All Questions
2,543 questions
13
votes
0
answers
718
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 ...
- 5,958
13
votes
0
answers
744
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 ...
- 52.9k
13
votes
0
answers
556
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^...
- 4,193
13
votes
0
answers
943
views
Beilinson-Bernstein localization in positive characteristic
This is a follow-up to this question; in particular, I'm wondering if anyone can expand upon the interesting answers given by Kevin McGerty and David Ben-Zvi there. (In particular, in this question I'...
- 3,637
12
votes
3
answers
4k
views
Books on reductive groups using scheme theory
Prof. Conrad mentioned in a recent answer that most of the (introductory?) books on reductive groups do not make use of scheme theory. Do any books using scheme theory actually exist? Further, are ...
- 19.6k
12
votes
2
answers
1k
views
To what extent does a torsor determine a group
Let $k$ be a field, and suppose $G$ is a group-scheme over $k$ (I am happy to assume that $k=\mathbb{Q}$ and that $G$ is affine). A $G$-torsor over $k$ is a non-empty $k$-scheme $T$ equipped with an ...
- 9,061
12
votes
2
answers
1k
views
Weil Conjectures for nonprojective algebraic varieties
If we replace projective variety with algebraic variety in the statement of the Weil conjectures what happens? To me it seems the statement still makes sense. But is it still true?
- 1,462
12
votes
2
answers
776
views
Distinguish smooth affine algebraic group by its underlying variety
If $G_1, G_2$ are two connected affine smooth algebraic groups over an algebraically closed field $k$, and $G_1 \cong G_2$ as algebraic varieties, must they be isomorphic as algebraic groups?
...
- 6,622
12
votes
2
answers
688
views
reductive group orbits in P(V)?
Say $G$ is a reductive group over $\mathbb{C}$. We can take a dominant highest weight $\lambda$ and look at the action of $G$ on $X = \mathbb{P} V(\lambda)$. The stabilizer of the class of the ...
- 3,968
12
votes
4
answers
2k
views
Finite subgroups of $PGL_2(K)$ in characteristic $p$
Let $K$ be a field of characteristic $p$. What are the finite subgroups of $PGL_2(K)$ whose orders are divisible by $p$? And if $G$ and $H$ are two such subgroups that are isomorphic, can one say when ...
- 1,199
12
votes
3
answers
1k
views
Is the Gelfand-Graev character isomorphic to a cohomology group for some sheaf on a Deligne-Lusztig variety?
Deligne-Lusztig theory
is awesome. You take a maximal torus $T$, you take a character $\theta$, construct a variety $X_T$$^*$, take etale cohomology, get a virtual character $R_T^\theta$, maybe it's ...
- 4,593
12
votes
1
answer
1k
views
isomorphism of abelian varieties
Let $A, B, C$ and $D$ be abelian varieties (over $\mathbb{C}$) such that $A \times B \cong C \times D$, and $A \cong C$. From the irreducibility of abelian varieties, we can say that $B$ and $D$ are ...
- 363
12
votes
1
answer
1k
views
Recovering classical Tannaka duality from Lurie's version for geometric stacks
In Lurie's paper Tannaka Duality for Geometric Stacks, it is essentially shown that specifying a morphism of geometric objects
$$ f \colon X \to Y$$
is equivalent to giving a corresponding pullback ...
- 805
12
votes
2
answers
929
views
Are representations of a linearly reductive group discretely parameterized?
Suppose $G$ is a linearly reductive group over a field (say $\mathbb C$). Does somebody know of a proof that any flat family of finite-dimensional representations of $G$ must be locally constant?
In ...
- 24.1k
12
votes
2
answers
1k
views
Reference request: Elementary proof of Lang's theorem
A few months ago, I read a nice elementary proof of Lang's theorem:
Theorem: Let $G$ be a connected linear algebraic group over $\overline{\mathbb{F}}_p$ and let $F : G \to G$ be a Frobenius map. Then ...
- 721
12
votes
1
answer
513
views
Littlewood–Richardson–Type Rule for Cohomology Ring of Grassmannians
$\DeclareMathOperator\GL{GL}$The ordinary Grassmannian of k-planes in n-space is a coset space for $\GL_n$.
It is $\GL_n$ mod a maximal parabolic. Here there is a nice basis given by Schubert ...
- 5,548
12
votes
1
answer
392
views
Non-conjugate subgroups that are conjugate in complexification
In trying to come up with a counter-example in my line of research, I would like to find an example as follows:
$G$ is a semisimple Lie group with complexification $G^{\mathbb{C}}$. $H_1, H_2 \...
- 471
12
votes
1
answer
879
views
Pointless groups III
This question is a sequel to Pointless groups, to which @DanielLitt produced an elegant and easy-to-understand counter-example, and Pointless groups II, where @R.vanDobbendeBruyn pointed out that my ...
- 12.9k
12
votes
1
answer
440
views
Arithmetic groups and integral points of integral structures
If $\mathbf{G}$ is an algebraic group defined over $\mathbb{Q}$, a subgroup of $\mathbf{G}(\mathbb{Q})$ is arithmetic if it is commensurable to $\mathbf{G}(\mathbb{Q}) \cap \operatorname{GL}_n(\mathbb{...
- 2,050
12
votes
2
answers
883
views
Residues of $\frac{1}{\prod_{i=1}^n (x-P_i)^{e_i}}$
This is a problem occurring in my research about deformations of $\mathbb{Z}/p^n$-covers over a ring of power series. Given an algebraically closed field $k$ of characteristic $p>0$, suppose $1< ...
- 245
12
votes
1
answer
472
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$ ...
- 1,453
12
votes
1
answer
860
views
Algebraic groups without torsors
If $G$ is an algebraic group such that $H^1(S, G) = 0$ for all schemes $S$, must $G$ be the trivial group?
My original motivation for the question is the rationale I always give students for studying ...
- 8,004
12
votes
1
answer
750
views
Vanishing theorems in positive characteristic
In the paper
Deligne, Pierre; Illusie, Luc (1987), "Relèvements modulo $p^{2}$ et décomposition du complexe de De Rham", Inventiones Mathematicae 89 (2): 247–270, doi:10.1007/BF01389078
I found the ...
- 8,998
12
votes
1
answer
2k
views
Replacement for derivations in characteristic p?
Let $k$ be a field.
If $f \in k[x]$ is a polynomial, and $d/dx\ f = 0$, then either
$f$ is constant, or
$char\ k = p$ and $f \in k[x^p]$.
So "annihilated by all derivations" is perhaps not the right ...
- 27.9k
12
votes
2
answers
660
views
What is the correct notion of representation for abelian varieties?
Zeroth question - am I right that in the "ordinary" sense an abelian variety does not possess any representations at all?
More precisely, a representation of an algebraic group $G$ (over an ...
- 18.4k
12
votes
1
answer
684
views
Is every connected semisimple linear Lie group the connected component of (the real points of) an algebraic group?
Is every connected semisimple linear Lie group the identity connected component of (the real points of) an algebraic group?
I was told some fact along this line is true but could not find any ...
- 511
12
votes
2
answers
587
views
Bounding weight multiplicities by number of certain Coxeter elements
This question concerns lower bounds of certain weight multiplicities in finite dimensional representations of algebraic groups (or Lie groups, Lie algebras).
Let's say $G$ is a simple algebraic group ...
- 313
12
votes
2
answers
3k
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 counter-...
12
votes
0
answers
555
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 ...
- 6,622
12
votes
0
answers
358
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 ...
- 2,928
12
votes
0
answers
273
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 ...
- 156k
12
votes
0
answers
729
views
Elkies' supersingularity theorem in higher dimension (in terms of the associated Newton polygon)
Elkies' supersingularity theorem: Given an elliptic curve $E$ over $\mathbb{Q}$, there are infinitely many primes $p$ such that $E$ is supersingular over $\mathbb{F}_p$.
I have seen another post on ...
- 3,539
12
votes
0
answers
216
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 ...
- 52.9k
12
votes
0
answers
968
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 ...
- 2,516
12
votes
0
answers
716
views
Lifting abelian varieties in (the closed fiber of) a fixed Neron model
Suppose that $R$ is a dvr with field of fractions $K$ and residue field $k$ and that $A_K$ is an abelian variety over $K$ with Neron model $A$ over $R$. Then the closed fiber $A_k$ is a smooth ...
- 1,609
11
votes
4
answers
979
views
Is the set of rational points of an (almost) simple algebraic group simple?
Let $G$ be an almost simple algebraic group defined over a field $K$. Then we know that, for $H = G/Z(G)$, the set of rational points $H(\overline{K})$ is a simple group (when considered with the ...
- 20.2k
11
votes
5
answers
2k
views
How do you switch between representations of an algebraic group and its Lie algebra?
I'm interested in the structures of categories like $Rep(GL_n), Rep(SL_n)$, etc. of algebraic representations of an algebraic group. I understand that there should be some relation between these and ...
- 25.6k
11
votes
3
answers
2k
views
Maximal compact subgroup of p-adic lie groups
Let $k$ be a number field and $S$ be a finite set of places of $k$.
Let $G$ be a connected semisimple algebraic group over $k$.
Let $k_S=\prod_{v\in S}k_v$
where $k_v$ is the completion of $k$ at $v$. ...
- 853
11
votes
2
answers
3k
views
Why are $S$-arithmetic groups interesting?
Let $K$ be a number field and $S$ a finite set of valuations of $K$, including $\infty$.
Define the $S$-numbers $K_S$ to be the direct product $\prod_{s \in S} K_s$ where $K_s$ denotes the completion ...
- 349
11
votes
2
answers
1k
views
Class groups of normal domains over finite fields
Let R be a local, normal domain of dimension 2. Suppose that R contains a finite field. I am interested in knowing when the class group of R is torsion. In characteristic 0, this is known to be ...
- 30.6k
11
votes
4
answers
3k
views
Classification of Tori of GL2, up to conjugation
Over an algebraically closed field $k$, every one-dimensional torus embedded (as a closed algebraic subgroup) into GL2 is diagonalisable, and the embedding is $t\mapsto (t^m,t^n)$ for some integers $m,...
- 7,722
11
votes
2
answers
2k
views
Partial (or complete) flag varieties as GIT quotients of affine spaces
I am looking for presentations of partial or complete flag varieties as GIT quotients of affine varieties spaces. That is, for a choice of of dimensions $0=d_1<d_2<\dots<d_k = n$, I would ...
- 142
11
votes
4
answers
2k
views
The category of finite locally-free commutative group schemes
I'm trying to understand the properties of the category $\mathcal{FL}/S$ of finite locally-free commutative group schemes over an arbitrary base-scheme $S$. I know it is not in general an abelian ...
- 111
11
votes
2
answers
589
views
To describe an invariant trivector in dimension 8 geometrically
$\newcommand\Alt{\bigwedge\nolimits}$Let $G=\operatorname{SL}(2,\Bbb C)$, and let $R$ denote the natural 2-dimensional representation of $G$ in ${\Bbb C}^2$.
For an integer $p\ge 0$, write $R_p=S^p R$;...
- 14.2k
11
votes
4
answers
3k
views
What does ramification have to do with separability?
Does ramification have anything to do with inseparability? It feels like an extension of Q in which p ramifies should somehow correspond to an extension of F_p(t). Does totally ramified <--> purely ...
- 15.4k
11
votes
2
answers
2k
views
Representation theory of the general linear group over a finite prime field
I am re-posting a question I asked on math.se here because I am unsatisfied with the answers I obtained.
The irreducible modules of $\operatorname{GL}_n(\mathbb C)$ over $\mathbb C$ are completely ...
- 4,902
11
votes
4
answers
2k
views
Constructing Affine Kac-Moody Groups
Does anyone know a simple construction for Affine Kac-Moody groups? There is a book by Kumar ("Kac-Moody groups, their flag varieties, and representation theory") that does the construction for the ...
- 5,548
11
votes
2
answers
938
views
Non-isomorphic complex Lie groups with the same exceptional Lie algebra for $\mathfrak{g_2,f_4,e_6,e_7,e_8}$?
An exceptional complex Lie algebra is a simple Lie algebra whose Dynkin diagram is of exceptional (nonclassical) type. There are exactly five such Lie algebras: $\mathfrak{g}_{2}$, ${\mathfrak {f}}_{4}...
- 2,147
11
votes
1
answer
527
views
What is a "non-trivial" example of a commutative algebraic group over $\mathbb{C}$?
Let $G$ be a commutative connected algebraic group over $\mathbb{C}$. A theorem of Serre says that there exists an exact sequence
$$1\to \mathbb{G}_a^n\times \mathbb{G}_m^m\to G\to A\to 1,$$
where $A$ ...
- 773
11
votes
2
answers
2k
views
Mostow's theorem on algebraic groups
In his classical 1956 paper
Fully reducible subgroups of algebraic groups
Mostow proves the following theorem:
Theorem 7.1.
Let $G$ be an algebraic group over a field $K$ of characteristic 0,
...
- 14.2k