Harish–Chandra's discrete-series criterion - Imported Knowlpedia Content

Statement

Let $G$ be a connected semisimple real Lie group with finite center, and let $K$ be a maximal compact subgroup. Harish–Chandra's discrete-series criterion states that the following are equivalent:

$G$ has a discrete series representation;
$G$ has a compact Cartan subgroup; and
$\operatorname{rank}G=\operatorname{rank}K$ .

Here $\operatorname{rank}G$ is the complex rank of $\mathfrak g_{\mathbb C}$ , while $\operatorname{rank}K$ is the dimension of a maximal torus of $K$ . Under equality, such a maximal torus is a Cartan subgroup of $G$ .

Meaning of the rank condition

Every maximal torus $T\subseteq K$ is compact. Equality of the two ranks means that its complexified Lie algebra is already a Cartan subalgebra of $\mathfrak g_{\mathbb C}$ ; equivalently, $T$ is a compact Cartan subgroup of $G$ . The criterion converts an analytic question about square-integrable matrix coefficients into this finite-dimensional structural condition Knapp, Chapter XII.

Examples

The group $\operatorname{SL}(2,\mathbb R)$ has $K=\operatorname{SO}(2)$ , and both ranks are $1$ , so it has discrete series. For $\operatorname{SL}(3,\mathbb R)$ , the complex rank is $2$ while $\operatorname{SO}(3)$ has rank $1$ , so no discrete series exists. Compact connected semisimple groups satisfy the condition automatically, with $K=G$ .

Scope of the finite-center hypothesis

Historical significance

Harish–Chandra did more than prove existence: he constructed and classified the discrete series and determined their characters. The compact Cartan provides the regular integral parameters, while quotienting by the relevant Weyl-group action removes equivalent parametrizations Harish–Chandra, part I and part II.

References

Harish–Chandra, “Discrete Series for Semisimple Lie Groups I: Construction of Invariant Eigendistributions,” Acta Mathematica 113 (1965), 241–318. DOI record. Relevant: construction and existence of the discrete series.
Harish–Chandra, “Discrete Series for Semisimple Lie Groups II: Explicit Determination of the Characters,” Acta Mathematica 116 (1966), 1–111. DOI record. Relevant: classification and character formulas.
Anthony W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton University Press, 1986. Author-maintained record. Relevant: Chapter XII, especially the equal-rank existence criterion.