28
votes
Accepted
What's the point of a Whittaker model?
This question is a bit like saying "what's the point of the theory of bases for vector spaces -- this just gives you an isomorphism of your space with $\mathbb{R}^n$. What is the point of defining ...
- 40.8k
23
votes
Accepted
What is a tamely-ramified Weil-Deligne representation?
$\def\R{\mathbf{R}}$
$\def\Z{\mathbf{Z}}$
$\def\Q{\mathbf{Q}}$
$\def\Qbar{\overline{\Q}}$
$\def\F{\mathbf{F}}$
$\def\GL{\mathrm{GL}}$
$\def\Gal{\mathrm{Gal}}$
Here are some thoughts on your question ...
- 957
17
votes
Accepted
Is it true that irreducible generic representations of $G_2(F)$ are self-dual?
Since $-1$ is in the Weyl group (over $F$, not just the algebraic closure) you might expect every irreducible representation to be self-dual. This is the case over $\mathbb R$. It is false over a $p$-...
- 2,429
15
votes
$GL_n(\Bbb Z_p)$ conjugacy classes in a $GL_n(\Bbb Q_p)$ conjugacy class
Is it always finite? No.
The matrices $u_n=\begin{pmatrix}1 & p^n\\0 & 1\end{pmatrix}\in\mathrm{GL}_2(\mathbf{Z}_p)$ are all conjugate in $\mathrm{GL}_2(\mathbf{Q}_p)$, and pairwise non-...
- 60.2k
14
votes
A question on representation theory of p-adic groups
It is indeed well-known that the category of smooth admissible representations of $G$ (and other reductive $p$-adic groups)
is not semi-simple. The principal series, that is the representations ...
- 25.7k
13
votes
Accepted
Examples to keep in mind while reading the book 'The Admissible Dual...' by Bushnell and Kutzko and the importance of Interwining of representations
Your question (f) makes me suspect that you don't really know any of the representation theory of $p$-adic groups at all. You definitely should not try to read the Bushnell--Kutzko book before ...
- 574
13
votes
Is $SL_n(\mathbb{Z}_p)$ virtually torsion free?
$SL_n(\mathbb{Z}_p)$ is virtually torsion free as it is $p$-adic analytic and therefore contains a uniformly powerful open subgroup.
- 5,200
11
votes
What's the point of a Whittaker model?
In addition to the other good answer, in the context of automorphic forms the uniqueness of local Whittaker models is specifically useful in computations of global integrals: when/if the integral can ...
- 22.6k
10
votes
Accepted
Topological dimension of $p$-adic manifolds
$p$-adic numbers are locally compact, Hausdorff and totally disconnected (see this nLab page), hence they are zero-dimensional. This means that---at least naively---topological dimension of $p$-adic ...
- 627
9
votes
Accepted
Naïve definition of parahoric subgroup
I'm not sure what it means to define parahoric subgroups "purely in terms of $B(G, F)$"; I would say that every definition boils down to taking integral points of integral models in one way ...
- 11.4k
9
votes
Accepted
Is the representation of $GL_n(\mathcal{O})$ in functions on Grassmannian multiplicity free?
Yes, this is due to Hill:
Hill, Gregory, On the nilpotent representations of (GL_ n({\mathcal O})), Manuscr. Math. 82, No. 3-4, 293-311 (1994). See especially Corollary 3.2.
This was generalised and ...
- 3,753
8
votes
Accepted
What are the special parahoric subgroups in unitary groups?
There's an explicit description of maximal compact subgroups of all unitary groups over local fields (not necessarily quasi-split) in section 3 of this paper:
Gan, Hanke, and Yu, "On an exact mass ...
- 36.1k
8
votes
The maximum order of torsion elements in ${\rm GL}_n(\mathbb{Z}_p)$ or ${\rm GL}_n(\mathbb{F}_p[[T]])$
Lemma Let $F/\mathbb Q_p$ be a $p$-adic field, with residue field $k_F$ of size $q$. Then the group of roots of unity in $F^\times$ is cyclic of order $p^e(q-1)$, where $e\ge0$ is the minimal integer ...
- 1,897
7
votes
Accepted
Is Howe's construction of tame supercuspidal representations independent of additive character?
The only real use of $\chi$ is to identify Moy–Prasad quotients with their character lattices; but notice that this is done twice, first to produce from $\theta$ an element $y$ (on p. 442), then to ...
- 11.4k
7
votes
Accepted
Finite dimensional irreducible representations of quasisplit p-adic groups
See Prop. 3.9 of http://math.stanford.edu/~conrad/JLseminar/Notes/L2.pdf for an optimal affirmative answer (no quasi-split condition needed: any connected reductive group over any non-archimedean ...
7
votes
Accepted
$GL_n(\Bbb Z_p)$ conjugacy classes in a $GL_n(\Bbb Q_p)$ conjugacy class
The answer is yes if $A \in \mathrm{GL}_n(\mathbb{Z}_p)$ is semisimple.
We may think of a matrix $A \in M_n(\mathbb{Z}_p)$ as a $\mathbb{Z}_p$-lattice of rank $n$ endowed with a $\mathbb{Z}_p$-linear ...
- 20.5k
7
votes
Is every compact simply-connected reductive p-adic group perfect?
At your request, I post my comment as an answer: the answer to the Related question is "no", i.e., all simple, anisotropic groups over a non-Archimedean local field are of type $\mathsf A$; ...
- 11.4k
7
votes
Accepted
Non-continuous group homomorphism from p-adic field to C*
Since $\mathbf C^\times$ is a divisible group, Zorn’s lemma tells us for every abelian group $G$ and subgroup $H$ that each group homomorphism $H\to \mathbf C^\times$ extends (somehow) to a group ...
- 49.6k
6
votes
Is it true that irreducible generic representations of $G_2(F)$ are self-dual?
$\DeclareMathOperator\PGSp{PGSp}\DeclareMathOperator\PGL{PGL}$Generic representations of a $p$-adic $G_2$ are indeed self dual. It suffices to prove this for super-cuspidal representations.
Observe ...
- 61
6
votes
Does local Langlands functoriality preserve genericity?
The general conjectural picture is the Gross-Prasad conjecture, found in Section 2 of Gross and Prasad's paper "On the decomposition of a representation of $SO_n$ when restricted to $SO_{n−1}$."
The ...
- 13.1k
6
votes
Accepted
Some question about cupidal automorphic representation and supercuspidal representation
It is not necessary that a cuspidal $\pi$ have a supercuspidal local component: for example, let $\Delta(\tau)$ be Ramanujan's delta function and let $\pi = \otimes' \pi_p$ denote the associated ...
- 271
6
votes
Is the representation of $GL_n(\mathcal{O})$ in functions on Grassmannian multiplicity free?
This may be redundant as a complete answer with references has already been posted.
Just in case this is still useful: I think multiplicity one can be proved by the usual Gefland trick: we need to ...
- 1,506
6
votes
Accepted
A question on linear algebra over non-Archimedean local field
$\def\FF{\mathbb{F}}\def\GL{\text{GL}}$This is basically the same thing YCor sketches in his comment: Let $R$ be the ring of integers of $\FF$. Let $K$ be the group of invertible matrices with entries ...
- 151k
6
votes
Non-trivial extension of representations have same central character
You probably should assume all representations are smooth.
You hope to show if $\pi_1$ and $\pi_2$ have two different central characters $\phi_1$ and $\phi_2$ then any extension $0\to \pi_1\to \pi\to \...
- 1,897
6
votes
Partition of unity for analytic manifolds over non-Archimedean local fields
It follows from Lemma 1 (part (2)) on page 7 in http://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/Bernst_Lecture_p-adic_repr.pdf
In the non-compact case, I think that you need to ...
- 2,581
5
votes
Accepted
How does Jacquet's "Generic Representations" classify tempered representations?
Do you assume that $\pi$ is irreducible ? You can find a proof in the book of David Renard, available on his webpage, p.343, VII.2.6.
- 128
5
votes
Dense subset in a product of $p$-adic groups?
Assuming that there are infinitely many primes $\ell$ of the form $\ell =(p^k-1)/(p-1)$, then no. For those primes $p^{n_i}$ takes $k$ distinct values so $\sum_{i=1}^M p^{n_i}$ takes at most $k^M$ ...
- 30.2k
5
votes
Accepted
Connections between representations of $\operatorname{SL}_n$ and $\operatorname{GL}_n$
The answer to your questions (with proofs) may be found in
C.J. Bushnell, P.C. Kutzko, The admissible dual of SL(N). I
Annales scientifiques de l'École Normale Supérieure, Série 4 : Volume 26 (1993)...
- 6,174
5
votes
Hecke algebra of GL(2,F)
A Hecke algebra, in the most common definition, is associated to a pair of groups $G$ and $K$, and is the convolution algebra of bi-$K$-invariant distributions on $G$.
As far as I know the left-...
- 137k
5
votes
Accepted
Hecke algebra of GL(2,F)
Let $G$ be a reductive $p$-adic group. First fixing a Haar measure on $G$, you can identify the algebra of distributions of $G$ with the ("big") Hecke algebra $H(G)$ of locally constant complex ...
- 6,174
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