Minimal parabolic subgroup - Imported Knowlpedia Content

Definition

Let $G$ be a connected real reductive Lie group. A minimal parabolic subgroup $P_0$ is a parabolic subgroup containing no proper parabolic subgroup of $G$ . Choose a Cartan decomposition $\mathfrak g=\mathfrak k\oplus\mathfrak p$ , a maximal abelian $\mathfrak a\subseteq\mathfrak p$ , and positive restricted roots. If $A=\exp(\mathfrak a)$ , $N$ has Lie algebra equal to the sum of the positive restricted-root spaces, and $M=Z_K(A)$ , then

P_0=MAN

is a minimal parabolic. Every minimal parabolic subgroup is conjugate to one obtained in this way.

Relation to the Iwasawa decomposition

The Iwasawa multiplication map $K\times A\times N\to G$ is a diffeomorphism. Replacing $K$ by its centralizer $M=Z_K(A)$ gives the subgroup $P_0=MAN$ , and the Iwasawa decomposition implies $G=KP_0$ . The quotient $G/P_0$ is therefore compact and identifies with $K/M$ . These structural facts are developed in Knapp, Chapter VI, §§4–5 and Chapter VII, §7.

Examples

For $G=\operatorname{SL}(n,\mathbb R)$ , the upper triangular matrices form a minimal parabolic subgroup. Here $A$ consists of positive diagonal matrices of determinant $1$ , $N$ consists of upper unitriangular matrices, and $M$ consists of diagonal sign matrices in $\operatorname{SO}(n)$ . For $\operatorname{SL}(2,\mathbb R)$ , this subgroup stabilizes a point of the projective line.

Conventions and uses

Minimal means minimal among parabolic subgroups, not among all nontrivial closed subgroups. A minimal parabolic need not be solvable because its compact factor $M$ can be nonabelian. It supplies both the induction subgroup for minimal principal series and the quotient model $G/P_0$ of the Furstenberg boundary.

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 on Iwasawa decomposition and Chapter VII, §7 on parabolics.
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 $KAN$ , $MAN$ , and compact homogeneous quotients.