For questions about groups which have additional structure as a algebraic varieties (the vanishing sets of collections of polynomials) which is compatible with their group structure. Consider using with the (group-theory) tag.

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1answer
57 views

How to show that $\mathcal{O}_q[U]$ is isomorphic to $U_q(\mathfrak{n})$?

Let $U$ be the positive unipotent radical of $SL_n$ and $\mathfrak{n}$ the Lie algebra of $U$. How to show that $\mathcal{O}_q[U]$ is isomorphic to $U_q(\mathfrak{n})$? Here $\mathcal{O}_q[U]$ is the ...
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0answers
20 views

Matrix logarithms for various algebraic groups

For any field $k$ of characteristic $p$ the sets $$\left\{g \in GL_n(k) \ \middle| \ g^p = 1\right\} \qquad \text{and} \qquad \left\{x \in \mathbb M_n(k) \ \middle| \ x^p = 0\right\}$$ are in ...
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0answers
17 views

Local systems on algebraic groups

Two questions about local systems: Given an algebraic group G over a perfect field $k$ and given a colection of vectorial spaces $(F_y)_{y\in G}$ why there is an unique local system $F$ such that the ...
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1answer
31 views

How to show that a representation is supercuspidal?

Let $G$ be a reductive group. If we know a representation $\pi$ of $G$ explicitly, how could we determine that $\pi$ is supercuspidal or not? Are there some references about this? Thank you very much. ...
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1answer
85 views

About the simply-connectedness of algebraic groups

Let $G$ be a simply-connected algebraic group. Is it necessarily true that its derived subgroup $G'$ is also simply-connected?
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2answers
133 views

Determining if these surjections have sections

Let $\pi: GL(2,k)\rightarrow PGL(2,k)$ be the canonical homomorphism, and pick some finite subgroup $G\subset PGL(2,k)$. Then we have an exact sequence $$1\rightarrow \{\alpha I\mid \alpha \in k\} ...
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0answers
16 views

Borel - Regular elements

In Borel's Linear Algebraic Groups (2ed) page 160 a regular element is defined in terms of its semisimple part, “thus $g$ is regular if and only if $g_s$ is regular.” A unipotent element $g$ has ...
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1answer
107 views

Are semisimplicity and regularity closed or open conditions in an algebraic group $G$?

Let $G$ be a connected algebraic group over an algebraically closed field. I'm trying to understand the phrase "the subvariety of semisimple elements in $G$ which are not regular." This tacitly ...
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1answer
67 views

Equivalent definitions of a regular element in an algebraic group

Let $G$ be a connected algebraic group over an algebraically closed field $k$. I'd like to see that the following two definitions are equivalent. An element $g\in G$ is regular if $\dim(C_G(g))$ is ...
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1answer
30 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.
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31 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 ...
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55 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 ...
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1answer
141 views

What is the unipotent radical of a parabolic subgroup?

What is the unipotent radical of a parabolic subgroup? For example, let $$ P=\{ \left( \begin{matrix} a & b & k & c \\ d & e & f & g \\ 0 & 0 & h & i \\ 0 & 0 ...
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0answers
43 views

Coset variety of an algebraic group

Let $k$ be an algebraically closed field of characteristic $p\geq0$. An affine algebraic group $G$ is an affine algebraic variety (a Zariski closed subset of $k^m$ for some $m$) such that ...
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1answer
50 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) ...
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1answer
67 views

Closed conjugacy classes in $M_n(k)$

Let $k$ be an algebraically closed field, $n$ a positive integer, and consider the action of $\mathrm{GL}_n(k)$ on $M_n(k)$ by conjugation. My professor tells me that semisimple conjugacy classes are ...
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1answer
121 views

How to prove that $\zeta*\zeta=\zeta$?

Let $F$ be a non-archimedean local field and $\mathcal{O}_F$ the ring of integers in $F$. Let $G_F=GL_2(F)$. Let $\pi_i$, $i=1,\ldots,n$,be non-equivalent finite dimensional irreducible ...
4
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1answer
50 views

Is $GL_2(\mathbb{Z}_p)$ a maximal compact subgroup of $GL_2(\mathbb{Q}_p)$?

Let $\mathbb{Q}_p$ be the set of all p-adic numbers and $\mathbb{Z}_p$ the set of all p-adic integers. Is $GL_2(\mathbb{Z}_p)$ a maximal compact subgroup of $GL_2(\mathbb{Q}_p)$? Thank you very ...
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0answers
33 views

Simple connected semi-simple group

Here is a question form springer's book Linear Algebraic Groups, 8.4.6(6) Let $G$ be semi-simple and simple-connected, $P$ a parabolic subgroup of $G$ with Levi group $L$, Prove the commutator ...
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2answers
65 views

Proving the associativity of a monoid with $a \circ b = a+b-ab$

For university, it was my excercise to proof the associativity of the monoid $$H=(\mathbb{Q},\circ)\text{ with } a \circ b := a+b-ab\quad(a,b \in \mathbb{Q})$$ The excercise instructor gave us the ...
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1answer
44 views

For an element $x$ in an algebraic group $G$, why do we have $\mathscr{L}(C_G(x))\subset\mathfrak{c}_{\mathfrak{g}}(x)$?

I'm reading Humphreys' Linear Algebraic Groups, trying to understand the following argument found on the top of pg. 76. Let $G$ be an algebraic group over some field $k$, with $x\in G$. Let ...
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1answer
153 views

Elements of order 3 in $PGL(4,\mathbb{R})$

I need to classify all elements of order 3 up to conjugation in $PGL(4,\mathbb{R})$. It's sufficient to give a representative of each conjugacy class. My thoughts: consider instead $GL(4,\mathbb{C})$ ...
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1answer
49 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 ...
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4answers
59 views

Index of $\langle a^4\rangle$ in the group $\langle a\rangle$

Let $a$ be an element of order 30 in a group $G$. What is the index of $\langle a^4\rangle$ in the group $\langle a \rangle$? The answer is 2, however I have no idea on how to obtain that answer. ...
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2answers
51 views

Does the quotient of an algebraic group by its neutral component always split?

Let $G$ be a complex algebraic group. Let $G^\circ\subseteq G$ be the connected component of $G$ which contains the neutral element $1\in G$. Then, it can be shown that $G^\circ\trianglelefteq G$ is a ...
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1answer
68 views

A question about reductive groups's structure.

There is a flaw somewhere in the following argument but I can't track it. Take a reductive connected affine algebraic group $G$ : by definition, its unipotent radical $R_u(G)$ is trivial. One have ...
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2answers
104 views

Dimension of a $G$-variety $X$ that is a finite union of $G$-orbits

Suppose that $G$ is an algebraic group acting on a variety $X$, and $X$ is a finite disjoint union of $G$-orbits $\mathcal{O}_i$, $i=1,\ldots,n$, under this action. Is it true that the dimension of ...
2
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1answer
71 views

Confusion about the quotient $G/B$

Let $G$ be an affine, complex, reductive algebraic group and $B$ a Borel of $G$. I have seen and understood the proof that $G/B$ is projective. Now, on the other hand, I have made the following ...
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1answer
77 views

Linearization of a group action: why the map is equivariant?

I'm using Dolgachev's book on invariant theory to learn linearizations of group actions. Here is a sketch of main construction: let linear algebraic group $G$ act on a quasi-projective variety $X$, ...
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1answer
53 views

Torus orbit closures and rank-1 subtori

Suppose I have a connected complex torus $K$ acting on a quasi-affine complex variety $X$. Suppose also that I have $p,q\in X$ such that the orbit $Kq$ is closed in $X$ and $q\in ...
2
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0answers
60 views

finite normal subgroup

$G$ is a subgroup of finite index in $SL(n,Z)$, $n\ge 3$, $N$ is finite normal subgroup of $G$, then I want to know why $N$ is a normal subgroup of $SL(n,Z)$. More generally, $A$ is an arithmetic ...
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0answers
41 views

Semisimple part of a nilpotent connected affine algebraic group

These notes on affine algebraic groups mention the following theorem. Let $G$ be a connected nilpotent affine algebraic group (over an algebraically closed field $k$), and denote $G_s$ and $G_u$ ...
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83 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 $\check\alpha(t) f_\alpha(x) \check\alpha^{-1}(t)= f_\alpha(x)$ ...
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0answers
152 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 ...
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2answers
142 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 ...
3
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1answer
36 views

Does the exponential map respect module actions?

Setup: Let $k$ be a field and $G \subseteq \mathrm{GL}_n(k)$ an algebraic group, reductive if that makes a difference. Let $\mathfrak g \subseteq \mathfrak{gl}_n(k)$ be the Lie algebra of $G$ with ...
2
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1answer
146 views

What is the group of $k$-rational points of an algebraic group?

Let $k$ be a field and $G$ a linear algebraic group over $k$. What is the group of $k$-rational points of $G$? By definition, $G$ is an algebraic variety. Suppose that $G$ is defined by polynomials ...
2
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1answer
94 views

The $\mathbb C((z))$-rational points of a complex semi-simple group $G$

By definition, if $R$ is a $\mathbb C$-algebra and $G$ is a $\mathbb C$-scheme then the set of $R$-valued points on $G$ is $G(R)=\hom_{\text{Sch}_{\mathbb C}}(\operatorname{Spec} R, G)$ In Ginzburg's ...
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249 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 ...
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152 views

a closed subset of an algebraic group with a constant tangent space is a coset

Let $G$ be an algebraic (not necessarily linear) group and let $Z \subset G$ be a Zariski closed irreducible subset. Since tangent bundle of $G$ is trivial, we may identify tangent spaces at all ...
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0answers
165 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 ...
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78 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 ...
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2answers
265 views

How many elements of order $k$ are in $S_n$?

I need to find how many elements of order $k$ are in $S_n$ (where $k \leq n$). So if $k$ is prime, it's easy: $k$ can't be the $\mathrm{lcm}$ of any integers besides itself and one's (which we're ...
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1answer
69 views

Factor/Quotient Group $M/\mathbb{Z}$

I've posted another question here a few days ago asking what a factor/quotient group is because I couldn't wrap my mind around it. Although I have an idea what it means, I still don't fully understand ...
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0answers
37 views

Closed subgroups of algebraic group have DCC?

Are there any infinite descending chains of closed subgroups of the general linear group over a field? More specifically, is my argument ok? Can you fill in some of the details? Prop: No. Proof: If ...
3
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1answer
182 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, ...
3
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1answer
68 views

Non-uniqueness of group structure for affine algebraic groups

We know that every abelian variety has a unique group structure, but in the affine case, is that every affine algebraic group has more than one (up to isomorphism) group structure?
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33 views

Baily Borel Compactification: choice of boundary

In Borel/Ji " compactifications of symmetric and locally symmetric spaces " the Baily Borel compactification of a locally symmetric space is defined as $$\Pi\backslash(X\coprod_{\bf{P}}X_{P,h})$$ ...
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56 views

Lie algebra of the unipotent radical of a standard parabolic subgroup in $GL_n$

Let $k$ be a field, and consider the algebraic group $G=GL_n(k)$. For any partition $n_1+n_2+\ldots+n_m=n$, we have a parabolic subgroup of the form ...
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0answers
50 views

Orbits and rational points in a $G$-variety

Let $K/k$ be a field extension, let $V_0$ be a variety over $k$, and let $V=V_0\times_k\mathrm{Spec}\;K$, so that we can speak of the $k$-rational points of $V$ as morphisms $\mathrm{Spec }\;k\to ...