All Questions
Tagged with algebraic-groups local-fields
31 questions
0
votes
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84
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is a linear algebraic group over an extension of $\mathbb{Q}_p$ a locally pro finite group?
Let $F$ be a non archimedean local field and let $G$ be linear algebraic group over $F$. I do not have a lot of experience with linear algebraic group, but it seems very obvious that $G$ inherits the ...
- 367
5
votes
1
answer
163
views
When is $G(R)\rightarrow G^{\textrm{ad}}(R)$ surjective?
Consider a reductive group $G$ over a field $k$. The adjoint group $G^{\textrm{ad}}$ is defined by the exact sequence $$1\rightarrow Z(G)\rightarrow G\rightarrow G^{\textrm{ad}}\rightarrow 1$$The ...
- 53
2
votes
0
answers
74
views
Arbitrary base change of a parahoric subgroup in split case
Assume $R\subset R'$ are henselien discretly valued rings with fraction field $K$ and $K'$, $G$ is a semisimple split group over $K$. Consider the parahoric group scheme $\mathcal{P}_F$ over $R$ ...
- 291
0
votes
1
answer
146
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Does an affine building associated to a group satisfy the axioms of building?
Now I am reading Groupes réductifs sur un corps local : I. Données radicielles valuées written by Bruhat and Tits in 1972. Let $\Phi$ be a root system, $G$ a group with root data $(T,(U_{a},M_{a})_{a\...
- 479
3
votes
0
answers
86
views
Cohomology of compact open subgroups of semisimple groups over local fields
Let $E$ be a local field, $\mathcal{O}$ its ring of integers, $k$ its residue field, and $G$ a split semisimple group over $\mathcal{O}$. Let $K$ be an open subgroup of $G(\mathcal{O})$; more ...
- 10.5k
0
votes
0
answers
129
views
Splitting of simply connected algebraic group
Let $k$ be a number field and let $G$ be a connected semisimple, simply connected algebraic group defined over $k$. Let $k'$ be a finite Galois extension over which $G$ splits. By the Chebotarev ...
- 223
9
votes
0
answers
440
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 ...
- 787
1
vote
0
answers
91
views
Why does norm map the $\sigma$-conjugacy classes to the conjugacy classes?
Let $E/F$ be a cyclic extension of order $\ell$ (not assumed prime) of fields of characteristic $0$, and $\Sigma$ its Galois group; we denote by $\sigma$ a generator of $\Sigma$. We denote by $G(E), G(...
- 479
4
votes
0
answers
157
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Reference request - conjugacy classes over local fields
Is there a nice reference for reductive groups over local fields, which for example contains discussion of things such as: Given a semisimple element in $G(F)$, its $G(F)$-conjugacy class is closed in ...
- 5,562
3
votes
0
answers
204
views
Miraculous Parahorics
Let $G$ be a connected simple group over a local field $k$. Let $I\subset G(\mathcal{O})$ denote an Iwahori subgroup of $G(k)$ with Lie algebra $\mathfrak{i}$. Let $P\supseteq I$ be any other ...
- 2,751
4
votes
1
answer
119
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Reference for nonquasi-split groups of type $E_6$ and $E_7$ over local fields
The semisimple groups over a local field have been classified by Tits, cf. [1] "Classification of algebraic semisimple groups" in Boulder and [2] "Reductive groups over local fields" in Corvallis.
In ...
- 991
4
votes
0
answers
99
views
Monoid cohomology of $\mathbb{N}$ for a linear algebraic group
Let $k$ be a finite field and $k_E:=k((X))$ denote the field of Laurent series over $k$. We define a Frobenius endomorphism on $k_E$ via $f(X)\mapsto f(X^p)$. We choose a lift $\varphi:k_E^{sep}\...
- 273
3
votes
0
answers
375
views
compact Zariski-dense subgroups of p-adic groups
Given an algebraic group $G$ defined over a $\mathbb Q_p$. It has two topologies: one is induced by the $p$-adic metric, the other is the Zariski topology. Let $C$ be a compact (w.r.t. the $p$-adic ...
- 453
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
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
7
votes
1
answer
684
views
Type of place versus type of unitary group
Let $F$ be a totally real number field, $E$ a totally imaginary quadratic extension over $E$, and $V$ an hermitian $n$-dimensional vector space over $F$. I assume $n=2m$ is even. Let $U$ be a unitary ...
- 5,424
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
9
votes
1
answer
617
views
Characters of simply connected semsimple algebraic groups over local fields
Let $G$ be a semisimple algebraic group over $\mathbb{Q}_p$. Then by definition $G$ admits no non-trivial algebraic characters, i.e. homomorphisms $G \to \mathbb{G}_m$.
However, it is quite possible ...
- 21.4k
8
votes
2
answers
1k
views
Examples to keep in mind while reading the book 'The Admissible Dual...' by Bushnell and Kutzko and the importance of Interwining of representations
I am a beginner in the field of representation theory. I was reading the book 'The Admissible Dual of $GL(N)$ Via Compact Open Subgroups' by Bushnell and Kutzko.
Let me first describe the book a ...
- 685
3
votes
0
answers
168
views
The left regular representation of the Jacobi groups over local fields of characteristic >2 is type I?
Let $K$ be a non-archimedean local field of characteristic $>2$. Consider the Jacobi group $G=H_{2n+1}(K)\rtimes Sp_{2n}(K)$, which is the semidirect product of the Heisenberg group $H_{2n+1}(K)$ ...
- 1,702
3
votes
0
answers
224
views
Metaplectic groups over non-archimedean local fields of characteristic>2
Let $K$ be a non-archimedean local field of characteristic $>2$. Consider the double cover metaplectic extension of symplectic groups
$p: Mp_{2n}(K)\rightarrow Sp_{2n}(...
- 1,702
0
votes
0
answers
138
views
Orbits of some action of SL2 on Pontryagin dual of the field of formal Laurent series
Let $K=\mathbb{F}_2((t))$ be the field of formal Laurent series over the finite field $\mathbb{F}_2$. Now consider $K^3$ as an additive group and its dual group $\hat{K^3}$, which consists of all ...
- 1,702
0
votes
1
answer
107
views
Existence of the double coset ring on paper of Ihara
In his paper "On discrete subgroups of the two by two projective linear group over $\mathfrak{p}$-adic fields", Yasutaka Ihara considers an abstract group $G$ together with a length function $l$ from $...
- 2,125
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
3
votes
1
answer
245
views
Is $G \rightarrow G/P$ surjective on $K$-points over a local field?
Let $K$ be a local field, $G$ a (connected) reductive $K$-group, and $P \le G$ a parabolic subgroup. Is the map $G(K) \rightarrow (G/P)(K)$ necessarily surjective, and, if so, then why?
- 1,103
11
votes
1
answer
2k
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On unramified p-adic groups
Let G be a reductive group over a local field F. Let O be the ring of integers of F.
The following are equivalent (and groups satisfying these conditions are called unramified):
(a) G is quasisplit ...
- 8,866
2
votes
2
answers
552
views
Compact elements in $G(K)$ for a reductive group $G$ over a nonarchimedean local field $K$
Let $K$ be a nonarchimedean local field and $G$ a (connected) reductive group over $K$, so that $G(K)$ carries a natural topology. An element $g \in G(K)$ is compact if it is contained in a compact ...
- 1,103
2
votes
0
answers
415
views
Algebraic characters and quasi-characters of reductive algebraic group over non-archimedean local field
Let $G$ be a reductive algebraic group over $F$, where $F$ is a non-archimedean local field.
Then $G(F)$ is a p-adic group.
Let $\Psi(G)$ be the lattice of algebraic characters.
Let $\Lambda_G$ be the ...
- 1,457
4
votes
1
answer
469
views
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
10
votes
4
answers
2k
views
Reference request: expository text on the structure of reductive groups over non-archimedean local fields
I am interested in an expository text in English, which summarizes the main results and aspects of the structure theory of reductive groups over local fields, in a hopefully not very technical manner (...
- 2,027
4
votes
0
answers
1k
views
Cartan decomposition for upper triangular matrices
Due to the comments, I have the impression that I have to be more precise.
Consider $G= GL_n(F)$ for a non-Archimedean field $F$ with ring of integers $o$.
Let $K= GL_n(o)$ and let $I$ the Iwahori ...
- 11.2k