Tagged Questions

0
votes
0answers
40 views

Why $k[\mathfrak{t}\oplus \mathfrak{t}]^{U\times U}\subseteq k[\det(a),a_{11}c_{22}+a_{22}c_{11},\det(c)]$?

Let $a= \left(\begin{array}{cc} a_{11} & 0 \\ 0 & a_{22} \\ \end{array}\right)$ and $c= \left(\begin{array}{cc} c_{11} & 0 \\ 0 & c_{22} \\ \end{array}\right)$ be in ...
4
votes
1answer
75 views

Characters of diagonalizable algebraic groups with no p-torsion

Let $G$ be a diagonalizable algebraic group and $X$ be the character group of $G$. Let $Y$ be a subgroup of $X$. We define $Y^{\perp}$ to be all the $x\in G$ such that $\chi(x)=1$ for all $\chi\in Y$. ...
5
votes
1answer
66 views

Dimension of the GL-orbit of d-forms in one less variable

Let $V:=k[x_0,\ldots,x_n]_d$ be the $k$-vector space of homogeneous polynomials of degree $d$. Let $G:=\mathrm{Gl}(n+1,k)$ act on $V$ induced by the canonical action on the linear forms: For ...
0
votes
1answer
57 views

A question about Lie algebras corresponding to Lie groups and algebraic groups

Lie groups and algebraic groups both correspond with Lie algebras, which are by definition the left invariant vector field. But the topology of Lie groups and algebraic groups are different. Are their ...
10
votes
2answers
364 views

References on Linear Algebraic Groups/Lie Theory

I am currently doing a course on Lie groups, Lie Algebras and Representation theory based on Brian Hall's book of the same name. We should cover upto chapter 4/5 in this book by the end of the ...
4
votes
1answer
92 views

Parabolic subgroups of $\mathrm{Sl}_n$ are the ones that stabilize some flag

I am looking for a reference for the above statement that every parabolic subgroup of $\mathrm{Sl}_n(\Bbbk)$ stabilizes some flag in $\Bbbk^n$. I have gone through a large pile of books and can't seem ...
3
votes
0answers
66 views

Semidirect product of reductive groups

Given two linearly reductive algebraic groups, is their semidirect product reductive again? By linearly reductive, I mean that any rational representation of the group is completely reducible. In ...
3
votes
1answer
67 views

Geometric difference between two actions of $GL_n(\mathbb{C})$ on $G\times \mathfrak{g}^*$

Let $G=GL_n(\mathbb{C})$. Scenerio 1: Let $G$ act on $T^*(G)=G\times \mathfrak{g}^*$ by $$ g.(x,y)=(gx,y). $$ Scenerio 2: Let $G$ act on $T^*(G)=G\times \mathfrak{g}^*$ by $$ ...
1
vote
2answers
68 views

Identifying $SL(2,\mathbb{C})/H$ with $\mathbb{C}^2\setminus \{ 0\}$

Let $G=SL(2,\mathbb{C})$ and let $H$ be the set of unipotent matrices $$ \left\{ \left[ \begin{array}{cc} 1 & b \\ 0 & 1 \\ \end{array} \right] : b\in \mathbb{C}\right\}. $$ I am ...
2
votes
1answer
106 views

Highest or positive weights (or roots)

Let $T= (\mathbb{C}^*)^2$ be embedded in $GL_2$ along its diagonal entries, and suppose $T$ acts on $M_2(\mathbb{C})$ via conjugation. Denote $\chi_i(g)=z_i$ where $$ g = \left( \begin{array}{cc} z_1 ...
0
votes
0answers
24 views

Relating two different sets of $1$-parameter subgroups $\mathop{Hom}(\mathbb{C}^*,G)$ and the exponential map $\exp$

I've been reading about algebraic groups and I came across something I do not yet fully understand. The following discussion is over the complex setting for simplicity. For an algebraic group $G$, a ...
3
votes
1answer
91 views

What's so special about unipotent groups

Why are they so important? I see them appear in Lie theory, algebraic geometry, etc. Can somebody elaborate? For example, can someone explain why they are such natural groups to consider in ...
0
votes
0answers
90 views

How can I generate $\mathrm{SL}(n,\mathbb Z)$ by the subgroup $\mathrm{SL}(n-1,\mathbb Z)$ and another Element of $\mathrm{SL}(n,\mathbb Z)$?

Let $\{z_1,...,z_n\}$ be the canonical Basis of $\mathbb{Z}^n$, such that $z_i$ equals the vector $(0,\dotsc,0,i,0,\dotsc,0)$ with a 1 in the $i$th position. I want to show that the ...
2
votes
0answers
56 views

symplectic representations: when could the center act trivially?

I'm considering a problem about symplectic representation of real reductive group, which fits more or less into the setting of symplectic representations discussed in Milne's survey ''Shimura ...
7
votes
2answers
422 views

Is the universal cover of an algebraic group an algebraic group?

Here algebraic group means affine algebraic group in both instances. Also I'm mainly interested in groups over $\mathbb{C}$. In fact I'm taking $\pi_1(G)$ to mean the fundamental group of $G_{an}$, ...