Why we need to study representations of matrix groups? For example, the group $\operatorname{SL}_2(\mathbb F_q)$, where $\mathbb F_q$ is the field with $q$ elements, is studied by Drinfeld. I think that these groups are already given by matrices. The representation theory is to represent elements in an algebra or group (or other algebraic structure) by matrices. Why we still need to study representations of matrix groups? Thank you very much.
[Math] Why we need to study representations of matrix groups
ag.algebraic-geometryalgebraic-groupsrt.representation-theory
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As Torsten points out, unipotent groups correspond naturally to nilpotent Lie algebras in characteristic 0. This is dealt with nicely on the scheme level, for example, in IV.2.4 of Demazure-Gabriel Groupes algebriques. They also treat in Chapter IV some questions about prime characteristic, which get quite tricky outside the commutative case. Both of your questions are more conveniently studied in the Lie algebra framework, I think, where standard Lie algebra methods for discussing forms in are available and where there is quite a bit of literature on structure, representations, and (in small dimensions) classification in characteristic 0. See for example Jacobson's 1962 book Lie Algebras.
Representation theory is potentially very complicated for nilpotent Lie algebras (say over the complex or real field), even in the finite dimensional situation: unlike the semisimple case, there is no nice general structure based on highest weights, etc. Dixmier and others have studied infinite dimensional representations extensively in connection with Lie groups. Classification of nilpotent Lie algebras is just about impossible in general, but up to dimension 7 or so there are lists. Anyway, there is a lot of literature out there. (Tori are on the other hand also studied a lot over fields of interest in number theory. They have at least the advantage of being commmutative.)
[ADDED] An older seminar write-up might be worth consulting, especially in prime characteristic, along with the relatively sparse literature published since then and best searched through MathSciNet: Unipotent Algebraic Groups by T. Kambayashi, M. Miyanishi, M. Takeuchi, Springer Lecture Notes in Math. 414 (1974). But as their treatment suggests, the main research challenges have occurred in prime characteristic. Over finite fields, there has been quite a bit of recent activity in studying the characters of finite unipotent groups related to the unipotent radical of a Borel subgroup.
The quadratic form whose matrix is $\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & I_n \\ 0 & I_n & 0 \end{pmatrix}$ gives an embedding of $\operatorname{SO}(2n + 1, \mathbb C)$ in $\operatorname{GL}(2n + 1, \mathbb C)$ whose derivative is your specified embedding $\mathfrak{so}(2n + 1, \mathbb C) \to \mathfrak{gl}(2n + 1, \mathbb C)$. Under this embedding,
- $\left\{\begin{pmatrix} 1 & 0 & 0 \\ 0 & A & 0 \\ 0 & 0 & A^{-\mathsf T} \end{pmatrix}\mathrel: \text{$A$ diagonal}\right\}$ is the image of a maximal (algebraic) torus in $\operatorname{SO}(2n + 1, \mathbb C)$,
- the image of a maximal unipotent subgroup of $\operatorname{SO}(2n + 1, \mathbb C)$ is generated by
- $\exp\begin{pmatrix} 0 & 0 & 0 \\ 0 & E_{ij} & 0 \\ 0 & 0 & -E_{ji} \end{pmatrix}$, where $i$ is less than $j$, and
- $\exp\begin{pmatrix} 0 & 0 & y \\ -y^{\mathsf T} & 0 & B \\ 0 & 0 & 0 \end{pmatrix}$, where $B$ is skew-symmetric, and
- the image of the Tits subgroup of $\operatorname{SO}(2n + 1, \mathbb C)$ is generated by
- $\begin{pmatrix} 1 & 0 & 0 \\ 0 & A & 0 \\ 0 & 0 & A^{-\mathsf T} \end{pmatrix}$, where $A = I - E_{ii} - E_{jj} + E_{ij} - E_{ji}$ with $i \ne j$, and
- $\begin{pmatrix} 1 & 0 & 0 \\ 0 & D' & D'' \\ 0 & D'' & D' \end{pmatrix}$, where $D'$ and $D''$ are diagonal $(0, 1)$-matrices such that $D' + D'' = I_n$.
(I originally made a comment to this effect, but I had the wrong quadratic form, and wrongly suggested that you had switched $x^{\mathsf T}$ and $y^{\mathsf T}$ in your embedding.)
Best Answer
I will answer a more general question.
Maybe algebraic geometers care about $X$, or maybe mathematical physicists, or maybe homotopy theorists, etc. Each discipline has techniques that can build a new object $X'$ out of $X$. These new objects are usually more inspiring (if less fundamental) than $X$ itself.
Maybe $X$ is projective $n$-space and algebraic geometers take $X'$ to be a Hilbert scheme. Or maybe $X$ is a manifold and mathematical physicists build the cotangent bundle of the cotangent bundle of $X$. Or maybe $X$ is a genus 3 surface and homotopy theorists build the loop space of the suspension of the space of maps from a genus 5 surface, or whatever.
The point is, these constructions preserve the $Aut(X)$ symmetry.
For some reason, every branch of mathematics seems to have a bunch of nice functors to the category of vector spaces over $\mathbb{C}$. Let $H$ be one such functor. Now $HX'$ will be a $\mathbb{C}$-vector space carrying information about the complicated object $X'$. But it will also be a representation of the group $Aut(X)$.
If the representation theory or $Aut(X)$ is already known, then we gain access to a powerful set of tools with which to study $HX'$. If it's not already known, then we should try to decompose $HX'$ anyway--this is the way to find irreducibles in the first place.
Knowing the representation theory of $Aut(X)$ lets us prepare for any possible construction of $X'$ and any possible functor $H$.