All Questions
2,543 questions
4
votes
3
answers
3k
views
Can every parabolic subgroup be conjugated to its opposite by an element of the Weyl group?
Given a minimal parabloic subgroup we know that conjugation by the longest element in the weyl group takes it to the opposite parabolic.
Can we do the same thing if we choose a standard parabolic ...
- 185
2
votes
1
answer
332
views
Ample bundle under Frobenius morphism
Let $k$ be an algebraically closed field of char($k$)=p>0, $X$ a smooth projective variety over $k$, $F:X\rightarrow X^{(1)}$ the relative Frobenius morphism. Let $E$ be an ample vector bundle on $X$. ...
- 85
6
votes
1
answer
2k
views
Margulis normal subgroup theorem
Margulis' normal subgroup theorem states that any normal subgroup of a higher rank lattice is either finite or of finite index. The obvious question is: can one classify finite normal subgroups of ...
- 96.4k
7
votes
1
answer
2k
views
Parametrization of 2-dimensional torus
The units with norm $+1$ in a pure cubic number field $K$ generated
by a cube root of $m = ab^2$, where $a$ and $b$ are coprime and
squarefree integers, correspond to integral points on the torus
$$ ...
- 32.6k
8
votes
1
answer
1k
views
Picard group of Schubert varieties
Let $G$ be a semisimple linear algebraic group, $P$ be a parabolic subgroup and $w$ be an element of the Weyl group of $G$. I want to calculate the Picard group of the Schubert variety $X_P(w):=\...
- 783
9
votes
2
answers
1k
views
Unitary groups over number fields
When defining unitary groups over number fields, one usually takes $F$ to be a totally real number field, $E$ a CM quadratic extension of $F$, and $V$ a hermitian space attached to $E/F$. Then $U(V)$ ...
- 467
9
votes
2
answers
1k
views
Relative Lie Algebra cohomology and sheaf cohomology
(I apologize in advance for the vagueness of my question). Let $G$ be a reductive algebraic group over $\mathbb C$ with Lie algebra $\frak g$ and Borel $B$. I have seen casual references to the fact ...
- 3,637
2
votes
1
answer
797
views
Rationality of flag varieties
Let $X$ be a (generalised) flag variety over an algebraically closed field $k$ of characteristic zero, that is to say, $X$ is a projective variety which is a homogeneous space for some algebraic group ...
- 21.4k
0
votes
0
answers
352
views
Liftability in positive characteristic
What clsses of algebraic varieties over field of positive characteristic can be lift to $W_2(k)$?
- 85
5
votes
2
answers
561
views
Diagonalizable subgroups in a simply connected group
This is a continuation of my previous question.
Let $G$ be a connected reductive group over an algebraically closed field $k$ of characteristic 0.
We assume that $\mathrm{Pic}\ G=0$.
This is the same ...
- 14.2k
16
votes
3
answers
2k
views
Diagonalizable subgroups of a connected linear algebraic group
Let $G$ be a connected linear algebraic group
over an algebraically closed field $k$ of characteristic 0.
Let $D\subset G$ be a closed diagonalizable subgroup of $G$
(a subgroup of multiplicative type)...
- 14.2k
7
votes
2
answers
513
views
Tameness for the Galois closure of a map of curves
Say we are working over some $K=\overline{K}$, of characteristic $p>0$. Let $\phi: Y\rightarrow X$ be a nonconstant map of smooth projective curves. To this map we can associate a map $\psi: Z\...
- 801
4
votes
2
answers
2k
views
Fixed points of the action of an algebraic group
Hello!
If a compact Lie group $K$ acts smoothly on a smooth manifold $M$, then the set $M^K$ of fixed points under this action is a smooth submanifold of $M$. This is proved for example in ...
- 2,756
9
votes
1
answer
977
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 here. For other root systems the original question should be modified, leading to a precise ...
- 52.9k
1
vote
1
answer
282
views
Spectral decomposition of parabolic induced for GL2(Zp)
Let $F$ be a number field and let $o$ be its ring of integers. Let $o_p$ resp. $F_p$ be the completion at a prime ideal $p$ in $o$. Let $B$ be the group of upper triangular matrices in $GL_2$. Let $\...
- 11.2k
7
votes
2
answers
2k
views
Are the $\Gamma(N)$ the only normal congruence subgroups of $\mathrm{SL}_2(\mathbb{Z})$?
The answer to the original question is no, see JSE's answer!
Are the $\Gamma(N)$ the only normal congruence subgroups of $\mathrm{PSL}_2(\mathbb{Z})$ with no finite subgroups (elliptic elements)? What ...
- 11.2k
6
votes
2
answers
1k
views
Cartan subgroups of p-adic groups.
Does anyone know about any reference describing the structure of Cartan subgroups in the case of connected p-adic reductive (or let's say semi-simple) groups?
I would like to know how different is ...
- 61
0
votes
0
answers
524
views
DeRham cohomology
The Poincare lemma fails in positive characteristic, since pth powers vanish under differention. My question is : is there still some kind of resolution of the local system k by considering some ...
- 1
14
votes
1
answer
1k
views
Double coset spaces of reductive groups and integral representations of L-functions
Let $G$ be a reductive group over a number field $k$, with center $Z$. Let $P$ be a parabolic subgroup. Let $H$ be a reductive subgroup of $G$. To what extent can we understand the double coset space $...
- 3,183
6
votes
1
answer
545
views
Is it true that no one-dimensional group variety acts transitively on $\mathbb{P}^1$?
This question may be trivial to people with the right background, but I do not see the answer.
Let $\Bbbk$ be an algebraically closed field. Can any one-dimensional group variety (over $\Bbbk$) ...
- 7,338
3
votes
2
answers
506
views
Parabolic induction for GL(2,Z/pn)
Fix a finite extension $F$ of $\mathbb{Q}_p$. Consider its ring of integers $\mathfrak{o}$ with maximal ideal $\mathfrak{p}$. Set $R_n = \mathfrak{o}/\mathfrak{p}^n$. Let $\mathrm{B}$ be the upper ...
- 11.2k
8
votes
5
answers
927
views
Why is p=2 special, if we want to classify cplx. representation of GL2(Zp)?
I am currently reading Shalika's article "Representation of the two by two unimodular group over local fields" and various other related articles, which deal with the classification of complex ...
- 11.2k
9
votes
0
answers
389
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 ...
- 1,645
20
votes
2
answers
2k
views
Frobenius splitting and derived Cartier isomorphism
Let $X$ be a smooth algebraic variety over an algebraically closed field $k$ of characteristic $p>\dim X$. The motivation for my question comes from the following results.
1. If $X$ is Frobenius ...
- 16.2k
6
votes
1
answer
825
views
More on universal homeomorphisms
I would like to understand this notion better; where could I find some examples? In particular, I am interested in the following questions (and references for the answers).
Is a universal ...
- 16.9k
4
votes
2
answers
339
views
Is the pre-image of a regular subscheme with respect to a universal homeomorphism of regular schemes regular?
Let $f:X\to Y$ be a universal homeomorphism of regular (excellent finite-dimensional) schemes, $Z\subset Y$ be a regular subscheme. Is $f^{-1}(Z)$ necessarily regular?
- 16.9k
6
votes
1
answer
393
views
finite quotients of fundamental groups in positive characteristic
For affine smooth curves over $k=\bar{k}$ of char. $p,$ Abhyankar's conjecture (proved by Raynaud and Harbater) tells us exactly which finite groups can be realized as quotients of their fundamental ...
- 4,265
9
votes
4
answers
1k
views
Structure of cuspidal Bernstein components—do non-commutative endomorphism rings ever really show up?
Let $F$ be a finite extension of $\mathbf{Q}_p$ with integers $\mathscr{O}$, let $\mathbb{G}$ be a connected reductive group over $F$ and let $G=\mathbb{G}(F)$ be its $F$-points. Let $X(G)=\...
- 41.4k
1
vote
2
answers
627
views
Zariski density of conjugates of subgroups by arithmetic subgroups?
Let $G$ be a linear algebraic $\mathbb{Q}$-group, which is assumed to be connected, $\mathbb{Q}$-simple, and of adjoint type, such that the Lie group $G(\mathbb{R})$ has no compact factor defined over ...
- 313
-1
votes
1
answer
2k
views
Semisimple elements of a lie algebra
Let $G\subset GL_n(\mathbb{C})$ be an algebraic group of dimension n, and let $\mathfrak{g}$ its Lie algebra.Is there a relations between the maximal number of independent semisimple elements of $G$ ...
- 671
7
votes
1
answer
901
views
Bruhat decomposition for G(R), R local ring or R=Z/p^r
Is there an invariant, which encodes the failure of the Bruhat decomposition to hold for a reductive group with coefficients in a local ring like the p-adic integers or the ring $\mathbb{Z}/\mathfrak{...
- 11.2k
2
votes
2
answers
548
views
The product of non-commuting semisimple matrices need not be semisimple
In general when one proves that the product of semisimple (i.e. diagonalizable) matrices is semisimple one assumes they commute and are thus simultaneously diagonalizable, and then the result follows. ...
- 2,108
7
votes
2
answers
1k
views
Character group of Frobenius kernels
Let $G$ be a semisimple algebraic group over an algebraically closed field $k$ of characteristic $p$ (e.g., $G=SL_n(k)$). Then $G$ is equal to its derived subgroup $[G,G]$. Consequently, the character ...
- 2,160
2
votes
1
answer
408
views
Is restriction of scalars of simply connected algebraic groups still SC?
Let $G$ be a simply connected semisimple algebraic $K$-group and $K$ be a finite extension of $k$.
Is $R_{K/k}G$ still a simply connected algebraic group?
We say $G$ is simply connected if for any ...
- 853
7
votes
1
answer
765
views
Frobenius splitting of affine flag varieties
NOTE: I am very much not an expert on affine Kac-Moody groups or ind-varieties, so the following may be vague.
The first question is: Has anyone developed a theory of Frobenius splitting for ind-...
- 3,637
7
votes
2
answers
736
views
What does a homogeneous space of a linear algebraic group know about the group?
Let $X=G/H$, where $G$ is a connected linear algebraic group over the field $\mathbf{C}$ of complex numbers
and $H\subset G$ is an algebraic subgroup.
In general, we can write the algebraic variety $X$...
- 14.2k
11
votes
2
answers
863
views
Valuations and separable extensions
Let $R$ be a valuation ring containing a field $k$, with residue field $F$ and quotient field $K$. Assume $F/k$ is separable. Is $K/k$ separable?
I have convinced myself that (for a positive answer) ...
- 19.6k
13
votes
3
answers
2k
views
Density question in algebraic group
Suppose G is an algebraic group defined over F, the algebraic closure of F is K. Consider the Zariski topology on G(K), is G(F) Zariski dense in G(K)?
- 131
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
3
votes
2
answers
2k
views
all parabolic subgroup of GL(3,K) and their Levi decomposition?
Who can tell me all parabolic subgroup of GL(3,K) and their Levi decomposition?
- 31
11
votes
1
answer
1k
views
Fontaine's classification of p-divisible groups
Let k be a finite extension of $\mathbb{F}_p$, and W its ring of Witt vectors. Write W[F,V] for the Dieudonn\'e ring.
Let G be a connected p-divisible group which is finite-dimensional over k, and ...
- 519
5
votes
1
answer
366
views
Differential of a nilpotent or semisimple element
Let $G$ be an algebraic connected subgroup of $GL_n(\mathbb{C})$ and let $\chi : G \to \mathbb{C}^*$ a character. Consider $d_e\chi : \mathfrak{g} \to \mathbb{C}$ the differential of $\chi$ at the ...
- 671
10
votes
2
answers
1k
views
For what reductive groups $G$ over $K$ are the inner forms classified by $H^1(K, G^{ad})$?
Suppose $G$ is a connected reductive algebraic group over an arbitrary field $K$; let $Z$ be the center of $G$. The inner automorphisms of $G$ are given by $\operatorname{Inn}(G) = G / Z = G^{\...
- 506
7
votes
5
answers
869
views
Principal bundles over groups
If we have an extension of groups (say algebraic groups or group schemes) $1\to F\to P\to G\to 1$, then $P$ is a principal $F$-bundle over $G$ (is it locally trivial?). How about going in the opposite ...
- 16.2k
3
votes
2
answers
1k
views
Connectedness of centralizers and regular elements in unipotent groups
Let $G$ be a connected linear algebraic group over an algebraically
closed field $k$ of characteristic $p$. An element $x\in G$ is
called regular if its centralizer has minimal dimension among
all the ...
- 3,823
7
votes
2
answers
429
views
Multiplication tables for H*(G/P)?
Hi everyone. My recent work has me developing software to compute in $H^\ast(G/P)$, where $G$ is a complex connected semisimple algebraic group and $P$ is a standard parabolic subgroup (usually, $B$ ...
- 195
18
votes
4
answers
9k
views
Longest element of Weyl groups
What is a reduced expression of the longest element of each type of Weyl group. For type $A_n$ it is just $s_n(s_ns_{n-1})...(s_n...s_1)$. I know for type $B_n,C_n,E_7,E_8$,$G_2$ and $D_n$ (n even) it ...
- 181
2
votes
1
answer
528
views
Is there an easy proof of the fact that the intermediate image functor respects weights?
It was proven in BBD (see Corollary 5.3.2) that for an open immersion $j$ the functor $j_{!*}$ preserves weights of mixed sheaves. The proof relies on several previous results; it is especially ...
- 16.9k
7
votes
2
answers
1k
views
Reference request: representations of unipotent groups have a fixed point.
I'm looking for a reference for the following standard result:
Let $U$ be a unipotent algebraic group over an algebraically closed field $k$ (of any characteristic); then any algebraic ...
- 3,032
0
votes
2
answers
2k
views
non discrete valuation ring [closed]
Hi,
I am looking for examples of non-discrete valuation rings. Could you help me?
Thanks
- 141