Restricted root system

Let $\mathfrak g=\mathfrak k\oplus\mathfrak p$ be the Cartan decomposition of a real reductive Lie algebra, and let $\mathfrak a\subseteq\mathfrak p$ be maximal abelian. For $\alpha\in\mathfrak a^*$ , set

\mathfrak g_\alpha=\{X\in\mathfrak g: [H,X]=\alpha(H)X\text{ for every }H\in\mathfrak a\}.

The restricted root system is

\Sigma(\mathfrak g,\mathfrak a) =\{\alpha\in\mathfrak a^*\setminus\{0\}:\mathfrak g_\alpha\ne0\}.

Its elements are restricted roots, and $m_\alpha=\dim\mathfrak g_\alpha$ is the root multiplicity. Unlike an ordinary reduced root system, $\Sigma$ may contain both $\alpha$ and $2\alpha$ .

Root-space decomposition

The commuting operators $\operatorname{ad}(H)$ , $H\in\mathfrak a$ , are simultaneously diagonalizable over $\mathbb R$ , giving

\mathfrak g=\mathfrak g_0\oplus \bigoplus_{\alpha\in\Sigma}\mathfrak g_\alpha.

Choosing a positive subsystem $\Sigma^+$ defines the nilpotent algebra $\mathfrak n=\bigoplus_{\alpha\in\Sigma^+}\mathfrak g_\alpha$ , the key ingredient in the Iwasawa decomposition.

Weyl group and chambers

The restricted Weyl group is

W(\mathfrak g,\mathfrak a)=N_K(\mathfrak a)/Z_K(\mathfrak a).

It is generated by the reflections associated with indivisible restricted roots and acts on $\mathfrak a$ . The connected components of the complement of the root hyperplanes are Weyl chambers. Their closures parametrize the radial part in the $KAK$ decomposition.

Example and comparison

For $\mathfrak{sl}_n(\mathbb R)$ , take $\mathfrak a$ to be the diagonal trace-zero matrices. The restricted roots are $\alpha_{ij}(H)=H_i-H_j$ , each with multiplicity one, so the system is of type $A_{n-1}$ . For other real forms, several complex roots can restrict to the same functional, producing higher multiplicities or a nonreduced system.

References

A. W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Birkhäuser, 2002. Publisher record. Relevant: Chapter VI, §4 on restricted roots.
S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, American Mathematical Society, 2001. DOI record. Relevant: Chapter VII on restricted-root decompositions.