Name for a particular subgroup of parabolic subgroups of the general linear groups. [duplicate]

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Name for a particular subgroup of parabolic subgroups of the general linear groups.

Let $V$ be vector space. The subgroup $P$ of $GL(V)$ consisting of all automorphisms stabilizing a flag $V=V_1\supset V_2\supset\cdots\supset V_1$ is called a parabolic subgroup of $GL(V)$. I am interested in the subgroup $Q$ of $P$ consisting of all automorphisms $h$ such that the induced automorphism $\overline{h}: V_i/V_{i+1}\rightarrow V_i/V_{i+1}$ is the identity for every $i$. My question is:

Is this subgroup $Q$ named somewhere yet? If not, can you recommend a name?

Similarly, I am also interested in the subgroup $T$ of $P$ consisting of automorphisms $h$ such that the group of induced automorphisms $\overline{h}: V_i/V_{i+1}\rightarrow V_i/V_{i+1}$ is the symmetric group (on a basis of $V_i/V_{i+1}$). Is this subgroup $T$ named somewhere yet? If not, can you recommend a name? Is there anyway to realize that $\overline{h}: V_i/V_{i+1}\rightarrow V_i/V_{i+1}$ is a symmetric group without looking a specific basis of $V_i/V_{i+1}$)?

Finally, I would appreciate very much if you have any reference on the study of these subgroups.

Answer 1

As David says $Q$ is the unipotent radical of $P$. The subgroup $T$ is a preimage of the Weyl group $W$ of the group $G_i\cong GL(V_i/ V_{i+1})$. This group $T$ looks a direct product of $Q$ with a big chunk of a Levi complement of $P$. The Levi complement is a direct product isomorphic to $G_1\times\cdots \times G_k$; to obtain the group $T$, you replace the $i$-th factor by the normalizer $N$ of a maximal split torus $T_0$ of $G_i$.

This is, in fact, the typical way to realize the Weyl group of $G_i$ -- $W$ is isomorphic to the quotient $N/T_0$ -- but this is effectively the same thing as your method of fixing a specific basis of $V_i/V_{i+1}$. The Weyl group rears its head in lots of different ways (most especially as a Coxeter group related to the Dynkin diagram of $G_i$) so this is certainly not the only way to realise it. I don't, however, see any other way to realise your group $T$ (although it depends what you mean by `realise'!).

As for references, it depends on what kind of approach you want. If you want a treatment of $GL_n$ as an algebraic group then I recommend anything by Carter or Humphreys, or else there is the book by Borel. All of these people work in much greater generality than $GL_n$ though. If you just want to understand $GL_n$, then standard algebra texts like the one of Jacobson might be your best bet. (I have e-copies of some of these. If you want them, email me.)