Parabolic subgroup

A parabolic subgroup of a linear algebraic group defined over a field is a subgroup , closed in the Zariski topology, for which the quotient space is a projective algebraic variety. A subgroup is a parabolic subgroup if and only if it contains some Borel subgroup of the group . A parabolic subgroup of the group of -rational points of the group is a subgroup that is the group of -rational points of some parabolic subgroup in and which is dense in in the Zariski topology. If and is the Lie algebra of , then a closed subgroup is a parabolic subgroup if and only if its Lie algebra is a parabolic subalgebra of .

Let be a connected reductive linear algebraic group, defined over the (arbitrary) ground field . A -subgroup of is a closed subgroup which is defined over . Minimal parabolic -subgroups play in the theory over the same role as Borel subgroups play for an algebraically closed field (see ). In particular, two arbitrary minimal parabolic -subgroups of are conjugate over . If two parabolic -subgroups of are conjugate over some extension of the field , then they are conjugate over . The set of conjugacy classes of parabolic subgroups (respectively, the set of conjugacy classes of parabolic -subgroups) of has (respectively, ) elements, where is the rank of the commutator subgroup of the group , and is its -rank, i.e. the dimension of a maximal torus in that splits over . More precisely, each such class is defined by a subset of the set of simple roots (respectively, simple -roots) of the group 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 of a group 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 -closed reductive subgroup, called a Levi subgroup of the group . Any two Levi subgroups in a parabolic subgroup are conjugate by means of an element of that is rational over . Two parabolic subgroups of a group are called opposite if their intersection is a Levi subgroup of each of them. A closed subgroup of a group is a parabolic subgroup if and only if it coincides with the normalizer of its unipotent radical. Each maximal closed subgroup of a group is either a parabolic subgroup or has a reductive connected component of the unit (see , ).

The parabolic subgroups of the group of non-singular linear transformations of an -dimensional vector space over a field are precisely the subgroups consisting of all automorphisms of the space which preserve a fixed flag of type of . The quotient space is the variety of all flags of type in the space .

In the case where , the parabolic -subgroups admit the following geometric interpretation (see ). Let be a non-compact real semi-simple Lie group defined by the group of real points of a semi-simple algebraic group which is defined over . A subgroup of is a parabolic subgroup if and only if it coincides with the group of motions of the corresponding non-compact symmetric space preserving some -pencil of geodesic rays of (two geodesic rays of are said to belong to the same -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 is a subgroup of the group that is conjugate to a subgroup containing . Each parabolic subgroup coincides with its normalizer. The intersection of any two parabolic subgroups contains a subgroup of that is conjugate to . In particular, a parabolic subgroup of a Tits system associated with a reductive linear algebraic group is the same as a parabolic subgroup of the group (see [3], [4]).

References

[1]	A. Borel, J. Tits, "Groupes réductifs" Publ. Math. IHES , 27 (1965) pp. 55–150
[2]	A. Borel, J. Tits, "Eléments unipotents et sous-groupes paraboliques de groupes réductifs I" Invent. Math. , 12 (1971) pp. 95–104
[3]	N. Bourbaki, "Groupes et algèbres de Lie" , Hermann (1975) pp. Chapts. VII-VIII
[4]	J.E. Humphreys, "Linear algebraic groups" , Springer (1975)
[5]	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

References

[a1]	A. Borel, "Linear algebraic groups" , Benjamin (1969)