**18**

votes

**0**answers

295 views

### Follow-up to Steinberg's problem (12) in his 1966 ICM talk?

Steinberg's lecture at the 1966 ICM in Moscow here surveyed his work on regular elements of semisimple algebraic groups, while also formulating a number of then-open questions as "problems" (with ...

**16**

votes

**0**answers

440 views

### Actions on ℍⁿ generated by torsion elements

Let $n$ be a large integer.
I am looking for a cocompact properly discontinuous isometric action on $n$-dimensional Lobachevky space which is generated by elements of finite order.
Or equivalently, ...

**16**

votes

**0**answers

599 views

### Should the Dynkin diagrams of types $A_1$ and $B_2$ be labelled $C_1$ and $C_2$?

The labels $A$--$G$ attached to connected Dynkin diagrams are of course arbitrary,
the result of historical accidents. In order to avoid repetitions, the four infinite
families $A_\ell, B_\ell, C_\...

**13**

votes

**0**answers

297 views

### p-groups as rational points of unipotent groups

Is it true that every finite p-group can be realized as the group of rational points over $\mathbb{F_p}$ of some connected unipotent algebraic group defined over $\mathbb{F_p}$? For abelian p-groups, ...

**13**

votes

**0**answers

671 views

### How much has been written down about Deligne's geometric approach to the order formula for a finite group of Lie type?

This is a follow-up to a recent mathoverflow question
34387
about computing the orders of finite unitary groups and the comments made there.
Between 1955 (Chevalley's Tohoku paper) and 1968 (...

**12**

votes

**0**answers

329 views

### Analog of Peter-Weyl theorem for $G[[t]]$

Let $G$ be a reductive group over ${\mathbb C}$ and let $G[[t]]$ denote the corresponding
group over the formal power series ring ${\mathbb C}[[t]]$. This is a group scheme, so one
can speak about its ...

**11**

votes

**0**answers

370 views

### Can an abelian variety/Q have no non-trivial points over Q_sol?

Let $A/\mathbb{Q}$ be an abelian variety. Must there be a finite solvable
extension $K/\mathbb{Q}$ such that $A(K)$ is nontrivial?
This follows from the conjecture that the maximal (pro-)solvable ...

**10**

votes

**0**answers

414 views

### Higher-dimensional algebraic subgroups of the proalgebraic Nottingham group?

Let $R$ be a commutative ring, and, for $n\ge0$,
${\mathcal{A}}_n={\mathcal{A}}_n(R)$ the group of series
$u(x)=\sum_0^\infty a_jx^{j+1}\in R[[x]]$ for which
$a_0\in R^\times$ and $u(x)\equiv x\pmod{x^...

**9**

votes

**0**answers

211 views

### Rationality of a certain real algebraic variety

Let $A_n$ denote the vector space of $n\times n$ antisymmetric matrices over ${\mathbb{Q}}$, where $n$ is even.
Let $A_n^*\subset A_n$ denote the affine ${\mathbb{Q}}$-subvariety of invertible ...

**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 ...

**9**

votes

**0**answers

252 views

### Earliest use of the term “linearly reductive”?

Recently a number of MO questions have referred to a "linearly reductive group", usually in a way that is out of focus. It's unclear to me why this terminology is so popular, since over a field of ...

**9**

votes

**0**answers

381 views

### Polynomial function from $S^3$ to $S^3$ and quaternions

I am searching the polynomial functions from $S^3$ to $S^3$.
($S^3$ is the set of vectors $x$ in $\mathbb{R}^4$ such that $\|x\|=1$)
We say $g$ is a polynomial function from $S^3$ to $S^3$, if there ...

**9**

votes

**0**answers

305 views

### Twisted Springer fibers

In the study of certain moduli spaces of $p$-divisible groups I came across the following twisted version of a Springer fiber, and I was wondering whether some expert on algebraic groups/algebraic ...

**9**

votes

**0**answers

381 views

### A uniform bound for a “true” non-congruence subgroup

Before stating my question, let me recall the Congruence Subgroup Property/Problem: Given simply connected absolutely and almost simple algebraic group $G$ with fixed realization as a matrix group one ...

**9**

votes

**0**answers

519 views

### Role of nontrivial component groups in Springer Correspondence?

Set-up for classical Springer Correspondence:
$G$ = reductive group (usually assumed to be semisimple of adjoint type) over $\mathbb{C}$, with Borel subgroup and
maximal torus $B \supset T$, Weyl ...

**8**

votes

**0**answers

131 views

### Smooth quotients of algebraic spaces that are varieties away from codimension $\ge 2$ subset

This is a question about when a smooth complex algebraic space that is very close to being an algebraic variety is actually an algebraic variety.
General question: Let $X$ be a smooth separated ...

**8**

votes

**0**answers

298 views

### Connection between two theorems on character values?

In a recent arXiv preprint here, Dipendra Prasad has revisited a 1976 theorem of Kostant (Theorem 2 in the paper On Macdonald's $\eta$-function formula, the Laplacian and generalized exponents, ...

**8**

votes

**0**answers

577 views

### Invariance of Euler characteristic under base change for sheaf cohomology of flag varieties

BACKGROUND:
Over an algebraically closed field of arbitrary characteristic, most of the basic structure theory of affine (= linear) algebraic groups can be developed concretely without quoting ...

**7**

votes

**0**answers

178 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 ...

**7**

votes

**0**answers

258 views

### Higher-dimensional generalization of Pink's theorem

Pink's theorem in the title of the question refers to the main theorem of Pink's paper "Compact Subgroups of Linear Algebraic Groups" that appeared in Journal of Algebra (206) in 1998. It essentially ...

**6**

votes

**0**answers

140 views

### Root-theoretic formulation of characteristic polynomial

Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra of rank $n$ over $\mathbb{C}$. Let $G$ denote the corresponding simple simply connected algebraic group. By Chevalley's Theorem, $\mathbb{...

**6**

votes

**0**answers

286 views

### What is miraculous about the mirabolic subgroup?

I recently asked this question about Euler subgroups and generalizing the automorphic theory of $\mathrm{GL}_n$ to a more general setting. My question here is more specific.
As mentioned there, the ...

**6**

votes

**0**answers

104 views

### Rational points with small denominator in $U(n)$

Fix integers $n,d>0$. (I'm probably thinking about $n\leq 6$ and $d\leq 2000$.) Let $X$ be the set of matrices $A\in U(n)$ such that the entries of $dA$ lie in $\mathbb{Z}[i]$.
Is there an ...

**6**

votes

**0**answers

960 views

### Closure of an orbit under the action of an algebraic group

Setting:
Fix some field $k$. I am not very prudent about the field - although I'd prefer to assume as little as possible, you may assume as much as you want, the case of primary interest being $k=\...

**6**

votes

**0**answers

236 views

### How to decide if two surfaces occurring in Springer theory are isomorphic?

In the study of a simple algebraic group (say over $\mathbb{C}$) and related geometry of its flag variety associated with the Springer correspondence, one encounters pairs of surfaces which have some ...

**6**

votes

**0**answers

330 views

### Real approximation for homogeneous spaces of linear algebraic groups

Let $X$ be a smooth geometrically integral variety over $\mathbf{Q}$, having a $\mathbf{Q}$-point.
We say that $X$ has the real approximation property if $X(\mathbf{Q})$ is dense in $X(\mathbf{R})$.
...

**6**

votes

**0**answers

274 views

### What is known about line bundles on the tangent bundle of a flag variety?

Let $G$ be a semisimple algebraic group over an algebraically closed field of arbitrary characteristic. (I'm most interested in the positive characteristic case). Let $B \subseteq G$ be a Borel ...

**5**

votes

**0**answers

313 views

### étale cohomology of rings of integers of number fields and Shafarevich-Tate groups

Let $K$ be a number field, $A$ an abelian variety over $K$.
Let $\mathcal{O}$ be the ring of integers of $K$, $\mathcal{A}$ the Néron model abelian scheme of $A$ over $\text{Spec}(\mathcal{O})$.
For ...

**5**

votes

**0**answers

176 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 ...

**5**

votes

**0**answers

223 views

### Hyperplane sections of principal homogeneous spaces

Let $P_i$ denote the $i$-th vertex in the Dynkin diagramm of an algebraic group. It symbolizes a parabolic subgroup of $G$ corresponding to the other vertices, meaning $G/P_i$ is a smooth, projective, ...

**5**

votes

**0**answers

130 views

### LS paths construction

Let $W$ be the Weyl group of a simple Lie algebra $\mathfrak L$, and for a dominant weight $\lambda$ denote by $W_{\lambda}$ the stabilizer of $\lambda$ in $W$. Let $\leq$ be the Bruhat order on $W/W_{...

**5**

votes

**0**answers

190 views

### Derived subgroup of rational points versus rational points of derived subgroup

Let $\mathbf G$ be a connected algebraic group defined over a field $\mathbb F_p$. If $q=p^n$, then the groups $\mathbf G^\prime (\mathbb F_q)$ and $\mathbf G (\mathbb F_q)^\prime$ are not always ...

**5**

votes

**0**answers

425 views

### Torsors and twists of algebraic groups

Let $G/S$ be an affine group scheme. Then the automorphism group of every $G$-torsor over $S$ is a twist of $G$, but it this functor isn't essentially surjective in general (It may be not full nor ...

**5**

votes

**0**answers

195 views

### Is a semiabelian algebraic space a scheme?

Let $S$ be a scheme and let $A$ be a commutative separated smooth $S$-group algebraic space of finite presentation each of whose geometric fibers is an extension of an abelian variety by a torus. Is $...

**5**

votes

**0**answers

177 views

### On Langlands Pairing and transfer factors

In the paper "On the definition of transfer factors" Langlands and Shelstad define a certain number of factors $\Delta_{I}$, $\Delta_{II}$,$\Delta_{III,1}$,$\Delta_{III,2}$, which are roots of unity.
...

**5**

votes

**0**answers

149 views

### On an interesting subalgebra of the functions on the cotangent bundle of the flag variety

Setup
Let $G$ be a semisimple algebraic group over a field $k$ of characteristic $p$ where $p = 0$ or $p > 0$ is a good prime for $G$. Fix a Borel subgroup $B \subseteq G$ corresponding to the ...

**5**

votes

**0**answers

397 views

### More familiar description of wonderful compactification of SL_n/S(GL_2 \times GL_n-2)

I am trying to learn a bit about spherical geometry and wonderful compactifications. Please correct any misconceptions. If I've understood http://www.springerlink.com/content/x62342v721707828/ ...

**5**

votes

**0**answers

855 views

### anisotropic and elliptic tori in GL(n)

Let $F$ be a commutative field and $n\geqslant 2$ be an integer. It is well known that the maximal anisotropic mod center tori in $G={\rm GL}(n,F)$ are of the form $T = {\rm Res}_{E/F}\; {\mathbb G}_m$...

**5**

votes

**0**answers

338 views

### Frobenius splitting of tangent bundles of flag varieties

BACKGROUND
Let $X$ be a variety over an algebraically closed field $k$ of positive characteristic $p$. Let $F : X \to X$ denote the absolute Frobenius morphism, i.e. the morphism that is the identity ...

**4**

votes

**0**answers

79 views

### Automorphisms of unipotent groups

I start with a hopelessly broad question: what is known about the structure of the automorphism group of a (smooth, connected) unipotent group (over a field), and particularly about the structure of ...

**4**

votes

**0**answers

152 views

### 'Noether normalization' for finite group schemes

Throughout let $p$ be a prime, and let $k$ be a field of characteristic $p$.
Let $G$ be a compact Lie group. Such a $G$ can always be embedded as a closed subgroup of $SU(n)$ for some $n$. This ...

**4**

votes

**0**answers

127 views

### Centralizer action on components of Springer fibers

Let $G$ be a complex adjoint group. Let $u\in G$ be unipotent. The group $A(u):=\pi_0(Z_G(u))$ acts on the set of components of the Springer fiber $\mathcal{B}_u$, the variety of Borel subgroups that ...

**4**

votes

**0**answers

169 views

### Is the class of commutative generalized Euclidean rings stable under quotient and localization?

Let $R$ be an associative ring with indentity and let $E_n(R)$ be the subgroup of $GL_n(R)$ generated by matrices obtained from the identity matrix by replacing an off-diagonal entry by some $r∈R$. ...

**4**

votes

**0**answers

164 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 ...

**4**

votes

**0**answers

146 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

**0**answers

86 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 ...

**4**

votes

**0**answers

94 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 ...

**4**

votes

**0**answers

196 views

### Correspondence between real forms and real structures on complex Lie groups

I asked this in MSE, but without success, so I hope, it will be suitable here.
E.B.Vinberg and A.L.Onishchik in their book give the following two definitions.
For a complex Lie group $G$ its real ...

**4**

votes

**0**answers

206 views

### Bruhat decomposition of $G/Q$

Let $G$ be a semisimple algebraic group over $\mathbb C$, $T$ be a maximal torus and $B$ be a Borel subgroup of $G$ containing $T$. Let $R^+$ be the set of positive roots with respect to $B$. Let $Q$ ...

**4**

votes

**0**answers

164 views

### An extension of group schemes admitting Neron models

Let $R$ be a discrete valuation ring, $K$ its field of fractions, and
$$ 0 \rightarrow G_K' \rightarrow G_K \rightarrow G_K'' \rightarrow 0$$
a short exact sequence of smooth $K$-group schemes of ...