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1
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0answers
97 views

Conjugacy scheme, fppf versus GIT

I would be glad to have some guidance in the following. Let $k$ be an algebraically closed field. Let $G$ be a connected reductive group over $k$. Denote by $\mathfrak{c}$ the Zariski spectrum of the ...
6
votes
2answers
315 views

First Galois cohomology of Weil restriction of $\mathbb{G}_m$

Let $L/K$ be a finite Galois extension, write $G:= Gal(L/K)$. Denote by $R = Res(\mathbb{G}_m)$ the Weil restriction of $\mathbb{G}_m$, from $L$ to $K$. I want to show that its first Galois cohomology ...
2
votes
2answers
156 views

odd length Chevalley relations (in rank two)

The unipotent radicals $\text{N}$ of the Borel subgroups of the complex algebraic groups of type $A_2$, $B_2$, and $G_2$ can each be abstractly presented using two one-parameter subgroups $x_1, x_2: ...
2
votes
1answer
95 views

What finite groups are stabilizers in Kirwan's desingularization construction?

Assume $X$ is a smooth projective curve of genus $g\geq 3$ over $\mathbb{C}$ and let $M$ be the (singular) moduli space of semistable rank two vector bundles with trivial determinant on $X$. Then ...
0
votes
2answers
158 views

Existence of $B$-reduction of a $G$-torsor on a curve

Let $k$ be an algebraically closed field, $X$ a connected smooth curve over $k$, $G$ a connected reductive group over $k$, and $B \subset G$ a Borel subgroup. Given a $G$-torsor $E$ on $X$ in the ...
3
votes
1answer
288 views

Galois cohomology of a non-abelian group over a function field

Let $k$ be an algebraically closed field, and $X$ a connected smooth projective curve over $X$. Let $F$ be the function field of $k$. Let $G$ be an algebraic group over $k$ (assume that it is smooth, ...
0
votes
0answers
98 views

Complete reducibility of representations of reductive algebraic groups

I need a reasonably detailed reference for the proof of the fact that, in characteristic 0, any linear representation of a reductive algebric group is completely reducible. I looked in Humphries and ...
17
votes
2answers
607 views

Solving equations in SO(3) : an open problem by Jan Mycielski

I am interested in a problem closely related to a problem stated by Jan Mycielski in his paper Can One Solve Equations in Group? (The American Mathematical Monthly, 1977, ...
2
votes
2answers
152 views

Intersection of Subspaces with $O(3)$

Sorry for the confusion from earlier. I tried to fix the thread. The old version can be found below. For $6$-dimensional subspaces $V$ of the space $\mathbb{R}^{3\times 3}$ of real three-times-three ...
4
votes
2answers
164 views

Describing Levi factors and unipotent radicals of parabolic subgroups in classical groups

I asked this question before at Math.SE (link) but got no answer. Let $G$ be an algebraic group over an algebraically closed field $k$ of characteristic $p \geq 0$. Then any parabolic subgroup $P$ ...
3
votes
0answers
135 views

Are all finitely generated subgroups of SL2 LERF? [closed]

Let $H \leq \mathrm{SL}_2(\mathbb{R})$ be a finitely generated subgroup. Must $H$ be LERF? A group $H$ is said to be LERF (locally extended residually finite), or subgroup separable, if its ...
6
votes
2answers
392 views

What are the basic possibilities for a tensor product of two fields?

Let $k$ be a field, with $F,k'$ field extensions of $k$. The ring $k' \otimes_k F$ is denoted by $F_{k'}$. In Borel's Linear Algebraic Groups, it is claimed (I believe erroneously) that "each of ...
9
votes
1answer
230 views

Invariant ring of $S_5$

The irreducible representations of the Symmetric group $S_5$ are classified by the partitions of $5$. For the standard representation which corresponds to the partition (4,1) the ring of invariants is ...
3
votes
1answer
163 views

Is there a covering of Prym variety?

$\mathstrut$Hi, guys! Let $C$, $C^\prime$ be projective smooth irreducible algebraic curves over an algebraically closed field $k$ ($\mathrm{char}(k) \neq 2$), $\phi : C$ $\to$ $C^\prime$ a ...
4
votes
1answer
252 views

What is known about the algebraic variety defined by the group determinant?

What is known about the algebraic variety $V_G$ defined by $det(X_G) = 1$ where $X_G$ is the group matrix $(x_{g_ig_j^-1})$ of a finite group $G$? It is known that two finite groups having the same ...
0
votes
1answer
81 views

Maximal split torus of universal chevalley group

Let $G$ be simply connected chevalley group over a field $K$. I am following the notations as in 'Lectures on Chevalley group' by Steinberg (Yale lectures). Let $H$ be the subgroup generated by ...
10
votes
2answers
226 views

Can one describe the multiplication of two Bruhat cells?

For $G$ a simple linear algebraic group and $B$ a fixed Borel subgroup, we have the Bruhat decomposition $G = \coprod_{w \in W} B\dot{w}B$, where $W$ is the Weyl group and $\dot{w}$ is any ...
2
votes
0answers
51 views

Characterizing subgroups of R^n with dense factors

It is well known that (additive) subgroups of $\mathbb{R}^n$ are products of discrete subgroups (lattices) by dense subgroups in subspaces. My question is the following: given a generator set of $p$ ...
3
votes
0answers
97 views

The left regular representation of the Jacobi groups over local fields of characteristic >2 is type I?

Let $K$ be a non-archimedean local field of characteristic $>2$. Consider the Jacobi group $G=H_{2n+1}(K)\rtimes Sp_{2n}(K)$, which is the semidirect product of the Heisenberg group $H_{2n+1}(K)$ ...
3
votes
0answers
92 views

Metaplectic groups over non-archimedean local fields of characteristic>2

Let $K$ be a non-archimedean local field of characteristic $>2$. Consider the double cover metaplectic extension of symplectic groups $p: Mp_{2n}(K)\rightarrow ...
3
votes
1answer
79 views

Global bound for number of vertices in Bruhat-Tits building

Let $G$ denote a semisimple linear algebraic group over $\mathbb Q$ and let $r$ be its absolute rank. For any prime $p$ let $v_p$ be a vertex of the Bruhat-Tits building of $G({\mathbb Q}_p)$ and let ...
1
vote
1answer
76 views

Concept of Facets in the structure of reductive algebraic groups

Where can I find a precise definition of Facet ? In some online notes it is stated that Facet is a maximal subset of co-characters having the same sign for every root. But shouldn't then every facet ...
5
votes
0answers
148 views

Homogeneous spaces of affine algebraic groups

Let $G$ be a reductive algebraic group over an algebraically closed field $K$ of characteristic zero (I am particularly interested in the case $G=GL_n(K))$. Let $H$ be a closed subgroup of $G$. It is ...
1
vote
1answer
69 views

Universal Chevalley group associated to $D_l$

Consider the simple Lie algebra $D_l$. Consider the universal Chevalley group $G$ over a field $K$ associated to it. Then $G$ is a subgroup of the orthogonal group $O_{2l}(K, f)$ where $f$ is the ...
4
votes
0answers
75 views

Local factors of Tamagawa measure

This is a reference request to some computations which I hope can be found in the literature somewhere. Let $G\subset GL_n$ be a semisimple linear algebraic group over $\mathbb Q$. The Tamagawa ...
1
vote
0answers
89 views

A question about the associative classes of parabolic subgroups

Let $\mathbb{A}$ denote the adele group of $\mathbb{Q}$. Suppose that $P_1$ and $P_2$ are parabolic subgroups of a reductive algebraic group $G$, and consider their Langlands decomposition $$ ...
0
votes
0answers
68 views

Orbits of some action of SL2 on Pontryagin dual of the field of formal Laurent series

Let $K=\mathbb{F}_2((t))$ be the field of formal Laurent series over the finite field $\mathbb{F}_2$. Now consider $K^3$ as an additive group and its dual group $\hat{K^3}$, which consists of all ...
1
vote
1answer
171 views

Maximal torus of Chevalley group $Sp(4)$

Consider a chevalley group a field $K$, with the right chevalley basis. Let $\alpha$ be a root. Let $x_{\alpha}(t)$ be the corresponding root space. Define ...
12
votes
0answers
356 views

If $k$ is an algebraically closed field of any characteristic, then the fundamental group of $A$ is abelian

This is a followup to my earlier question, see here. I reproduce it as follows. Let $A$ be an abelian variety over a field $k$ of characteristic $0$. How do I prove, without using transcendental ...
5
votes
2answers
265 views

Conjugate transpose and discreteness, for Kleinian groups

Let $G=\langle g_1,\dots g_n\rangle<\mathrm{PSL}_2(\mathbb{C})$ be discrete, i.e. a finitely generated Kleinian group. Let $H=\langle g^{\dagger}g\mid g\in G\rangle$, (the group generated by the ...
20
votes
1answer
600 views

Which philosophy for reductive groups?

I am just beginning to look further into trace formulas and automorphic forms in a quite general setting. For long I have noticed that the natural assumption on the groupe $G$ we work on is to be ...
6
votes
1answer
145 views

Are the integer matrices in SO(3,2) “boundedly generated”?

Let $G$ be the subgroup of integer matrices in $\mathrm{SO}(3,2)$. (The invertible linear maps from a $5$ dimensional real vector space to itself which leave invariant a nondegenerate symmetric ...
1
vote
1answer
59 views

Existence of the double coset ring on paper of Ihara

In his paper "On discrete subgroups of the two by two projective linear group over $\mathfrak{p}$-adic fields", Yasutaka Ihara considers an abstract group $G$ together with a length function $l$ from ...
10
votes
2answers
290 views

Conjugacy of matrices of order three in $PGL(2,k)$, where $k$ is any field

We know that every matrix of order three in $PGL(2,\mathbb Z)$ is conjugate to the following matrix $$ \left( \begin{array}[cc] &1 & -1 \\ 1&0 \end{array} \right) $$ I want to know if ...
2
votes
0answers
38 views

is there a criterion for a two-generator subgroup of $PL(2,K)$ to be a cocompact lattice?

In the case of the group $SO(n,1)$ there is a criterion known for whether or not two given elements of the group generate a cocompact lattice. Is any similar criterion known in the case of $PL(2,K)$ ...
1
vote
0answers
68 views

Conjugacy classes of involutions in Kac-Moody groups

Let $A=(a_{s,s'})_{s,s'\in S}$ be a generalized Cartan matrix. Let $G=G(A)$ be the corresponding simply connected complex Kac-Moody group with Cartan subgroup $H$ and Weyl group $W$ acting on $H$. ...
3
votes
1answer
96 views

Unipotent radical of minimal parabolic subgroup of a unitary group over an arbitrary field

I am looking for an explicit description of the unipotent radical of a minimal parabolic subgroup of a unitary group, i.e. the group of isometries of a hermitian form, over an arbitrary field. In his ...
2
votes
1answer
160 views

base extension of algebraic groups

Let $\bf{G}$ be a simple and simply connected algebraic group over $\mathbb{Q}$. Is it true that the base extension of $\bf{G}$ over $\bar{\mathbb{F}}_p$, i.e., ${\bf ...
6
votes
1answer
223 views

General Linear Group as a Direct Product?

Let $K$ be a field and consider the surjective determinant homomorphism $\mathrm{GL}_n(K)\to K^\times$. Since the kernel is the special linear group $\mathrm{SL}_n(K)$ we obtain a short exact sequence ...
4
votes
1answer
155 views

Simple groups of Lie type

What I am going to ask is probably simple and maybe trivial. But I want to be sure that I am not missing any point. Let ${\bf G}\subseteq \mathrm{GL}_n(\mathbb{C})$ be a simple and simply connected ...
3
votes
1answer
176 views

Chevalley devissage

Let $G$ be an algebraic group over a perfect field $k$. Then it is know that it can be written as an extension of an affine algebraic group and a proper algebraic group. Is there a similar result for ...
6
votes
1answer
266 views

Etale fundamental of a parahoric group scheme

Let $p:X\rightarrow Y$ be a double cover of curves, denote by $$SU_n:=(p_*SL_n(\mathcal O_X))^{\tilde{\sigma}}$$ i.e. the $\tilde{\sigma}-$invariant part, the action of $\tilde{\sigma}$ is given by ...
4
votes
0answers
88 views

Norm variety for n=5, p=2 not isomorphic to a quadric

In the paper "Motivic construction of cohomological invariants", the author displays a list of known norm varieties for several $n,p$ on page $11$. For $p=2, n=5$ it says that a norm variety is given ...
0
votes
0answers
78 views

Non-split simple groups

Let $G$ be a simple group over a field $F$, and let $K$ be an extension of $F$ such that $G$ base changed to $K$ is split. Does this mean that $K$ splits a maximal torus of $G$? Context: I want to ...
7
votes
2answers
229 views

What is the most efficient way to factor a matrix into a given set of generators?

I am studying finite index subgroups of certain finitely presented groups. The particular conditions on my groups make this problem easier than I phrase it here, but I am curious about a more general ...
9
votes
0answers
113 views

Failure of surjectivity in Hotta-Springer specialization: examples for special unipotents?

Last weekend's workshop on Springer theory and its generalizations at UMass demonstrated how far the subject has expanded over four decades, but the original set-up for the Springer correspondence ...
10
votes
1answer
154 views

Growth of dimension of fixed spaces in $GL_n(\mathbb{Q}_p)$-representations

Let $\pi$ be a generic irreducible admissible representation of $GL_n(L)$, where $L$ is a $p$-adic field, $R$ is its ring of integers, and $\mathfrak{p}$ is its prime ideal. The conductor of $\pi$ ...
2
votes
1answer
110 views

Reference request: groups of multiplicative type are closed under extensions

I remember reading (quite a while ago, and I can't remember where!) that linear algebraic groups of multiplicative type over a field of characteristic zero are closed under extensions. This is ...
3
votes
1answer
98 views

Orbits in the adjoint representation of $SU(2,1)$

How can one describe the orbits of the Lie group $G=\mathrm{SU}(2,1)$ in its Lie algebra $\mathfrak{g}=\mathfrak{su}(2,1)$ with respect to the adjoint representation?
5
votes
1answer
317 views

What is the algebraic fundamental groups of $SO(n)$ and $Sp(2n)$?

Let $k$ be an algebraically closed field of characteristic zero. and let $$\sigma: SL_n(k)\rightarrow SL_n(k)$$ be an involution. My questions are: How could one calculate the fundamental group of ...