Real rank of a reductive group - Imported Knowlpedia Content

Definition

Let $G$ be a connected real reductive Lie group with Lie algebra $\mathfrak g$ , and choose a Cartan decomposition $\mathfrak g=\mathfrak k\oplus\mathfrak p$ . If $\mathfrak a\subseteq \mathfrak p$ is maximal abelian, the real rank of $G$ is

\operatorname{rank}_{\mathbb R}G=\dim_{\mathbb R}\mathfrak a.

All maximal abelian subspaces of $\mathfrak p$ are conjugate under the maximal compact subgroup, so this number is independent of the choices. It is the rank of the restricted roots when $G$ is semisimple. For reductive $G$ , split central directions contribute to $\dim\mathfrak a$ even though every restricted root vanishes on them.

Equivalent split-torus formulation

For a linear real algebraic group, $\operatorname{rank}_{\mathbb R}G$ is the dimension of a maximal $\mathbb R$ -split torus. On the Lie-group side, the connected subgroup $A=\exp(\mathfrak a)$ is the corresponding maximal split abelian subgroup. Conjugacy of the possible $\mathfrak a$ 's and the associated Cartan and Iwasawa structure are treated in Knapp, Chapter VI, §§2–5.

Examples

The compact group $\operatorname{SO}(n)$ has real rank $0$ . $\operatorname{SL}(n,\mathbb R)$ has real rank $n-1$ , witnessed by the trace-zero diagonal matrices in $\mathfrak p$ . $\operatorname{SO}_0(p,q)$ , with $p\leq q$ , has real rank $p$ . Thus real rank distinguishes compact groups, rank-one groups, and higher-rank groups even when their complexified Lie algebras have comparable root data.

Conventions and consequences

For a semisimple group, real rank is zero exactly when the group is compact up to finite covering and finite center. For a reductive group, a noncompact split center contributes to the displayed definition; “real semisimple rank” sometimes means the real rank after removing that center. Real rank should not be confused with the rank of a maximal torus or with complex rank.

References

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Progress in Mathematics 140, Birkhäuser, 2002. Author-maintained record. Relevant: Chapter VI, §§2–5 on Cartan and Iwasawa decompositions, and Chapter VI, §11 on restricted roots.
Sigurdur Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, corrected reprint, Graduate Studies in Mathematics 34, American Mathematical Society, 2001. DOI record. Relevant: Chapter VI on symmetric spaces of noncompact type and their rank.