All Questions
Tagged with local-fields reductive-groups
19 questions
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
9
votes
1
answer
331
views
Tempered Iwahori-spherical representations
Consider a local field $F$ of characteristic 0 and $G=GL_n(F)$.
It is well known (for example Cartiers article in Corvallis) that an admissible, irreducible representation $\pi$ of $G$ has a ...
- 163
9
votes
2
answers
626
views
Invariant functor for admissible representations of reductive groups over local fields
Hello,
I have a question concerning a certain functor between represention categories. I'm rather sure this is already known, but I could not find a reference.
Let $F$ be a local non-archimedean ...
- 223
8
votes
2
answers
725
views
Is every connected reductive group over a local field already defined over a global field?
Let $K$ be a local field, e.g. $\mathbb{Q}_p$ or $\mathbb{F}_p((t))$. Let $G$ be a connected reductive group over $K$. Is it true that $G$ is already defined over a global field? More precisely, does ...
- 113
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
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
7
votes
1
answer
851
views
Understanding the structure of unitary groups
I would like to understand precisely the structure of unitary groups.
Let $F$ be a global number field, $E$ a quadratic extension of $F$, and $U$ a unitary group on $E$ (i.e. the group of ...
- 5,424
6
votes
2
answers
299
views
Is every compact simply-connected reductive p-adic group perfect?
Let $k$ be a nonarchimedean local field and $G$ a reductive $k$-group,
which we assume to be semisimple and simply-connected. Recall that an abstract group $H$ is perfect if it is generated by ...
- 835
6
votes
1
answer
598
views
Clarification about Tits' article in the Corvallis
I am studying Tits' article in the Corvallis wherein he defines the apartment in the general case (not necessarily split). I wish to know what he means about the filtration of the groups $U_a(K)$ (...
- 899
5
votes
0
answers
231
views
Question on the unramified local Langlands conjecture
I'm working on the unramified local Langlands conjecture and there is something that I don't understand if it is true or not. I want to start by saying that I don't care about endoscopic transfer or ...
- 151
4
votes
1
answer
751
views
How to understand the representation theory of $SL(n)$ from $GL(n)$?
Let $F$ be a local field. Consider the group extension (split)
$$ PSL(n,F) \rightarrow PGL(n,F) \rightarrow F^\times / (F^\times)^n.$$
What knowledge about $PGL(n)$ is necessary in order to understand ...
- 11.2k
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
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
3
votes
0
answers
85
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
2
votes
1
answer
123
views
F-points of product of closed subgroups vs. product of F-points, F a local field, reference?
Let $F$ be a finite extension of $\mathbb Q_p$, where p is an odd prime. Let $G$ be a connected reductive group defined over $F$. Let $M, H$ be closed $F$-subgroups of $G$ (in particular, I'm ...
- 41
2
votes
0
answers
258
views
Is a reductive group scheme always parahoric?
Let $R$ be complete (or, more generally, Henselian) discrete valuation ring with fraction field $K$. Let $G$ be a reductive $R$-group scheme. Is $G$ a parahoric in the sense of Bruhat-Tits? If so, ...
- 407
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
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
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