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).