All Questions
2,543 questions
2
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169
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Relation between a conjecture of Pink and semi-abelian varieties
A conjecture of Pink says that in a mixed Shimura variety, every Hodge-generic point is Galois generic (conjecture 6.8 of "A Combination of the Conjectures of Mordell-Lang and Andr\'e-Oort). One can ...
2
votes
0
answers
463
views
Center of a split unipotent group
Let $N$ be a unipotent algebraic group over a field $k$ of characteristic $p>0$.
Assume that $N$ is split (i.e. it admits a filtration whose quotients are isomorphic to the additive group).
In ...
- 991
2
votes
0
answers
948
views
Description of the center of a reductive group using absolute and relative roots
Let $G$ be a connected, reductive group over a field $k$. Let $T \subseteq B$ be a maximal torus and Borel subgroup of $G$ with corresponding base $\Delta \subseteq X(T)$. Then $T$ contains $Z(G)$, ...
- 6,180
2
votes
0
answers
178
views
Absolute and Relative Coroots
$G$ is a connected reductive group over a field $k$. $T$ is a maximal torus and $S \subset T$ is a maximal $k$-split torus. We have an embedding $X_*(S) \hookrightarrow X_*(T)$. Is it true that if $\...
- 953
2
votes
0
answers
82
views
What is the classification of this group?
Let $K=\mathbb C((t))$ and $O=\mathbb C[[t]]$, and $n\geq 1$. Consider the matrix $$J_{2n}=\begin{pmatrix} 0& I_n \\ -I_n & 0\end{pmatrix},$$ And let $\Psi : K^{2n}\times K^{2n}\rightarrow K$ ...
- 1,891
2
votes
0
answers
885
views
Why is the radical of a reductive group equal to the connected component of the center?
If $G$ is a connected reductive group over a perfect field $k$ (The definition given in Milne's "Algebraic Groups": $G$ is a connected group variety containing no non-trivial connected unipotent ...
2
votes
0
answers
436
views
Central isogenies differ by an element of the maximal torus
Let $G, G'$ be connected, reductive groups over an algebraically closed field $k$, and let $T$ be a maximal torus of $G$. A central isogeny is a surjective morphism of algebraic groups $\phi: G \...
- 6,180
2
votes
0
answers
115
views
Converging sequence of base change
Here is a natural question that I hope will be of interest to some.
Let $\mathbf{F}_p(\!(T)\!)$ be the field of formal Laurent series over $\mathbf{F}_p$. An automorphism of $\mathbf{F}_p(\!(T)\!)$ ...
- 1,017
2
votes
0
answers
215
views
About the representation theory of $SL_2$ in positive characteristic
Let $k$ be an algebraically closed field of positive characteristic, let $G=SL_2(k)$ acting naturally on $V=k^2$, and let $S^r V$ be the $r$-th symmetric power of the standard representation $V$.
Is ...
- 21
2
votes
0
answers
232
views
Didactic (counter-)examples in algebraic groups and groups schemes
Algebraic groups are very rich objects. As such, a large bag of examples against which one can test his intuition can be very helpful in learning the general theory.
What are some good didactic (...
- 7,799
2
votes
0
answers
286
views
Does the sheaf of locally exact differential forms splitting in positive characteristic
Let k be an algebraically closed field of characteristic $p>2$, $X$ a smooth projective curve of genus $g>1$ over $k$, and $F_X:X\rightarrow X$ be the absolute Frobenius morphism. Let $B^1_X$ be ...
- 85
2
votes
0
answers
233
views
Naive question on quotients of algebraic groups and moduli spaces
I asked this question on MathStackExchange and was suggested to ask here.
I started studying GIT theory, and I am stuck with the following problem.
Let $\textbf{Sch}$ be the category of schemes over ...
- 558
2
votes
0
answers
419
views
How can this argument calculating the Haar measure on a parabolic subgroup be generalized to the non-split case?
Let $\mathbf G$ be a connected, reductive group over a local field $F$. Assume there is a maximal torus $\mathbf T$ which is split over $F$. Let $\mathbf P$ be a parabolic subgroup of $\mathbf G$ ...
- 6,180
2
votes
0
answers
47
views
Rational isotropic space stable by a real split torus
Let $G$ be a semisimple algebraic group defined over $\mathbb{Q}$ and $(\rho,V)$ be an irreducible $\mathbb{Q}$-representation of $G$ preserving a sympectic form $w$ defined over $\mathbb{Q}$.
Can ...
- 474
2
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0
answers
571
views
Elementary question: Sheaf on quotient is locally free
I'm sorry for the following elementary question.
Things are algebraic/holomorphic over $\mathbb{C}$.
I'm reading the book "Vector bundles on Complex Projective Spaces" by Okonek et al. and the ...
- 297
2
votes
0
answers
355
views
Parabolic Subgroups: dimension of double coset $PwN$?
Let $G$ be a quasisplit connected, reductive group over a field $F$. Let $A_0$ be a maximal $F$-split torus, and $P$ a maximal $F$-parabolic subgroup containing $A_0$. Let $W = N_G(A_0)/Z_G(A_0)$ be ...
- 6,180
2
votes
0
answers
109
views
What does equality modulo $p$ of $p$-adic linear groups imply?
$\newcommand{\GL}{\mathrm{GL}}\newcommand{\dbZ}{\mathbb{Z}}\newcommand{\dbF}{\mathbb{F}}\newcommand{\dbN}{\mathbb{N}}$
Hello.
I have a small question regarding closed subgroup of $\GL_n(\dbZ_p)$, ...
- 993
2
votes
0
answers
150
views
Why do proper morphisms induce covariant maps on Grothendieck groups?
Let $X$ be a separated scheme of finite type over a field $k$, and let $G$ be a smooth linear algebraic group over $k$ acting on $X$.
Let $K_G(X)$ denote the Grothendieck group of coherent $(G,\...
- 7,547
2
votes
0
answers
345
views
Examples of semi-stable models of curves
Let $R$ be a discrete valuation ring with fraction field $K$ of characteristic zero and residue field $k$ of characteristic $p>0$. Assume $k$ is algebraically closed. I want to produce examples of ...
- 2,323
2
votes
0
answers
142
views
Unipotent characters of (disconnected) centralizers of semisimple elements: Why these two definitions are equivalent?
Assume that $\mathcal{G}$ is a simple simply-connected algebraic group over $k$, where $k$ is algebraic closure of a finite field of characteristic $p>0$, and $F$ is a Frobenius endomorphism. Let $(...
- 143
2
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0
answers
142
views
Iwahori subalgebra as maximal solvable
I think the following is true, but haven't came up with a proof myself. Thanks in advance!
Let $G$ be a semisimple (to avoid more words) algebraic group over $\mathbb{C}$. Write $F=\mathbb{C}((t))$ ...
- 1,856
2
votes
0
answers
148
views
Purely inseparable $k$-rational dominant maps between an absolutely irreducible $k$-surface and $\mathbb{P}^2$
Let $X$ be an absolutely irreducible surface over an algebraically closed or finite field $k$ of characteristic $p$. Is it true that the following conditions are equivalent?
There is a purely ...
- 1,833
2
votes
0
answers
386
views
Bruhat-Tits building for $PGL_2(F)$ and repressentation theory
I am trying to understand representation theory of $GL_2(F)$ where $F$ is a finite extension of $\mathbb{Q}_p$. I am using the following paper of Colmez to understand it:
https://webusers.imj-prg.fr/~...
- 685
2
votes
0
answers
97
views
Can we write an element in a super Grassmannian as a pair of matrices?
Super Grassmannians are introduced by Manin, see for example.
Elements in a grassmannian can be written as matrices, see for example.
Can we write an element in a super Grassmannian as a pair of ...
- 6,211
2
votes
0
answers
476
views
Does the numerical equivalence relation coincide with the homological one for 1-cycles (in positive characteristic)?
Is the Grothendieck's Standard Conjecture D (stating that the numerical equivalence relation for algebraic cycles with rational coefficients coincides with the homological one) known to be true for ...
- 16.9k
2
votes
0
answers
131
views
Connectedness of Centralisers in Unitary group
I want to understand centralizers of semisimple elements in unitary groups.
Let us begin with example of $GL_n(k)$. Centralizers of semisimple elememts are a product of smaller $GL_m(k)$ thus ...
- 661
2
votes
0
answers
139
views
Centralizer of a dense subgroup in a maximal subgroup of a reductive group
I am looking for a reference to the following statement
"Let $G$ be a reductive algebraic group and $K$ a maximal compact subgroup of $G$. If $H$ is a dense subgroup in $K$, then the centralizer of $H$...
- 21
2
votes
0
answers
975
views
Complete reducibility of representations of reductive algebraic groups
I need a reasonably detailed reference for the proof of the fact that, in characteristic 0, any linear representation of a reductive algebric group is completely reducible. I looked in Humphries and ...
- 21
2
votes
0
answers
213
views
Characterizing subgroups of R^n with dense factors
It is well known that (additive) subgroups of $\mathbb{R}^n$ are products of discrete subgroups (lattices) by dense subgroups in subspaces. My question is the following: given a generator set of $p$ ...
- 51
2
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0
answers
111
views
A question about the associative classes of parabolic subgroups
Let $\mathbb{A}$ denote the adele group of $\mathbb{Q}$. Suppose that $P_1$ and $P_2$ are parabolic subgroups of a reductive algebraic group $G$, and consider their Langlands decomposition
$$
P(\...
- 21
2
votes
0
answers
50
views
is there a criterion for a two-generator subgroup of $PL(2,K)$ to be a cocompact lattice?
In the case of the group $SO(n,1)$ there is a criterion known for whether or not two given elements of the group generate a cocompact lattice. Is any similar criterion known in the case of $PL(2,K)$ ...
- 2,125
2
votes
0
answers
105
views
Conjugacy classes of involutions in Kac-Moody groups
Let $A=(a_{s,s'})_{s,s'\in S}$ be a generalized Cartan matrix.
Let $G=G(A)$ be the corresponding simply connected complex Kac-Moody group with Cartan subgroup $H$ and Weyl group $W$ acting on $H$.
...
- 14.2k
2
votes
0
answers
142
views
maximal elementary abelian p-subgroups of finite groups of Lie type
Let $G$ be a simple algebraic group over an algebraically closed field k of characteristic $p$. Let $G(\mathbb{F}_{p^r})$ be the corresponding finite group of Lie type. Is it true that every maximal ...
- 21
2
votes
0
answers
223
views
Form over $ \mathbb{Z} $ of non-split simple algebraic groups over non-archimedean local fields
Here is a basic observation :
On page 68 of his article Reductive groups over local fields, Jacques Tits writes:
All types of groups listed (...) exist over an arbitrary [non-archimedean local] ...
- 1,017
2
votes
0
answers
245
views
Reference request: proofs of the theorems in the paper "On the representation of the group GL(n, K) where K is a local field"
In the paper On the representation of the group $GL(n, K)$ where $K$ is a local field by Gelfand and Kazhdan, it is said that the proofs of the theorems in the paper are published in some other papers....
- 6,211
2
votes
0
answers
275
views
From algebraic group actions to group scheme actions
I am trying to understand the basic results of geometric invariant theory. I want to pull off the band aid and use Mumford, but am a neophyte with respect to scheme theory. Thus, I have been trying to ...
- 1,089
2
votes
0
answers
156
views
Extension of the Hilbert-Mumford Criterion
Let $X$ be a smooth variety, $L$ a line bundle on $X$ and $G$ a reductive group actin on $X$ with a linearization of the action to $L$. Say we are over the complex numbers.
Both the concept of GIT ...
- 2,384
2
votes
0
answers
659
views
Constant group scheme and torsors
Let $X$ be a scheme and $G$ a (commutative) constant group scheme. Consider a $G$-torsor $Y$ for $X$, by which I mean that there is a canonical isomorphism:
$$g_Y \colon Y \times_X Y \cong Y \...
- 781
2
votes
0
answers
317
views
Dimension of affine Springer fiber and its functor of points as an ind-scheme
Let $k$ be a finite field and let $F = k( (t))$ with ring of integers $\mathfrak{o} = k[ [t]]$. Let $G$ be a connected linear algebraic $k$-group with Lie algebra $\mathfrak{g}$. Suppose that $\gamma\...
user1437
2
votes
0
answers
192
views
Analogies, Riemann surfaces and Algebraic groups
Let G be a complex simple Lie group of adjoint type. Then, it is well known that every such $G$ contains, unique up to conjugacy, an irreducibly embedded copy of $PSL(2,\mathbb{C}).$ This fact seems ...
- 3,020
2
votes
0
answers
132
views
$\mathbb{Q}$-forms of $\mathrm{SL}_2\times \mathrm{SL}_2$
I am learning something about lattices in algebraic groups. Consider the algebraic group $\mathrm{SL}_2\times \mathrm{SL}_2$. What are the $\mathbb{Q}$-forms of such groups?
- 21
2
votes
0
answers
311
views
Generators of the algebra of invariant polynomials on a Lie algebra and the root-space decomposition
Let $G$ be a connected, simply-connected complex semisimple group with Lie algebra $\mathfrak{g}$. Fix a pair $T\subseteq B\subseteq G$ of a maximal torus and Borel subgroup, and let $\mathfrak{t}$ ...
- 4,920
2
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0
answers
853
views
polynomials with roots on the unit circle
Suppose $P(x) \in \mathbb{Z}[x]$ is irreducible, and such that at least one of its roots has modulus $1.$ Is there anything we can say about the reduction of $P(x)$ modulo primes? Do these have some ...
- 96.4k
2
votes
0
answers
232
views
Intuition for the structure theorem for connected solvable algebraic groups over an algebraically closed field of positive characteristic
In the theory of algebraic groups one of the fundamental results is the structure theorem for connected solvable groups. I never understood the proof in any of the standard textbooks, so I just moved ...
- 3,830
2
votes
1
answer
141
views
Characteristic polynomials of reductive subgroup over C
Can any one provide a hint to prove the following statement? :
Let $H$ be a complex reductive subgroup (not necessarily connected) contained in $SO(n,\mathbb{C)})$. Consider the map $H \rightarrow ...
- 601
2
votes
0
answers
118
views
Symmetric spaces which are compact modulo the unipotent radical are compact
Is the following true?
Let $X = G/H$ be a symmetric space of a reductive group over a p-adic field $F$.
Let $X^0$ be an open orbit w.r.t. the action of the minimal parabolic $B$ of $G$.
Let $U$ be ...
- 2,649
2
votes
0
answers
376
views
Parahoric group schemes over curves
Let $X$ be a smooth projective curve over $\mathbb{C}$ and Let $G$
be a complex reductive group. By a parahoric group scheme $\mathcal{G}$
over $X$, I mean a smooth group scheme over $X$ whose ...
- 135
2
votes
0
answers
172
views
Springer Isomorphisms for Adjoint Simple Exceptional Groups
I'm trying to understand explicitly a construction of Springer isomorphisms for adjoint exceptional groups given by Bardsley and Richardson. Their construction is as follows. Let $G$ be an adjoint ...
- 2,902
2
votes
0
answers
123
views
What is classified by $H^1(\mathbb{R},SO(p,q))$ and by $H^1(\mathbb{R},SU(p,q))$?
We denote by $F^{\mathbb{R}}_{p,q}$ the quadratic form over the field ${\mathbb{R}}$
$$
F^{\mathbb{R}}_{p,q}(x)=x_1^2+\dots+x_p^2-(x_{p+1}^2+\dots+x_{p+q}^2)
$$
on the vector space $V^{\mathbb{R}}:={\...
- 14.2k
2
votes
0
answers
116
views
Cohomology and quotients for the canonical topology
Recall that for any category $\mathcal C$, there is a unique finest topology, the canonical topology on $\mathcal C$ for which all representable functors are sheaves. I am interested in the example $\...
- 5,759