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 |
Comments
References
[a1] | A. Borel, "Linear algebraic groups" , Benjamin (1969) |
Parabolic subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parabolic_subgroup&oldid=16195