Questions tagged [algebraic-groups]

Algebraic varieties with group operations given by morphisms, or group objects in the category of algebraic varieties, the category of algebraic schemes, or closely related categories.

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35
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Homology of $\mathrm{PGL}_2(F)$

Update: As mentioned below, the answer to the original question is a strong No. However, the case of $\pi_4$ remains, and actually I think that this one would follow from Suslin's conjecture on ...
29
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938 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" (...
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858 views

Is every $p$-group the $\mathbb{F}_p$-points of a unipotent group

Let $\Gamma$ be a finite group of order $p^n$. Is there necessarily a unipotent algebraic group $G$ of dimension $n$, defined over $\mathbb{F}_p$, with $\Gamma \cong G(\mathbb{F}_p)$? I have no real ...
19
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527 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, ...
19
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712 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_\...
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521 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, ...
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854 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 (...
14
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783 views

What goes wrong with this alternate proof of Dirichlet's Theorem?

I had an idea for an alternate proof of Dirichlet's theorem, but something goes wrong. Dirichlet's theorem on primes in arithmetic progression says that for $ m,a \in \mathbb{N} $ which are ...
14
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421 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 ...
13
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516 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^...
12
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535 views

Representation theory of finite groups with additional structures

Let $H$ be a finite group, representation theory of $H$ over $\Bbb C$ essentially determines $\operatorname{Hom}(H,GL_n(\Bbb C))$ up to conjugation action of $GL_n(\Bbb C)$ for each $n$. If we replace ...
12
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287 views

Is the quotient of two linear group schemes linear?

Let $S$ be an affine scheme. Call a group scheme $G\to S$ linear if there exists an $S$-group morphism $G\to \mathrm{GL}_{n,S}$ with trivial kernel. Assuming this, suppose $H\to S$ is a central closed ...
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259 views

Has Kac's conjecture (*), from "Infinite root systems, representations of graphs and invariant theory", been proved?

Let $k$ be an algebraically closed field of characteristic zero, $V$ a finite dimensional $k$ vector space, $V^{\ast}$ the dual space, and $G$ an algebraic subgroup of $GL(V)$. Let $V_0$ be the points ...
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517 views

Uniform proof of Hasse principle for algebraic groups?

Let $G$ be a simply connected semi-simple linear algebraic group over a global field $k$. The Hasse principle for algebraic groups states that the map $$H^1(k,G)\rightarrow\prod_vH^1(k_v,G)$$ is ...
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439 views

Sheaf-theoretic Grothendieck groups

Let $S$ be a scheme, $M\to S$ a commutative monoid object in algebraic $S$-spaces, ie. an algebraic $S$-space such that, functorially on $S$-schemes $T$, $M(T)$ is a commutative monoid with neutral ...
11
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539 views

Is there a reasonable way to define "reductive Lie algebra" in prime characteristic?

Among the finite dimensional Lie algebras over a field of characteristic 0, there is a sensible definition of "reductive Lie algebra" going back at least to the 1960 first chapter of N. Bourbaki's ...
11
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299 views

Mysterious "raison d'être" of filtrations of congruence subgroups

I wonder for long why congruence subgroups seem to arise so naturally in certain filtrations. Everything below is on a local field $F_p$. Filtration for $GL_n$. Casselman and later Jacquet, Piatetski-...
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191 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 ...
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478 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 ...
11
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744 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 ...
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192 views

Quotients by algebraic group actions at the level of the Grothendieck ring

$\DeclareMathOperator\SGro{SGro}\DeclareMathOperator\Gro{Gro}\DeclareMathOperator\GL{GL}$For an algebraically closed field $K$, the Grothendieck semiring of $K$ consists of, say, quasi-projective $K$-...
10
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499 views

What is the mirror of an algebraic group?

Background: Kontsevich's homological mirror symmetry conjecture posits the existence of pairs $(X,\check X)$ with an equivalence of dg/$A_\infty$-categories $$\mathcal F(X)=\mathcal D^b(\check X)$$ ...
10
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304 views

What are the analogs of a Levi/Parabolic/Borel/Bruhat over the field with 1 element?

This is inevitably an imprecise question, but there are already several questions like this on the site so I thought i'd try anyway. If I understand correctly, for any reductive algebraic group $G$ ...
10
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368 views

Lazard's theorem and Hopf structures on the polynomial algebra

Let $k$ be an algebraically closed field of characteristic $0$. A well-known result of Lazard's states that an algebraic group which is isomorphic as a variety to an affine space is unipotent (M. ...
10
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406 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 ...
10
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816 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 ...
10
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452 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
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372 views

Full measure properties for Zariski open subsets in $p$-adic situation

Let $F$ be a $p$-adic field and let $X$ be a smooth integral variety over $F$ (I am chiefly interested in the case when $X$ is a connected reductive group over $F$). Let $U$ be a non-empty open subset ...
9
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210 views

Decomposition of linear groups into free products

I recently learnt about Nagao's theorem which states $SL_2(k[t])\cong SL_2(k)\ast_{B(k)}B(k[t])$ for a field $k$. I read in "A.W. Mason, Serre's generalization of Nagao's theorem" that a theorem of ...
9
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351 views

Naive question about classification of unipotent character sheaves

Let $G$ be a connected reductive algebraic group over (say) $\mathbb{C}$. The set $\hat{G}_u$ of isomorphism classes of unipotent irreducible character sheaves has some complicated classification in ...
9
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259 views

Why the hyperoctahedral group is a ``reductive'' group?

Sorry for the misleading title, I actually mean the following: The $n$-th hyperoctahedral group, also known as the Weyl group of $\mathrm{Sp}_{2n}$ and of $\mathrm{SO}_{2n+1}$, is isomorphic to the ...
9
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169 views

Demazure’s principle for tower of reductive groups and analogy with Teichmuller tower

Grothendieck in his Esquisse states (page 6/7 of the English translation) that the “principle” that the Teichmuller tower, i.e., the system of profinite fundamental groupoids $\hat{T}_{g,\nu}$ of ...
9
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235 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
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280 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
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367 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
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636 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
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221 views

Does Borel fixed-point theorem hold for Deligne-Mumford stacks?

Let $X$ be a proper Deligne-Mumford stack over $\mathbb{C}$ with an action by a complex torus $T$. Let $X^T$ denote the fixed locus. Question: Is the following statement true? ...
8
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247 views

Degree of parametrization of $\textrm{SO}_n$?

Let $G=\mathrm{SO}_{2 n}$ (or $G=\mathrm{SO}_{2n+1}$, $G=\mathrm{Sp}_{2 n}$ …) defined over some field $K$. Consider $G$ as an affine subvariety of the space of matrices. (Warm-up question) What are ...
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238 views

Are the braid groups good in the sense of Toën?

In this question it is asked whether the mapping class groups are 'good' in relation to pro-finite completion. Helpfully, one of the answers gives a link to a proof that the braid groups are good ...
8
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0answers
154 views

Why is the set of lifts of a p-divisible group canonically the same as the set of lines that span $M(G)/FM(G)$?

Let $M$ be the Dieudonne module of a p-divisible group $G_0$ over $k$, and let a lift of $G_0$ to $A$ be a p-divisible group $G$ over $A$ such that $G \otimes_A k \simeq G_0$. Let $\omega_G$ be the ...
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149 views

Can semisimple orbits be written $\exp(\mathfrak{g})\cdot x$?

Let $\mathfrak{g}$ be a complex semisimple Lie algebra and let $G$ be its adjoint group. If $x\in\mathfrak{g}^{rs}$ is a regular semisimple element, is its orbit $$G\cdot x=\{\mathrm{Ad}_gx:g\in G\}$$ ...
8
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0answers
307 views

Significance of half sum of non-simple positive roots

In representation theory, there are plenty of places that a $\rho$-shift makes an appearance, where $\rho$ is the half sum of positive roots. See, for instance, this post for some discussions of the ...
8
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0answers
137 views

Semisimple Lie groups admitting a free algebra of invariants

Assume we work over an algebraically closed field of characteristic zero. I know that for a connected semisimple algebraic group there is an upper bound for the number of isomorphism classes of ...
8
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0answers
498 views

When is GIT quotient of a projective variety smooth?

It might be very basic, but my google search could not find an answer for this, nor the search in previous related questions. Let $X$ be a smooth projective variety, and let $G$ be a reductive group ...
8
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183 views

Mapping class groups of algebraic varieties

Let $X$ be a projective algebraic variety over a (perfect) field $k$. Let $Aut(X):k \text{-}Alg \to Grp$ be the functor of points defined by $$Aut(X) : A \mapsto Aut_{Spec (A)}(X \times_{k} Spec (...
8
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0answers
389 views

Is a quotient of a solvable group always affine?

After several useful comments, I understood that I meant two questions rather than one. Sorry for messing it up, and feel free to answer any of them or vote for closure! If $G$ is a connected, ...
8
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0answers
173 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
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0answers
384 views

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

Let $R$ be an associative ring with identity 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 $r∈R$. Let ...
8
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0answers
372 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
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158 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 ...

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