All Questions
2,543 questions
0
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245
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compact subset in linear algebraic group over local field
Let $F$ be a local field of characteristic zero, and $G$ a linear algebraic group with finite connected components over $F$. We will consider the $G(F)$, and give it $p$-adic topology. Let $C$ be some ...
- 12.1k
4
votes
1
answer
469
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Semisimple group not split by an unramified extension?
Let $F$ be a nonarchimedean local field. Does there exist a semisimple algebraic group over $F$ which is not split over a maximal unramified extension of $F$ ?
- 1,704
1
vote
0
answers
190
views
Compactifications of group schemes
Let $G$ be a group scheme over a scheme $S$ which is the spectrum of a discrete valuation ring. Let $\eta$ (resp. $s$) be the generic (resp. closed) point. Assume that the generic fiber $G_{\eta}$ is ...
- 3,359
3
votes
0
answers
235
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 ...
- 2,508
9
votes
1
answer
579
views
Deformation theory of octonion algebras?
In Grothendieck's Brauer group papers, he uses deformation theory to bootstrap the theory of central simple algebras over a field to the theory of Azumaya algebras over rings (and schemes). I am ...
- 375
6
votes
1
answer
494
views
Decomposing tensor products of modules for the orthogonal/symplectic groups in characteristic zero
I would like to know if there is a perfect analogue of the classical Littlewood-Richardson rule for decomposing tensor products of simple modules for the orthogonal/symplectic groups in characteristic ...
- 31.2k
5
votes
1
answer
322
views
Is SL_n/S(GL_k x GL_n-k) symmetric?
Background: a symmetric variety is a homogeneous space $G/H$ associated to an involution $\theta$ of a semisimple algebraic group $G$ and $\{g | \theta(g) = g\} = G^\theta \subset H \subset N_G(G^\...
- 3,968
25
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0
answers
1k
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Status of the Euler characteristic in characteristic p
In the introduction to the Asterisque 82-83 volume on `Caractérisque d'Euler-Poincaré, Verdier writes:
Enfin signalons que la situation en caractéristique positive est loin
d'être aussi ...
- 7,338
4
votes
1
answer
596
views
counting the parabolic subgroups that contain a fixed semisimple element
Let $G$ be a finite group of Lie type , and let $P$ be a parabolic subgroup of $G$ containing a proper Borel subgroup $B$. Let $s$ be a semisimple element of $G$. I want to count the number of ...
- 225
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0
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501
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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 ...
CommunityBot
- 1
3
votes
1
answer
611
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How to translate the representation theory of semisimple to reductive groups?
I am aware of the following question: Definitions of Reductive and Semisimple Groups
So let me phrase a precise question:
Is there a standard technique by which one can translate the unitary/...
CommunityBot
- 1
6
votes
1
answer
1k
views
Group scheme over a DVR whose special fibre is the image of points under reduction mod p
Let $R$ be a complete discrete valuation ring with maximal ideal
$\mathfrak{p}$ and algebraically closed residue field $k$. Denote
the field of fractions of $R$ by $F$. Let $G$ be an affine flat
group ...
- 3,823
1
vote
1
answer
242
views
Smooth map to the stack of G-bundles
Let $G$ a semisimple group and $B$ a Borel subgroup.
We denote by $Bun_{G}$ the stack of G-bundles.
Is it true that a certain open subset $Bun_{B,r}$ maps smoothly to $Bun_{G}$?
My question comes ...
- 4,111
8
votes
0
answers
182
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 ...
- 56.9k
1
vote
0
answers
307
views
For what fields is $GL_n(k)$ a rational variety?
I know that every linear algebraic group is rational over algebraically closed fields. To what extent is that true for other fields? For example: is $GL_n(\mathbb{Q}_p)$ a rational variety? Are there ...
- 11
5
votes
2
answers
761
views
Equivariant Cohomology of a Complex Projective Variety
Suppose that I have a complex projective variety $X$ endowed with an algebraic action of a complex torus $T$. Suppose also that the set $X^T$ of fixed points is finite. I would like to relate the ...
- 27.9k
1
vote
1
answer
347
views
Reference on elements of finite order in principal congruence subgroups of symplectic groups
We should start with the definition of the symplectic group for an arbitrary ring $R$.
The symplectic group $Sp(g,R)$ is the subgroup of $SL(2g,R)$ such that all elements satisfy $M=J_g^t M J_g$ with $...
- 44.4k
3
votes
1
answer
754
views
Quotient of algebraic groups in the étale topology
Let $G$ be an affine algebraic group over $\mathbb{C}$. According to SGA3, any closed normal subgroup $N$ is representable by an affine algebraic group, as is the quotient $G/N$.
These statements ...
- 3,396
3
votes
1
answer
295
views
differential of the characteristic polynomial
Let $\chi:GL_{n}(\mathbb{C})\rightarrow \mathbb{C}^{n}$ the map given by the coefficients of the characteristic polynomial.
Let $A$ a regular semisimple matrix, do we have a formula for the ...
- 52.9k
3
votes
1
answer
1k
views
On the $F$-rational points of the derived group of a connected reductive algebraic group
Let $F$ be a local non-archimedean field and let $G$ be a connected reductive algebraic group defined over $F$. Let $G_{der}$ denote the algebraic derived group of $G$; this is connected and ...
- 639
1
vote
0
answers
88
views
open immersion, affine grassmanian and negative loop group
Let $G$ a semisimple group over $k=\bar{k}$.
Let the $k$-indgroup, $LG^{-}\subset G(k[t^{-1}])$ be the kernel of the reduction. We know by Faltings that the multiplication map:
$LG^{-}\times G(k[[t]]...
- 3,472
2
votes
2
answers
668
views
Why are these parabolic subgroups opposed?
I am reading notes of Michel Brion on spherical varieties.
Consider a reductive group $G$, a Borel $B$ in $G$, a finite dimensional $G$-module $M$ and a closed orbit $Y$ of $G$ in $\mathbb{P}(M)$. ...
- 52.9k
6
votes
3
answers
1k
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Naive question about the representation theory of algebraic groups and hopf algebras
I have been learning some representation theory and have some questions about the following pattern:
Instance 1: If we have a finite group $G$ and a field $k$, a representation of $G$ over $k$ ...
- 7,426
11
votes
1
answer
1k
views
Counting conjugacy classes in simple groups of Lie type
Finite groups of Lie type include those obtained as rational points of a connected simple algebrraic group over a finite field $k = \mathbb{F}_q$ of characteristic $p$: these are split or quasi-split. ...
CommunityBot
- 1
5
votes
0
answers
223
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 :
$$...
- 991
1
vote
0
answers
102
views
do commutative groups torsors have a point in an Abelian extension of the base field?
Let $A$ be a principal homogeneous space for a commutative algebraic group defined over a field $k$ that contains all roots of unity. Is it true that $A$ has a $K$-point for an extension $K \supset k$ ...
- 4,070
5
votes
2
answers
586
views
Quotient of a reductive group by a non-smooth subgroup
This is a continuation of my question Quotient of a reductive group by a non-smooth central finite subgroup.
Let $G$ be a smooth, connected, reductive $k$-group over a field $k$ of characteristic $p&...
CommunityBot
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2
votes
0
answers
404
views
Conjugacy classes of centralizers of semisimple elements in a finite group of Lie type
Let $G$ be a finite group of Lie type. By Deriziotis' and Carter's articles we know that conjugacy classes of connected centralizers of semisimple elements are parametrized by $(J,[w])$ where $J$ is a ...
- 19.6k
2
votes
0
answers
255
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Lang isogeny for group stacks
Let $G$ be a commutative algebraic group stack over $\mathbb{F}_q$ (I don't really care about the precise definition: I'm secretly thinking about the Picard stack of a projective curve). To what ...
- 3,605
3
votes
1
answer
252
views
affine weyl group and affine schubert cells
Let $G$ a connected reductive split group over $k=\bar{k}$, $(B,T)$ a split Borel pair. Let $F:=k((t)))$. Let $\tilde{W}$ the extended Weyl group, $\tilde{W}=N_{G}(T(F))/T(O)$.
By Iwasawa ...
- 27.9k
1
vote
1
answer
387
views
weights and exceptional root systems
Let $G$ a simple simply connected group over $\mathbb{C}$ and $W$ his Weyl group.
Let $\lambda$ a minuscule or quasiminuscule weight.
For which types and for which weights do we have that:
$\forall ...
- 52.9k
2
votes
0
answers
187
views
Generators of $Rep(G)$
Let $G$ be a reductive group over $\mathbb{C}$ and $Rep(G)$ the category of rational representations.
Is there a "nice" (let's say combinatorical) description of the generators of $Rep(G)$ as a tensor ...
- 2,554
5
votes
1
answer
774
views
Weyl group of the restriction of scalars of split reductive group
Let $G$ be a connected algebraic group defined over a field $E$ of characteristic $0$. Suppose $G$ reductive $E$-split and let $T \subset G$ a maximal (split) torus defined over $E$.
Set $G' = Res_{E/...
- 52.9k
3
votes
0
answers
139
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.
...
- 31
0
votes
0
answers
109
views
solve the singularities of parabolic orbits of schubert cells
Let G a semsisimple connect'ed group over $k$, $B$ a Borel and $P$ a parabolic subgroup of $G$ with Weyl group W_{P}.
For $w\in W_{P}\backslash W/W_{P}$, how can we solve the singularities of $X_{w}=\...
- 3,472
4
votes
0
answers
136
views
A subring of the Serre Swinnerton -Dyer ring of level N modular power series
Suppose ell is prime and (N,ell)=1. Consider those power series over Z that are expansions at infinity of modular forms for gamma_0 (N) of weight a multiple of ell-1. I'll say that an element of (Z/...
- 5,422
3
votes
2
answers
409
views
Connectedness of Springer Fibers
Let $G$ be a connected, simply-connected, complex semisimple Lie group with Lie algebra $\frak{g}$. Let $\mu:T^*\mathcal{B}\rightarrow\mathcal{N}$ be the Springer resolution of $\mathcal{N}$. If $G=\...
- 1,201
3
votes
2
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540
views
group generated by Coxeter elements
Let $G$ a connected semisimple simply connected group over $\mathbb{C}$ and $W$ his Weyl group.
What can be said about $W'$, the subgroup of $W$ generated by the Coxeter elements of $W$?
- 52.9k
3
votes
1
answer
552
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On the Cartan decomposition of unitary group
Hello. I have some question on Cartan decomposition of unitary group, especially $U(2)$.
I am interested in local situation, that is p-adic or archimedian.
Let $F$ be a local field and $E$ be its ...
CommunityBot
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6
votes
1
answer
1k
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A question about the proof of Beilinson-Bernstein localisation
I'm trying to understand the proof of the Beilinson-Bernstein localisation theorem at the moment, but there's just one point where I'm having a mental block, and was wondering if anybody could clarify ...
- 105
2
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1
answer
1k
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Thom-Gysin Sequences and Stratifications
Let $X$ be an affine algebraic variety over $\mathbb{C}$, and let $G$ be a semisimple complex linear algebraic group acting by variety automorphisms with finitely many orbits. The decomposition of $X$ ...
- 40.3k
3
votes
2
answers
2k
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Is there an almost-direct product decomposition for disconnected reductive algebraic groups?
$\textbf{Some definitions:}$
Let $G$ be an algebraic group (for me that is the complex points of an affine algebraic group). We say $G$ is reductive if its unipotent radical (maximal connected normal ...
CommunityBot
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4
votes
1
answer
369
views
Gauss mapping in finite characteristic
Suppose that $X\subset\mathbb P^n$ is a $d$-dimentional smooth projective variety (not a linear subspace) over an algebraically closed field. If $\gamma\colon X\to\mathrm{Gr}(d,\mathbb P^n)$ is Gauss ...
- 1,939
2
votes
2
answers
408
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a conjugacy question in quasi-split reductive groups
I have a somewhat technical question about conjugacy in
quasi-reductive groups.
Let $k$ be a field (in my main case interest, $k$ is finite), $G$ be a connected
quasi-split reductive group over $k$....
- 52.9k
1
vote
1
answer
257
views
arithmetic group over function fields and its fundamental domain
Let $G$ be a semi-simple algebraic group defined over a global function field $K$.
Let $S$ be a finite set of places of $K$. For a place $v$ of $K$ let $K_v$ be the completion under $v$. We take $K_S=\...
- 52.9k
1
vote
1
answer
2k
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About isomorphism of $PGL(2)$ and $SO(3)$ [closed]
I need to prove that $PGL_2(\mathbb{R})\cong SO_3(\mathbb{R})$. Abstract considerations show that both can be identified with the group of projective motions of a conic curve. But maybe there is more ...
- 44.4k
2
votes
0
answers
606
views
Kawamata-Viehweg Vanishing Theorem for Excellent Surfaces
Is there any version of Kawamata-Viehweg vanishing theorem that holds for excellent surfaces? I will be very glad if there is one such, but if not then is it at least true for a surface over a non ...
- 165
3
votes
0
answers
184
views
Integral conjugacy vs. Rational conjugacy
Let $G$ be an algebraic group over a field $F$. Let $g\in G(F)$, and write $C(g)$ for the centralizer of $g$ in $G$. Conjugacy over $F$ is of course not necessarily the same thing as conjugacy over an ...
- 407
4
votes
0
answers
814
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Adjunction Formula for Weil Divisors on a Normal Variety X
Let $X$ be a normal variety over an algebraically closed field $k$ of characteristic $p>0$ and $S$ be a prime Weil divisor on $X$ which is normal too. Now if $K_X+S$ is NOT $\mathbb{Q}$-Cartier, ...
- 165
2
votes
0
answers
129
views
Seeking a generalization of group embedding of symmetric varieties
I am looking for generalizations of the following construction.
Let $G$ be a connected, reductive group and let $\theta : G \rightarrow G$ be an involution. Let $H = G^{\theta}$ be the subgroup of $\...
- 1,780