**4**

votes

**0**answers

83 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

**0**answers

92 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
$$
P(\...

**0**

votes

**0**answers

74 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

**1**answer

178 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 $w_{\alpha}(t)=x_{\alpha}(t)x_{-\alpha}(-t^...

**5**

votes

**2**answers

277 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 $g^...

**21**

votes

**1**answer

719 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

**1**answer

153 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

**1**answer

61 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

**2**answers

300 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

**0**answers

39 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 $PL(2,K)$ ...

**2**

votes

**0**answers

85 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

**1**answer

100 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

**1**answer

168 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 G}\times_{\mathrm{Spec(\mathbb{Z})...

**6**

votes

**1**answer

246 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

**1**answer

165 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

**1**answer

186 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

**1**answer

273 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

**0**answers

91 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

**0**answers

85 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

**2**answers

256 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

**0**answers

118 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

**1**answer

170 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

**1**answer

114 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

**1**answer

113 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

**1**answer

322 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 $...

**7**

votes

**1**answer

184 views

### Structure of Deligne-Lusztig representations $R_{T,\theta}$ for ministropic $T$ and cuspidal representations

Let $G$ be a reductive group over a finite field $k$, let $F$ be a Frobenius morphism on $G$.
I'll start with a somewhat vague question and make my question more specific further down:
How do ...

**2**

votes

**1**answer

111 views

### When is the image of the adjoint representation of a real algebraic group Zariski closed?

Let $\operatorname{Ad}:\operatorname{SL}_n(\mathbb{R}) \to \operatorname{GL}(\mathfrak{sl}_n(\mathbb{R}))$ be the adjoint representation (i.e. $\operatorname{Ad}(g)X=gXg^{-1}$) of $SL_n(\mathbb{R})$. ...

**6**

votes

**2**answers

214 views

### “Interesting” projective varieties being quotients of $\mathbb{A}^n\setminus \{0\}$ by an action of an algebraic group?

The algebraic (multiplicative) group $G^m$ acts on $\mathbb{A}^n$ (diagonally) and the quotient of $\mathbb{A}^n\setminus \{0\}$ by $G_m$ is $\mathbb{P}^{n-1}$ (which is a proper variety). I would ...

**2**

votes

**0**answers

75 views

### maximal elementary abelian p-subgroups of finite groups of Lie type

Let $G$ be a simple algebraic group over an algebraically closed field k of characteristic $p$. Let $G(\mathbb{F}_{p^r})$ be the corresponding finite group of Lie type. Is it true that every maximal ...

**7**

votes

**0**answers

177 views

### A conjecture of Lubotzky on ranks of subgroups of special linear groups over the integers

In a 1985 paper named "Dimension function for discrete groups" Lubotzky conjectured that:
For any integer $n \geq 3$ the group $\mathrm{SL}_n(\mathbb{Z})$
contains infinitely many finite index ...

**1**

vote

**0**answers

103 views

### Maximal $k$-split $k$-tori are $G(k)$-conjugate, but maximal $k$-tori are not?

Suppose $G$ is a reductive algebraic group defined over a field $k$.
Is this right?
If so, it means for each maximal $k$-split $k$-tori $T$ there are many maximal $k$-anisotropic ones $T'$ that ...

**4**

votes

**1**answer

172 views

### Parametrization of Schubert varieties in isotropic Grassmannians by partitions

Let $X=\mathbb{G}_Q(l,p)$ be the isotropic Grassmannian, where $l\leq p-2$. Let $q=p-l$. Let $W^P$ be the set of minimal length representatives. Let $\tilde{\mathcal{Q}}(l,p)$ be the set of partition ...

**1**

vote

**2**answers

136 views

### Jordan decomposition of elements in non-connected component of algebraic group

Let $G$ be a Linear Algebraic Group (over algebraically closed field). We know that the connected component $G^o$ is a normal subgroup of finite index in $G$. Let $g$ be an element of $G$ which is not ...

**6**

votes

**1**answer

282 views

### Do representations of real semisimple algebraic group have to be algebraic?

If $G$ is the real points of a semisimple algebraic group and $\rho:G\to GL(n,\mathbb R)$ is continuous representation. Is $\rho$ an algebraic morphism?

**2**

votes

**1**answer

175 views

### Is it possible to describe the action of the Weyl group on the cohomology of the fibers of the Grothendieck-Springer resolution?

I am confused about the following: can one describe the action of the Weyl group on the cohomology of each fiber of the Grothendieck-Springer resolution? I only need the case of ${\mathfrak sl}_n$. ...

**1**

vote

**0**answers

145 views

### Does the functor of taking invariants commute with tensor products? [closed]

Suppose that $G$ is a group acting on a commutative ring $R$, inducing an action on each $R$-module. For any $R$-module $M$, let $M^G$ denote the collection of elements of $M$ invariant under the $G$-...

**3**

votes

**1**answer

319 views

### Maximal ideal of group ring

Let $R$ be a finite commutative ring with identity and $G$ an finite abelian group. Is there any more conditions (on $R$ or on $G $) under which we can characterize maximal ideals of group ring $RG $, ...

**8**

votes

**2**answers

304 views

### Parahoric Group Scheme

I am looking for the definition of a parahoric group scheme in the sense of Bruhat and Tits? I couldn't find a reference for that? at least a "clear" reference!
thanks

**6**

votes

**1**answer

137 views

### Pullback of line bundles and existence of divisors representing line bundles

I am studying the proof of Lemma 7.2. on page 108 in Dolgachev's "Lectures on invariant theory". It states (everything is done over the field $k = \overline{k}$):
Let $X$ be a normal affine ...

**2**

votes

**0**answers

135 views

### Form over $ \mathbb{Z} $ of non-split simple algebraic groups over non-archimedean local fields

Here is a basic observation :
On page 68 of his article Reductive groups over local fields, Jacques Tits writes:
All types of groups listed (...) exist over an arbitrary [non-archimedean local] ...

**4**

votes

**2**answers

174 views

### Questions about $\mathbb{C}[G/U^-]$ and $\mathbb{C}[B]$

Let $G = GL_n$. By algebraic Peter-Weyl theorem, we have
$$
\mathbb{C}[G] = \bigoplus_{\lambda} V_{\lambda} \otimes V_{\lambda}^*,
$$
where $\lambda$'s are dominant weights. Let $U^-$ be the ...

**2**

votes

**1**answer

230 views

### Any representation is a sub representation of direct sum of regular representation

I need a reference for the following statement:
Let G be a linear algebraic group over algebraically closed field k. Let V be a finite dimensional G-module. The V is sub representation of k[G]^n for ...

**6**

votes

**1**answer

272 views

### Is there a non-smooth algebraic group scheme in char $p$, all of whose defining relations have degree less than $p$?

Let $k$ be an algebraically closed field of characteristic $p>0$.
All the examples of non-smooth algebraic group schemes over $k$ that
I have seen (apart from "artificial" examples; see below) have ...

**1**

vote

**1**answer

301 views

### Classification of finite group schemes over a field

What is known about the classification of finite group schemes over a field? By a finite group scheme I mean $Spec A$ where $A$ is a finite-dimensional algebra over a field.
Is there a full ...

**6**

votes

**2**answers

357 views

### How simple does a $\mathbb{Q}$-simple group remain after base change to $\mathbb{Q}_{\ell}$?

Of course the general answer to the question in the title is: not very simple.
I could not think of a better title, so let me explain my question in more detail.
I have a number field $E/\mathbb{Q}$, ...

**0**

votes

**0**answers

52 views

### Relation between $\Gamma$-percuspidal parabolic subgroups and split parabolic subgroups of real semisimple Lie groups

Let $G$ be a reductive algebraic group defined over $\mathbb{Q}$. Let $\Gamma$ be a lattice in $\mathcal{G}:= G(\mathbb{R})$. I am interested in knowing under what conditions on either of $\mathcal{G}...

**1**

vote

**1**answer

249 views

### Generalization of a theorem of Steinberg

Steinberg has a beautiful theorem counting $F$-stable maximal tori in a reductive group. Here's the version of the result that you can find as Theorem 3.4.1 of Carter's Finite groups of Lie type:
...

**2**

votes

**0**answers

71 views

### Connected components of a certain real homogeneous space

Let $m>0$ be a natural number.
Consider the following semisimple algebraic groups over ${\mathbb{R}}$:
$$ G={\mathrm{SU}}(2m,4m),\ \ H={\mathrm{SU}}(2m,2m)\times{\mathrm{SU}}(2m). $$
We embed $H$ ...

**4**

votes

**1**answer

160 views

### Matrix from a homomorphism of simply connected groups

Let $H$ be a simple algebraic group of type $\mathbf{G}_2$ over $\mathbb{C}$.
Let $\rho$ be the standard 7-dimensional complex representation
$$ \rho\colon H=\mathbf{G}_2\to \mathrm{SO}_7.$$
We ...

**2**

votes

**1**answer

108 views

### How to associate a proper parabolic subgroup of a real s.s Lie group $G$ to a non-trivial unipotent element in a non uniform lattice in $G$?

Let $G$ be a real semi-simple Lie group. Let $\Gamma$ be a non-uniform lattice in $G$.
Then it is known that $\Gamma$ contains a non-trivial unipotent element. When $\mathbb{R}$-rank of $G$ is 1, it ...