-2
votes
1answer
39 views

reduced space of coadjoint orbit

Let $G$ be a compact Lie group and $\lambda\in \frak{g}^*$$=(Lie G)^*$ and $O_\lambda$ be the Coadjoint orbit through $\lambda\in \frak{g}^*$ and $\mu:O_\lambda\to\frak{g}^*$ be the moment map, ...
5
votes
0answers
72 views

Closed orbits of complete flags in $\mathbb{C}^n$

Let $B$ be a symmetric (or antisymmetric) non-degenerate bilinear form on $\mathbb{C}^n$ and let $G$ be the associated group of automorphisms $O(n)$ (resp. $Sp(n)$). What can we say about the ...
5
votes
2answers
114 views

What is the coordinate ring of $G/U$?

Let $G$ be an algebraic group and $U$ its subgroup consisting all upper triangular matrices. For example, $G=GL_n(k)$ and $U$ the subgroup consisting of all upper triangular unipotent matrices in ...
1
vote
0answers
74 views

P/B is isomorphic to the projective line $\mathbb{P}^1$

Suppose that $P \subset G$ is a parabolic subgroup containing a Borel subgroup $B$. Moreover, let $P$ be a minimal parabolic subgroup properly containing B, i.e., one corresponding to a single root ...
3
votes
3answers
173 views

What is the dimension of this Grassmannian?

Why is $2\times 3$ the dimension of $Gr_2(\mathbb{R}^5)$? and can one use the dimensions of Lie groups to derive this dimension? Note: $Gr_2(\mathbb{R}^5)$ denotes the Grassmannian of all ...
0
votes
1answer
51 views

What is this dimension?

What is the real dimension of the cone of $2$ by $2$ Hermitain matrices with at lease one eigenvalue that is $0$?
0
votes
0answers
28 views

parabolic subalgebra

Let $G$ be a semisimple lie group, a parabolic subgroup of $P$ is a connected subgroup that contains a conjugate of $B$, (which $B$ is Borel subgroup of $G$) then I can not see why lie algebra of $P$ ...
2
votes
0answers
151 views

Construction of line bundles on the flag variety

Edit: The following is phrased in terms of algebraic geometry, but can be thought of analytically as well. Hence I added some tags... I am a bit confused about the subject in the title. For ...
4
votes
2answers
214 views

Complex algebraic group is reductive $\iff$ it is the complexification of a compact Lie group?

By a complex reductive algebraic group I mean the group of complex points of a (possibly disconnected) affine algebraic group defined over $\mathbb{C}$ whose unipotent radical (maximal connected ...
7
votes
1answer
135 views

Is taking inverse automatically well-defined?

By the usual definition, Lie group is a manifold $G$ with a group structure on it such that the multiplication $m\colon G\times G\to G$ and taking inverse $i\colon G\to G$ are both smooth maps. But it ...
8
votes
1answer
331 views

Learning Roadmap for Borel - Weil - Bott Theorem

Next semester I may study a course where the ultimate goal is to get to the Borel - Weil - Bott (BWB) Theorem, if not at least try to understand it in the case that we have $G = \text{SL}_n$. I have ...
8
votes
1answer
133 views

Trivial summand of a representation's symmetric power

The following comes from Exercise 13.17 of Fulton and Harris's book, Representation Theory: A First Course. Let $V$ denote the standard representation of $\mathfrak{sl}_3\mathbb{C}$, with weights ...
4
votes
1answer
96 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 $\chi\in Y$. ...
3
votes
1answer
74 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 ...
8
votes
1answer
458 views

What is reductive group intuitively?

I am studying Geometric invariant theory and wonder how I should understand linearly reductive algebraic group. We say that an affine algebraic group $G$ is linearly reductive if all finite ...
1
vote
2answers
260 views

Connections of Geometric Group Theory with other areas of mathematics.

I'm a master's student in the Turin University. At the end of my studies, I have to write a master thesis. My main interest is geometric group theory, but it is not a research area of the Turin's ...
3
votes
1answer
145 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 ...
4
votes
1answer
609 views

Orbit-stabilizer theorem for Lie groups?

Let $G$ be a finite-dimensional lie group, with a transitive action on the points of a smooth finite-dimensional manifold $S$. Let $p$ be some point of $S$ and let $T$ be the stabilizer of $p$ in $G$. ...
2
votes
1answer
318 views

Homogeneous binary quartic forms to elliptic curves

Broadly I would like to know what is the connection between homogeneous binary quartic forms and elliptic curves. I see that the invariants on the first space under the action of $SL(2,\mathbb{C})$ ...
9
votes
2answers
596 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 $G_{an}$, ...