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398 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, ...

**13**

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278 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**

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508 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**

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228 views

### Which p-adic algebraic groups are type I?

It was proved by Jacques Dixmier (Sur les représentations unitaires des groupes de Lie algébriques, Annales de l'institut Fourier, 7 (1957), p. 315-328, doi: 10.5802/aif.73, MR 20 #5820 , Zbl ...

**11**

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292 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 ...

**10**

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329 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 ...

**9**

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372 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**

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430 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, ...

**8**

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208 views

### Examples of non-split algebraic groups

I am interested in knowing various examples of non-split (added hypothesis reductive) reductive linear algebraic groups. In particular, I would like to collect the following examples in my ...

**8**

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159 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 ...

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250 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 ...

**8**

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463 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 ...

**8**

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350 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 ...

**8**

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391 views

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

Set-up for classical Springer Correspondence:
$G$ = reductive group over $\mathbb{C}$, with Borel subgroup and
maximal torus $B \supset T$, Weyl group $W=N_G(T)/T$.
Fix a unipotent $u \in G$ with ...

**7**

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**0**answers

249 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, ...

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229 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 ...

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97 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**

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**0**answers

718 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 ...

**6**

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291 views

### Kazhdan-Lusztig graph for the Springer fiber of the minimal special unipotent class?

This graph was determined in the case of simply-laced root systems by Igor Dolgachev and Norman Goldstein: "On the Springer resolution of the minimal unipotent conjugacy class" (J. Pure Appl. Algebra ...

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259 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 $X(\mathbf{R})$.
...

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218 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 ...

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139 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.
...

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139 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 ...

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273 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/ ...

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605 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 ...

**5**

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282 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**

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122 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 ...

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138 views

### The Killing form on quantized enveloping algebras and reduction to the classical case

Let $U_q$ be the quantized enveloping algebra associated to a semisimple Lie algebra $\mathfrak g$. It is a result due to Tanisaki (see here; also see Chapter 6 of Jantzen's book Lectures on Quantum ...

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180 views

### On a resolution of sections of line bundles on the cotangent bundle of a flag variety

Background
Let $G$ be a semisimple algebraic group over an algebraically closed field $k$ of characteristic 0. Let $B \subseteq G$ be a Borel subgroup and let $U \subseteq B$ be its unipotent ...

**4**

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148 views

### Exotic Chains for Group Cohomology of a Complex Lie Group

Related Question: Exotic Chains for Group Homology of a Complex Lie Group
Let's take the group cohomology of a affine algebraic group over $\mathbb C$ (with its discrete topology). The natural ...

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819 views

### Cartan decomposition for upper triangular matrices

Due to the comments, I have the impression that I have to be more precise.
Consider $G= GL_n(F)$ for a non-Archimedean field $F$ with ring of integers $o$.
Let $K= GL_n(o)$ and let $I$ the Iwahori ...

**4**

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191 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 ...

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169 views

### Lattices in Hermitian spaces over local fields

Let $F$ be a $p$-adic field, $E / F$ a quadratic extension, and $n \ge 1$. Let $V = E^n$ with the obvious diagonal Hermitian form,
$$ \langle (u_1, \dots, u_n), (v_1, \dots, v_n) \rangle = \sum_{i = ...

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304 views

### about ℓ-adic and Perverse Stuff and ℓ-adic cohomology with compact support

this question is trivial.
We know from this paper link text, Springer constructed rep of the Weyl group $W$
on the cohomology of the Springer fibre. Also, Deligne-Lusztig constructed the linear rep ...

**4**

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300 views

### Tannakian categories equivalent as abelian categories

Suppose $A = Rep_k(G)$ and $B=Rep_k(H)$ are tannakian categories and $F: A\to B$ is an equivalence of abelian categories with $F(1_A) = 1_B$ (but not a $\otimes$-equivalence). What can I say about $G$ ...

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325 views

### Étale cohomology of linear groups

This is in a sense a follow up question to the answer here Analytic tools in algebraic geometry
Let $k$ be an algebraically closed field of positive characteristic and let $R$ be the result of ...

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41 views

### points with small U stabilizer on a spherical variety

Let $(G,H)$ be a spherical pair (i.e. $G$ is a reductive group, $H$ is a closed subgroup and the Borel subgroup $B$ of $G$ has a finite number of orbits on $G/H$). Let $U$ be the unipotent radical of ...

**3**

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135 views

### determine if a toric variety is Gorenstein

Let $G$ a simply connected group over $k$ and $car(k)=0$.
Let $T_{+}=(T\times T)/Z_{G}$ we consider the closure $\overline{T}_{+}$ of the torus $T_{+}$ in $\prod ...

**3**

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**0**answers

56 views

### on Neron defect of smoothness for groups schemes

Let $G$ a semisimple simply connected group over $\mathbb{C}$.
Let $\gamma\in G(\mathbb{C}[[t]])$ such that $\gamma$ is regular semisimple on $G(\mathbb{C}((t)))$.
We consider $I_{\gamma}$ the group ...

**3**

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84 views

### Is a proconstructible subsemigroup of $M_n(\mathbb{C})$ an intersection of constructible subsemigroups?

Let $S$ be a proconstructible subsemigroup of $M_n(\mathbb{C})$, that is a subsemigroup which is an intersection of constructible sets. Is $S$ an intersection of constructible subsemigroups?
The ...

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164 views

### K-theory of categories of group schemes and abelian varieties

Let $k$ be a field (perfect, or characteristic zero if you want - I'm especially interested in when $k$ is a number field). Consider the categories $\mathsf{G}_k=\{\text{commutative affine group ...

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121 views

### Jordan decomposition for non algebraically closed fields

Let $G$ be a (linear?) algebraic group defined over some field $k$ (not necessarily algebraically closed). For $g\in G$ we have the Jordan decomposition $g=su$ in the semisimple part $s$ and the ...

**3**

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110 views

### Product of Fixed points and kernel of Frobenius morphism

If $G$ is a reductive algebraic group over an algebraically closed field of positive characteristic $p$, and $G$ is defined over the prime field, we have the Frobenius morphism $F: G\to G$, which for ...

**3**

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**0**answers

137 views

### How to think about non-connected reductive groups

Suppose someone knows well the theory of connected reductive groups, over an algebraically closed field or more generally over any field, say for instance most of the content of Borel-Tits.
Is ...

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245 views

### Are principal bundles isotrivial?

Let $U$ be a $k$-scheme, where $k$ is a field. Let $G$ be a smooth affine $k$-group. Recall that a principal $G$-bundle over $U$ is a smooth surjective $U$-scheme $E$ with an action of $G$ on $E$ such ...

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130 views

### Decomposition of k-split tori of p-adic reductive groups

Let $G$ be a reductive group over a $p$-adic field $k$, $S \subset G$ a maximal $k$-split torus, $\Phi(G,S)$ the relative root system and $\Delta$ a basis of $\Phi$.
There is a group homomorphism :
...

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126 views

### ideal generated by highest weight vectors

Let $S$ be a polynomial ring which carries the action of a semi-simple linear algebraic group $G$ (I'm interested in a product of $GL$'s). Take $S$ and $G$ to be over an algebraically closed field.
...

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217 views

### Reductive Lie Groups and Complexification

Let $G$ be a complex Lie group (not necessarily connected) with reductive Lie algebra $\frak{g}$. (We may assume that $G$ has finitely many connected components and is linear-algebraic.) Of course, ...

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163 views

### Classify cross-sections of the adjoint quotient for a semisimple algebraic group?

[This question arises from trying to understand an incompletely formulated earlier question
here.]
Let $G$ be a semisimple algebraic group over an algebraically closed field $K$ of
characteristic ...

**3**

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**0**answers

221 views

### finite etale group scheme over a field

Could I have an example of a finite etale group scheme over a field k which is not a constant group scheme?
I just know that the category of etale group schemes over a k is equivalent to the category ...