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Parabolic subgroup

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2010 Mathematics Subject Classification: Primary: 14L Secondary: 20G [MSN][ZBL]

A parabolic subgroup of a linear algebraic group $G$ defined over a field $k$ is a subgroup $P\subset G$, closed in the Zariski topology, for which the quotient space $G/P$ is a projective algebraic variety. A subgroup $P\subset G$ is a parabolic subgroup if and only if it contains some Borel subgroup of the group $G$. A parabolic subgroup of the group $G_k$ of $k$-rational points of the group $G$ is a subgroup $P_k\subset G_k$ that is the group of $k$-rational points of some parabolic subgroup $P$ in $G$ and which is dense in $P$ in the Zariski topology. If $\textrm{char}\; k = 0$ and $\def\fg{\mathfrak{g}}$ is the Lie algebra of $G$, then a closed subgroup $P\subset G$ is a parabolic subgroup if and only if its Lie algebra is a parabolic subalgebra of $\fg$.

Let $G$ be a connected reductive linear algebraic group, defined over the (arbitrary) ground field $k$. A $k$-subgroup of $G$ is a closed subgroup which is defined over $k$. Minimal parabolic $k$-subgroups play in the theory over $k$ the same role as Borel subgroups play for an algebraically closed field (see ). In particular, two arbitrary minimal parabolic $k$-subgroups of $G$ are conjugate over $k$. If two parabolic $k$-subgroups of $G$ are conjugate over some extension of the field $k$, then they are conjugate over $k$. The set of conjugacy classes of parabolic subgroups (respectively, the set of conjugacy classes of parabolic $k$-subgroups) of $G$ has $2^r$ (respectively, $2^{r_k}$) elements, where $r$ is the rank of the commutator subgroup $[G,G]$ of the group $G$, and $r_k$ is its $k$-rank, i.e. the dimension of a maximal torus in $[G,G]$ that splits over $k$. More precisely, each such class is defined by a subset of the set of simple roots (respectively, simple $k$-roots) of the group $G$ in an analogous way to that in which each parabolic subalgebra of a reductive Lie algebra is conjugate to one of the standard subalgebras (see , ).

Each parabolic subgroup $P$ of a group $G$ is connected, coincides with its normalizer and admits a Levi decomposition, i.e. it can be represented in the form of the semi-direct product of its unipotent radical and a $k$-closed reductive subgroup, called a Levi subgroup of the group $P$. Any two Levi subgroups in a parabolic subgroup $P$ are conjugate by means of an element of $P$ that is rational over $k$. Two parabolic subgroups of a group $G$ are called opposite if their intersection is a Levi subgroup of each of them. A closed subgroup of a group $G$ is a parabolic subgroup if and only if it coincides with the normalizer of its unipotent radical. Each maximal closed subgroup of a group $G$ is either a parabolic subgroup or has a reductive connected component of the unit (see , ).

The parabolic subgroups of the group $\def\GL{\textrm{GL}}\GL_n(k)$ of non-singular linear transformations of an $V$-dimensional vector space $k$ over a field $k$ are precisely the subgroups $P(\nu)$ consisting of all automorphisms of the space $V$ which preserve a fixed flag of type $\nu=(n_1,\dots,nt)$ of $V$. The quotient space $\GL_n(k)/P(\nu)$ is the variety of all flags of type $\nu$ in the space $V$.

In the case where $k=\R$, the parabolic $\R$-subgroups admit the following geometric interpretation (see ). Let $G_\R$ be a non-compact real semi-simple Lie group defined by the group of real points of a semi-simple algebraic group $G$ which is defined over $\R$. A subgroup of $G_\R$ is a parabolic subgroup if and only if it coincides with the group of motions of the corresponding non-compact symmetric space $M$ preserving some $\R$-pencil of geodesic rays of $M$ (two geodesic rays of $M$ are said to belong to the same $\R$-pencil if the distance between two points, moving with the same fixed velocity along their rays to infinity, has a finite limit).

A parabolic subgroup of a Tits system $(G,B,N,S)$ is a subgroup of the group $G$ that is conjugate to a subgroup containing $B$. Each parabolic subgroup coincides with its normalizer. The intersection of any two parabolic subgroups contains a subgroup of $G$ that is conjugate to $T=B\cap N$. In particular, a parabolic subgroup of a Tits system associated with a reductive linear algebraic group $G$ is the same as a parabolic subgroup of the group $G$ (see [Bo], [Hu]).

References

[Bo] N. Bourbaki, "Groupes et algèbres de Lie", Hermann (1975) pp. Chapts. VII-VIII MR0682756 MR0573068 MR0271276 MR0240238 MR0132805 Zbl 0329.17002
[Bo2] A. Borel, "Linear algebraic groups", Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[BoTi] A. Borel, J. Tits, "Groupes réductifs" Publ. Math. IHES, 27 (1965) pp. 55–150 MR0207712 Zbl 0145.17402
[BoTi2] A. Borel, J. Tits, "Eléments unipotents et sous-groupes paraboliques de groupes réductifs I" Invent. Math., 12 (1971) pp. 95–104 MR0294349 Zbl 0238.20055
[Hu] J.E. Humphreys, "Linear algebraic groups", Springer (1975) MR0396773 Zbl 0325.20039
[Ka] F.I. Karpelevich, "The geometry of geodesics and the eigenfunctions of the Laplace–Beltrami operator on symmetric spaces" Trans. Moscow Math. Soc., 14 (1967) pp. 51–199 Trudy Moskov. Mat. Obshch., 14 (1965) pp. 48–185
How to Cite This Entry:
Parabolic subgroup. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Parabolic_subgroup&oldid=25649
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article