0
votes
0answers
6 views

How to show that $GL_n/U$ is birationally isomorphic to $B^-$?

It is said that $GL_n/U$ is birationally isomorphic to $B^-$. Here $U$ acts by right multiplication on $GL_n$. I think that $GL_n/U$ consisting of cosets. Two matrices in the same coset if any two ...
3
votes
2answers
66 views

T//W for adjoint type group PGL3

Let $G$ be a reductive algebraic group and $T$ a maximal torus (over $\mathbb{C}$). It is well known that if $G$ is simply connected type then $T//W = \mathbb{A}^r$. I want to verify that the ...
2
votes
1answer
25 views

Maximal tori in $SO(n,\mathbb{C})$

What are maximal tori in $SO(n,\mathbb{C})$? (not $SO(n,\mathbb{R})$) Can a maximal torus in $SO(n,\mathbb{C})$ be written as $T\cap SO(n,\mathbb{C})$ for some maximal torus $T$ in ...
1
vote
1answer
38 views

How to characterize elements in the Bruhat open cell?

This might be an elementary question. For simplicity, let's assume $G=GL(n,F)$, where $F$ is a local field. Let $U$ be the subgroup of upper triangular unipotents, $A$ the subgroup of diagonal ...
6
votes
2answers
184 views

Lie Groups/Lie algebras to algebraic groups

I am reading some lie groups/lie algebras on my own.. I am using Brian Hall's Lie Groups, Lie Algebras, and Representations: An Elementary Introduction I was checking for some other references on ...
0
votes
0answers
22 views

Are there some examples of Kac-Moody groups which are not reductive?

Are there some examples of Kac-Moody groups which are not reductive? Thank you very much.
1
vote
1answer
29 views

Conditions for a matrix in $Bw_0B$.

Let $B$ be the set of all upper triangular matrices in $GL_n$. What are the conditions for a matrix in $GL_n$ lies in $Bw_0B$ (What do the matrices in $Bw_0B$ look like)? Thank you very much.
0
votes
0answers
26 views

Question about a notation.

I have a question about the notation in the paper. On page 8, the 3-rd line from bottom, it is said that $h_J(t)$ is the corresponding diagonal matrix in $GL_4$. I think that $J$ is some set such that ...
0
votes
0answers
46 views

Questions about unipotent matrices.

I have a question about the notation in the paper. On page 8, the 5-th line from bottom, it is said that $u_J(t)$ denotes the upper triangular unipotent matrix with $t$ in position corresponding to ...
1
vote
0answers
28 views

Unimodular groups

Let F be a non-archimedean local fields of characteristic p (for p any given prime number): the field of formal Laurent series Fq((T)) over a finite field Fq (where q is a power of p). Is GL(n,F) ...
0
votes
1answer
41 views

Orthogonal invariants of an irredubile GL-representation

Let $n\in 2\mathbb Z$ be an even number. Let $G=\operatorname{GL}_n(\mathbb{C})$ and $V_\lambda$ the irreducible complex $G$-module corresponding to the partition ...
4
votes
0answers
80 views

Making the definition of dual root unambiguous

In 5.4 of his book Lectures on Invariant Theory, Igor Dolgachev introduces the dual of a root by requiring that $\check\alpha(t) f_\alpha(x) \check\alpha^{-1}(t)= f_\alpha(x)$ ...
2
votes
0answers
55 views

Confused about Borel-Weil theorem

I am trying to understand the Borel-Weil theorem, but I am very confused because of the different conventions used in different sources. I am especially confused about two things: (1) the definition ...
5
votes
2answers
121 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 ...
0
votes
0answers
102 views

Why Bruhat decomposition in $GL_n$ case is the Gauss decomposition?

Gauss decomposition of a matrix is also called LU decomposition. Let $A$ be a matrix. Then $A=LU$ for some lower triangular matrix $L$ and upper triangular matrix $U$. This can be obtained using Gauss ...
1
vote
0answers
85 views

Questions about affine Weyl group and extended affine Weyl group for SL2.

Let $G=SL_2$. Then the Weyl group is generated by $s_1$. On page 3 of the lecture notes, it is said that the affine Weyl group is generated by $s_0, s_1$. (1) The element $s_0s_1$ can be identified ...
1
vote
0answers
75 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 ...
0
votes
0answers
30 views

What is a quotient of a building by a lattice?

For an algebraic group $G$, we may defined a building associated to $G$. Let $B = B(G)$ be the corresponding building. I don't understand much about the concept quotient $B/\Gamma$ of a building $B$ ...
3
votes
1answer
83 views

Minimal parabolic subgroups of a reductive group - Bruhat type decomposition

Let $G$ be a reductive group, $B$ a Borel subgroup, $P$ a minimal parabolic subgroup having a Levi decomposition $P = UL$, let $\alpha$ be one of the two roots of $L$ relative to $T$, and $U_\alpha, ...
0
votes
0answers
31 views

References request about exponentials in Lie algebras.

I saw two formulas about Lie algebras. Let $G$ be an algebraic group over $k$ and $\mathfrak{g}$ its Lie algebra. For any $x \in \mathfrak{g}$, $a \in k$ and $g \in G$, we have $$ g \exp(ax) g^{-1} = ...
4
votes
0answers
92 views

Role of the discrete subgroups of Lie groups

This is a question I don't believe is too vague to admit a sensible answer: In what directions can the structure theory of the discrete subgroups of real and p-adic Lie groups applied? What ...
1
vote
0answers
42 views

Isomorphism types of stabilizers of vectors in linear representations of the special linear group

Suppose we have a linear representation of the group $SL_d$ over $\mathbb{C}$. i.e. a finite dimensional vector space $V$ with a linear action of $SL_d$ on it. Let $v\in V$ be some vector and let ...
2
votes
1answer
101 views

Mal'cev completion of nilpotent groups

Is the $\mathbb{R}$-Mal'cev completion of a finitely generated torsion free nilpotent group connected and simply connected?. Thanks!
2
votes
0answers
160 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
223 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 ...
1
vote
0answers
35 views

Complexification of the inclusion $\text{U}_n\subset \text{GL}_n(\mathbb{C})$

What is the map $\text{GL}_n(\mathbb{C}) \to \text{GL}_n(\mathbb{C}) \times \text{GL}_n(\mathbb{C})$ named in the title? I guess it has something to do with the polar decomposition, but I can't manage ...
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 ...
4
votes
1answer
70 views

Is Bruhat cell dense in p-adic topology?

I've seen in literature a statement like 'there exists an open and dense Bruhat cell'. In $GL(2,F)$ for example, where $F$ is a p-adic field, let $\omega=\begin{pmatrix} & 1 \\ 1 & ...
2
votes
0answers
68 views

What is the Weyl group of this group?

Let $G$ be the group $GL_{n_1}(q^{l_1})\times GL_{n_2}(q^{l_2})\times GL_{n_3}(q^{l_3})$. Here $GL_{n_1}(q^{l_1})$ is the rational points of $GL(n,\bar {\mathbf{F}}_q)$. My question iis what is the ...
9
votes
0answers
127 views

p-adic Lie groups vs. algebraic groups over $\mathbb{Q}_p$

I am somewhat confused about the following two concepts and the relations between them- One concept is a Lie group $G$ over the $p$-adic field. This is defined in a similar fashion to a (real) Lie ...
4
votes
1answer
97 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 ...
0
votes
1answer
85 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 ...
11
votes
2answers
774 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
196 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
1answer
71 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
83 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
168 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 ...
3
votes
1answer
148 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
109 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
80 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 ...
9
votes
2answers
610 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 $G_{an}$, ...