Question

please help with exercise 6

5 Prove the following generalization of Lemma 3: If P is a parabolic subgroup of G which is the stabilizer of the flag (W. , W), then the W, are the only subspaces of V (F) left invariant by P. (cont.) Using Exercise 5, show that any parabolic subgroup of G is self-normalizing.

Answer 1

Let P be a parabolic subgroup of GL(V ). By the definition, this means that
the quotient GL(V )/P is a complete variety.
Equivalently, this means that P is
the stabilizer in GL(V ) of a flag. The
parabolic subgroup uniquely determines
the flag, and thus the dimension sequence.
Conversely, two parabolic subgroups
have the same dimension sequence if and
only if they are conjugate. A parabolic
subgroup is a Borel subgroup if it equals the stabilizer of a full flag. Equivalently,
it is minimal among the parabolic
subgroups. Equivalently, it is maximal
among the solvable subgroups. Each
parabolic subgroup contains a Levi factor L, which
is a maximal among the maximal rank reductive subgroups of P. The subgroup
L is not uniquely determined by P, but is unique up to P-conjugacy. We can
choose L as the direct product of groups L = GL(Wf1)· · · GL(Wfs) where Wfi+1
is chosen as a subspace of Wi+1 such that Wi+1 = Wi ⊕ Wfi+1. Note that
dim GL(V ) = n
2