-group representation theory that the library does contain is not a substitute. A finite-dimensional unitary representation has relatively compact image; it can be faithful for a noncompact group (even with dense image in a torus), but it cannot realise that group as a closed noncompact matrix subgroup. The unitary dual and its topological representation theory are therefore not captured by the compact theory.

The boundary is recorded here so that no consumer of this page treats the Vogan/Satake classification of real forms as a classification of representations. The relevant foundational constructions for the analytic theory are documented in the Historical Notes for Chapter VII of the source.