Restricted Weyl group - Imported Knowlpedia Content

Definition

Let $G$ be a connected real reductive Lie group, let $K$ be a maximal compact subgroup, and let $\mathfrak a$ be a maximal abelian subspace of the noncompact part of a Cartan decomposition. The restricted Weyl group is

W(G,\mathfrak a)=N_K(\mathfrak a)/Z_K(\mathfrak a),

where the numerator normalizes $\mathfrak a$ under the adjoint action and the denominator fixes it pointwise. It acts faithfully on $\mathfrak a$ and on the restricted root system. The group is finite and is generated by the reflections in the hyperplanes orthogonal to the restricted roots.

Chambers and orbit representatives

The reflecting hyperplanes cut $\mathfrak a$ into Weyl chambers, and the restricted Weyl group acts simply transitively on the chambers. A closed positive chamber is a fundamental domain for this action. Consequently many conjugacy questions in $G$ reduce to finite symmetry on $\mathfrak a$ ; for example, the Cartan decomposition parametrizes double cosets $K\backslash G/K$ by a closed chamber. See Knapp, Chapter VI, §§4–5 and Chapter VII, §3.

Comparison with the complex Weyl group

The restricted roots are the nonzero restrictions to $\mathfrak a$ of complex roots, and distinct complex roots may have the same restriction. Accordingly, $W(G,\mathfrak a)$ need not be the full Weyl group of the complexified Lie algebra. It is the symmetry group adapted to the real form and is often called the little Weyl group.

Example

For $G=\operatorname{SL}(n,\mathbb R)$ , take $\mathfrak a$ to be the trace-zero diagonal matrices and $K=\operatorname{SO}(n)$ . Permutation matrices, adjusted when necessary to have determinant $1$ , normalize $\mathfrak a$ ; after quotienting by the diagonal centralizer, one obtains $W(G,\mathfrak a)\cong S_n$ , acting by permuting diagonal entries.

References

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Progress in Mathematics 140, Birkhäuser, 2002. Author-maintained record. Relevant: Chapter VI, §§4–5 and Chapter VII, §§3 and 7.
Sigurdur Helgason, Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions, Mathematical Surveys and Monographs 83, American Mathematical Society, 2000. AMS record. Relevant: Chapter I on symmetric spaces and restricted Weyl groups.