Real reductive Lie group - Imported Knowlpedia Content

Definition

A real reductive Lie group in the Harish–Chandra class is a real Lie group $G$ such that its Lie algebra $\mathfrak g$ is reductive, $\operatorname{Ad}(G)$ lies in the group of inner automorphisms of $\mathfrak g_{\mathbb C}$ , $G$ has finitely many connected components, and the connected analytic subgroup with Lie algebra $[\mathfrak g,\mathfrak g]$ has finite center. These global conditions are part of the definition: reductivity of $\mathfrak g$ alone does not control the component group or the kernel of the adjoint action. This is the standard setting for Harish–Chandra’s representation theory.

Why the global hypotheses matter

The Lie algebra determines local structure but not the topology of the group. The finite-component condition prevents unrelated discrete phenomena, while the finite-center condition on the derived subgroup gives the semisimple part the global finiteness needed for admissibility and harmonic analysis. A noncompact central vector group is still allowed. Thus groups such as $\mathrm{GL}_n(\mathbb R)$ fit the convention even though their centers need not be compact Knapp, Chapter VII.

Cartan structure

Such a group admits a global Cartan involution. Its fixed-point subgroup is a maximal compact subgroup $K$ , and its Lie algebra has a Cartan decomposition $\mathfrak g=\mathfrak k\oplus\mathfrak p$ . These data lead to both the global Cartan decomposition and the Iwasawa decomposition.

Conventions and examples

Terminology varies: some authors use “real reductive group” for linear algebraic groups over $\mathbb R$ , for closed transpose-stable matrix groups, or for Lie groups with slightly different covering hypotheses. The Harish–Chandra-class qualifier therefore records substantive assumptions. Connected real semisimple Lie groups with finite center, their finite extensions satisfying the adjoint condition, and $\mathrm{GL}_n(\mathbb R)$ are basic examples.

References

A. W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton University Press, 1986. DOI record. Relevant: Chapter VII, especially the standing hypotheses for real reductive groups.
N. R. Wallach, Real Reductive Groups I, Academic Press, 1988. Publisher record. Relevant: Chapter 2 on the structure of real reductive groups.