Parabolic subgroup

2020 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 $\nu=(n_1,\dots,n_t)$ 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