**5**

votes

**2**answers

147 views

### Cartan subspaces for general algebraic representations

So I feel like asking the following likely open-ended question: What good generalizations of the notion of Cartan subspace do we have?
To be precise, let $G\curvearrowright V$ be an algebraic ...

**18**

votes

**1**answer

224 views

### How many ways can I factor a matrix (over $\mathbb{Z}$)?

Let $A$ be a fixed matrix in $M_2\mathbb{Z}$ with determinant $n \neq 0$.
Question 1 How many ways can I write $A = XY$ for $X, Y \in M_2\mathbb{Z}$?
The answer to this question is pretty clearly ...

**4**

votes

**0**answers

163 views

### Descent of line bundles to the quotient

If a finite group acts $G$ on a variety $X$, consider the quotient $X/G$. I would like to understand which line bundle on $X$ descends to $X/G$. The action is not free. Can anyone direct me to some ...

**3**

votes

**1**answer

192 views

### system of complex equations

I am working on a system of complex equations The question is the following:
Let $a_1,a_2,\ldots,a_N\in \mathbb{C}$ such that
$$\sum_{j=1}^N \sum_{q=0}^{N-1-k} {N-1 \choose q} {N-1 \choose k+q} |a_j|...

**2**

votes

**0**answers

105 views

### Quotients of quasi affine varieties and extension of scalars

I have some questions about GIT quotients and extensions of scalars of categorical quotients:
1) Let $X$
be a complex algebraic quasi-affine variety, $G$
an algebraic reductive group over $\...

**0**

votes

**1**answer

79 views

### presentation for a nilpotent group associated to the square of a coxeter element

This question is related to one asked earlier about inductive presentations of unipotent radicals in Kac-Moody groups.
Let $\Gamma$ be a coxeter diagram --- i.e. an unoriented graph with $r$ vertices ...

**4**

votes

**0**answers

144 views

### Affine Steinberg groups vs Steinberg groups over Laurent polynomials

Let $R$ be a commutative ring and $\Phi$ be a finite (also called spherical) reduced irreducible root system of rank $\geq 2$. I will denote by $\mathrm{St}(\Phi,R)$ the Steinberg group of type $\Phi$ ...

**4**

votes

**1**answer

176 views

### Is the Luna slice theorem valid for any orbit with a reductive stabilizer?

The Luna slice theorem states that if a reductive group $G$ acts on an affine space $X$ and $O$ is a closed orbit, then (in the etale topology) there exists a $G$-invariant negihborhood of $O$ with a ...

**4**

votes

**1**answer

187 views

### inductive construction of unipotent radicals

Consider a directed coxeter diagram $\vec{\Gamma}$, i.e. a finite graph where each edge is decorated with one of the integer weights $\big\{3,4,6\big\}$ and those edges with weights $4$ or $6$ are ...

**0**

votes

**0**answers

100 views

### Action on algebraic variety and adjoint bundles

Let $X$ be a complex algebraic variety and let $G$ be a complex algebraic group; I mean that $X$ is a reduced, separated scheme of finite type on $\operatorname{Spec}\mathbb{C}$, and the underlying ...

**3**

votes

**0**answers

122 views

### reduction mod $p$ of Weyl modules

Let $G$ be a reductive algebraic group defined over a non-Archimedean field $F$.
Let $k_F$ be its residue field, of characteristic $p$.
Assume $G$ is unramified over $F$, then it admits a hyperspecial ...

**11**

votes

**1**answer

228 views

### Property of bundles with connections on abelian variety doesn't hold for additive or multiplicative group?

This question is a followup to two of my previous questions, see here and here.
Let $A$ be an abelian variety over a field $k$ of characteristic $0$. How do I prove, without using ...

**6**

votes

**2**answers

143 views

### Can we count the number of simple modules for a reduced enveloping algebra?

Let $G$ be a reductive algebraic group over a field of positive characteristic $p$, which I'll assume to be very good for $G$. Then the Lie algebra $\mathfrak{g}$ is restricted and each simple $\...

**0**

votes

**0**answers

59 views

### Connectedness of Centralisers in Unitary group

I want to understand centralizers of semisimple elements in unitary groups.
Let us begin with example of $GL_n(k)$. Centralizers of semisimple elememts are a product of smaller $GL_m(k)$ thus ...

**3**

votes

**2**answers

202 views

### open subgroup scheme closed

Let $G/S$ be a group scheme and $H \leq G$ an open subgroup scheme. Is $H \subseteq G$ closed? I want to apply this to $G^0 \leq G$ (see SGA 3, VI_B, Théorème 3.10) for $G$ commutative.
(*) If $S = \...

**2**

votes

**0**answers

88 views

### Centralizer of a dense subgroup in a maximal subgroup of a reductive group

I am looking for a reference to the following statement
"Let $G$ be a reductive algebraic group and $K$ a maximal compact subgroup of $G$. If $H$ is a dense subgroup in $K$, then the centralizer of $H$...

**3**

votes

**0**answers

74 views

### Do character sheaves split over the Lang isogeny?

Let $G$ be a smooth commutative connected algebraic group over a finite field $\mathbb{F}_q$. For my purposes a character sheaf on G is a rank one $\ell$-adic local system $\mathcal{L}$ on $G$ ...

**4**

votes

**1**answer

112 views

### Quotient of a reductive group by a Levi subgroup and locally triviality

Suppose $G$ is a connected reductive algebraic group over an algebraically closed field $k$, and suppose $L$ is a Levi subgroup (of some parabolic subgroup of $G$), is it always true that the ...

**5**

votes

**1**answer

256 views

### If $G$ is absolutely simple simply connected, why is G(F_v) quasisimple for almost every valuation v?

Let $G$ be an absolutely simple simply connected and connected algebraic group defined over a global field $k$ with ring of integers $\mathcal{O}$. Fix an embedding of $G$ into $GL_n$. Given $v$ a non-...

**3**

votes

**2**answers

161 views

### Replacement for Lie-algebra complements

All groups are linear algebraic over some fixed field $k$.
I believe that it is true that, in characteristic $0$, if $G'$ is a reductive subgroup of $G$, then there is a $G'$-invariant complement to $...

**6**

votes

**3**answers

303 views

### Simple lie algebras, (almost-)simple groups of Lie type

Take an algebraic group $G$ defined over a finite field $K$. Suppose its Lie algebra $\mathfrak{g}$ is simple. It should follow that $G$ is almost-simple. (By this I mean not that $G(K)$ is simple -- ...

**4**

votes

**1**answer

172 views

### Smooth algebraic stacks with precisely two $\mathbb C$-objects

In my quest of "understanding" stacks, I recently tried to figure out the structure of a smooth algebraic stack of finite type $\mathcal X$ over $\mathbb C$ with affine diagonal and precisely one $\...

**6**

votes

**1**answer

211 views

### Are Picard stacks group objects in the category of algebraic stacks

I've been wondering about what a "group algebraic stack" should be, and ran into the notion of a Picard stack.
I'm slightly confused by the terminology here.
Given an algebraic stack $\mathcal X$ ...

**2**

votes

**1**answer

285 views

### When is a $\overline{\mathbb{Q}}_{\ell}$-local system the inverse image of a $\overline{\mathbb{Q}}_{\ell}$-local system?

I am trying to learn character sheaf theory, and encounter the following question:
(*) Let $f\colon X\rightarrow Y$ be a morphism of quasi-projective smooth varieties over $\overline{\mathbb{F}}_q$, ...

**1**

vote

**1**answer

244 views

### Zariski-closed subgroups of ${\mathbf G}_{\mathbf a}^n$

Let's work over an algebraically closed field $K$. A $1$-dimensional Zariski-closed connected subgroup of ${\mathbf G}_{\mathbf a}^n$ is isomorphic to ${\mathbf G}_{\mathbf a}^1$. If $K$ has ...

**3**

votes

**0**answers

122 views

### For which fields are the 1-dimensional algebraic groups known?

Given an algebraically closed field, or even a perfect one, a connected 1-dimensional algebraic group $G$ over the field $K$ is isomorphic to either $\mathbf G_a$ or $\mathbf G_m$.
For which fields $...

**2**

votes

**1**answer

317 views

### One-dimension Algebraic groups

I am searching for a possible analogue of a result in algebraic groups in a non-commutative setting, so I am looking for different proofs of the following :
Let $K$ be an algebraically closed field. ...

**1**

vote

**0**answers

111 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

**2**answers

412 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 ...

**4**

votes

**2**answers

278 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

**1**answer

111 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

**2**answers

171 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

**1**answer

317 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

**0**answers

146 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

**2**answers

654 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, http://www.jstor.org/stable/...

**2**

votes

**2**answers

158 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 ...

**5**

votes

**2**answers

216 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

**0**answers

148 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

**2**answers

432 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 [$...

**10**

votes

**1**answer

257 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

**1**answer

174 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 two-...

**4**

votes

**1**answer

260 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

**1**answer

92 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 $\{h_{\...

**10**

votes

**2**answers

238 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

**0**answers

53 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

**0**answers

108 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

**0**answers

107 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 Sp_{2n}(...

**3**

votes

**1**answer

93 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

**1**answer

110 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

**0**answers

175 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 ...