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15
votes
0answers
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
votes
0answers
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
votes
0answers
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
votes
0answers
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
votes
0answers
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
votes
0answers
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
votes
0answers
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
votes
0answers
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
votes
0answers
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
votes
0answers
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 ...
8
votes
0answers
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
votes
0answers
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
votes
0answers
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
votes
0answers
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
votes
0answers
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, ...
7
votes
0answers
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 ...
6
votes
0answers
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
votes
0answers
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
votes
0answers
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 ...
6
votes
0answers
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 $X(\mathbf{R})$. ...
6
votes
0answers
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 ...
5
votes
0answers
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. ...
5
votes
0answers
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 ...
5
votes
0answers
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/ ...
5
votes
0answers
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
votes
0answers
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
votes
0answers
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 ...
4
votes
0answers
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 ...
4
votes
0answers
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
votes
0answers
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 ...
4
votes
0answers
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
votes
0answers
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 ...
4
votes
0answers
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 = ...
4
votes
0answers
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
votes
0answers
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 $G$ ...
4
votes
0answers
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 ...
3
votes
0answers
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
votes
0answers
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
votes
0answers
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
votes
0answers
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 ...
3
votes
0answers
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 ...
3
votes
0answers
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
votes
0answers
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
votes
0answers
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 ...
3
votes
0answers
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 ...
3
votes
0answers
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 : ...
3
votes
0answers
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. ...
3
votes
0answers
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, ...
3
votes
0answers
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
votes
0answers
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 ...