Questions tagged [p-adic-groups]

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Question on the geometric lemma in $p$-adic representation theory

$\DeclareMathOperator\GL{GL} \DeclareMathOperator\Sp{Sp} \DeclareMathOperator\Ind{Ind}\DeclareMathOperator\B{B} $ Let $F$ be a $p$-adic field and $\Sp_{2n}$ the symplectic group over a $2n$-...
Andrew's user avatar
  • 835
3 votes
0 answers
92 views

Representations of $\mathrm{GL}_n(\mathcal{O})$ in functions on Grassmannians

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Gr{Gr}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers. The natural representation of the group $\GL_n(\...
asv's user avatar
  • 20.9k
1 vote
0 answers
49 views

Simplest way to classify reducibility of principal series for $p$-adic $\mathrm{SL}_2$

Let $F$ be a $p$-adic field and $G_1=\mathrm{SL}_2(F)\subset \mathrm{GL}_2(F)=G$. For simplicity, we assume $p>2$. Denote by $|\cdot|$ the normalized absolute value on $F$. Here I shall focus on ...
youknowwho's user avatar
5 votes
0 answers
195 views

Subgroups of ${\rm GL}_{2}(\mathbb{Z}/2^{n} \mathbb{Z})$

Assume that $n \geq 3$. Is there a subgroup $H \leq {\rm GL}_{2}(\mathbb{Z}/2^{n} \mathbb{Z})$ whose order is a power of $2$ so that $\bullet$ $\det : H \to (\mathbb{Z}/2^{n} \mathbb{Z})^{\times}$ is ...
Jeremy Rouse's user avatar
  • 20.2k
2 votes
0 answers
57 views

Simple question on the genericity of induced representation

$\DeclareMathOperator\GL{GL} \DeclareMathOperator\Sp{Sp} \DeclareMathOperator\Ind{Ind}$ Let $F$ be a $p$-adic field and $\Sp(2n)$ symplectic group over 2n dimensional symplectic space over $F$. Let $B=...
Andrew's user avatar
  • 835
1 vote
0 answers
87 views

Question on the unramified representation

$\DeclareMathOperator\GL{GL}$Let $F$ be a $p$-adic field and $\chi$ be an unramified character of $\GL_1(F)$. Consider an induced representation $\pi$ of $\GL_2(F)$ induced from the character $\chi|\...
Andrew's user avatar
  • 835
4 votes
0 answers
184 views

Gelfand-Kazhdan criterion, exposition by Paul Garrett

Here is Paul Garrett's exposition on the Gelfand-Kazhdan Criterion. In page 4 of the exposition, he showed the following lemma. Lemma (Page 4): Let $B, t, S$ be as above and for $\alpha, \beta$ in $...
L-JS's user avatar
  • 151
2 votes
1 answer
155 views

Orbital integrals of $\operatorname{SL}_2$ and the fundamental lemma

$\DeclareMathOperator\SL{SL}$When I was checking some orbital integral computations of Sally-Shalika's The Plancherel Formula for $\SL_2$ over a Local Field, Proceedings of the National Academy of ...
youknowwho's user avatar
1 vote
1 answer
63 views

Compact subgroups of a linear group over non-Archimedean local field

$\DeclareMathOperator\GL{GL}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers. Is it true that any compact subgroup of $\GL_n(\mathbb{F})$ is conjugated to ...
asv's user avatar
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1 vote
0 answers
91 views

Jacquet module for characteristic p

Let $F$ be a finite extension of $\mathbb{Q}_p$ and let $G=\operatorname{GL}_2(F)$ (or $\operatorname{GL}_n$ or any reductive group). I consider smooth representations of $G$ in characteristic $p$. ...
re'em waxman's user avatar
4 votes
1 answer
119 views

Partition of unity for analytic manifolds over non-Archimedean local fields

I am looking for a reference to the following fact which, I hope, is correct. Let $X$ be a compact analytic manifold over a non-Archimedean local field. Let $X=\cup_\alpha U_\alpha$ be a finite open ...
asv's user avatar
  • 20.9k
5 votes
0 answers
116 views

Is $\mathbf{C}_p(X)$ self-dual?

Let $X$ be a set. Consider $\mathbf{Q}_p$ and $\mathbf{Z}_p$ as the $p$-adic numbers and $p$-adic integers, respectively. For any finite subset $F \subseteq X$, one can construct the topological ...
Luiz Felipe Garcia's user avatar
2 votes
1 answer
198 views

Question on the modulus character of classical p-adic group

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Sp{Sp}$It is well known for the formula of the computation of modulus character of general linear groups. For example, for the standard Borel subgroup $...
Andrew's user avatar
  • 835
2 votes
0 answers
82 views

Hereditarily just-infinite pro-$2$ groups

An infinite profinite group $G$ is called just-infinite if all non-trivial closed normal subgroups of $G$ have finite index. A profinite group is called hereditarily just-infinite if every open ...
stupid boy's user avatar
3 votes
1 answer
216 views

Pro-unipotent radical (in Bruhat-Tits) vs. unipotent radical (and reference request)

I'm a beginner in Bruhat-Tits theory, and the following phenomenon makes me puzzled. So let $F$ be a $p$-adic field, with $\mathfrak{o}\supset \mathfrak{p}$ its ring of integers and maximal ideal, and ...
youknowwho's user avatar
2 votes
0 answers
135 views

Prime-to-$p$ quotients of ${\rm PSL}_{2}(\mathbb{Z}_{p})$

Let $p$ be a prime and $\mathbb{Z}_p$ the ring of $p$-adic integers. Let ${\rm PSL}_{2}(\mathbb{Z}_{p})={\rm SL}_{2}(\mathbb{Z}_{p})/\{\pm 1\}$ be the projective special linear group over $\mathbb{Z}...
stupid boy's user avatar
2 votes
0 answers
158 views

$\mathrm{Ext}^i(\pi_1, \pi_2)\neq0$ implies same central character

If $\pi_1$ and $\pi_2$ are two smooth admissible representations of $\operatorname{GL}_2(\mathbb{Q}_p)$ over $\overline{\mathbb{F}}_p$ with central characters. I want to prove that if $\pi_1$ has ...
user avatar
1 vote
1 answer
177 views

Non-split extension of representations of $\mathrm{GL}_2$ and $\mathrm{Hom}$

Let $0\to V_1\to V\to V_1\to0$ be a sequence of representations of $\mathrm{GL}_2(\mathbb{Q}_p)$ over $\overline{\mathbb{F}}_p$, where $V_1$ is irreducible, smooth and admissible. Assume that this ...
user avatar
5 votes
2 answers
471 views

Non-trivial extension of representations have same central character

Let $\pi_1, \pi_2$ be two irreducible complex representations of $G=\mathrm{GL}_2(\mathbb{Q}_p)$ and assume that there exists a non-split extension $0\to\pi_1\to \pi\to\pi_2\to0$ of representations ...
user avatar
3 votes
0 answers
103 views

(Non)complete abelian groups in the “transfinite p-adic topology”

For an abelian group $A,$ a prime $p$ and an ordinal $\alpha,$ we recursively define $p^\alpha A$ as a subgroup of $A$ such that $p^0A=A,$ $$p^{\alpha+1}A=p(p^\alpha A) \hspace{5mm} \text{and} \...
Sergei Ivanov's user avatar
1 vote
0 answers
101 views

Does $\tilde{\mathrm E}_{6,3}^{(2)6}$ exist over a p-adic field?

Does a form of $\tilde{\mathrm E}_6^{(2)}$ with this Satake-Tits diagram exist over a p-adic field?
Daniel Sebald's user avatar
5 votes
1 answer
192 views

Two different local Langlands parameters for quadratic extension

Let $E/F$ be a quadratic extension of nonarchimedean local field, and let $G$ be a reductive group over $E$, and $G(E)$ the $E$-points of $G$. (For my question, one may just assume $G=\operatorname{GL}...
Windi's user avatar
  • 833
1 vote
0 answers
119 views

Explicit construction of $T$-orbits of generic characters of unitary groups

Let $F$ be a $p$-adic field. Let $E$ be a quadratic extension of $F$ and $G$ be a quasi-split unitary group $U(2n)$ or $U(2n+1)$ over with respect to $E/F$. Let $N_{E/F}$ be a norm map. Let $B=TU$ be ...
Andrew's user avatar
  • 835
1 vote
2 answers
217 views

How to assign the $L$- and $A$-parameters for the trivial representation

Let $G$ be a (split) reductive group over a $p$-adic field $F$, and $\mathbf{1}$ the trivial representation of $G(F)$. Under the (conjectural) Langlands correspondence, this should correspond to an $L$...
Windi's user avatar
  • 833
2 votes
1 answer
234 views

Non-continuous group homomorphism from p-adic field to C*

Let $F$ be a p-adic field. A character on $F^\times$ is defined as a continuous group homomorphism $F^\times\longrightarrow\mathbb{C}^\times$. But is there any way to construct a non-continuous group ...
Windi's user avatar
  • 833
1 vote
1 answer
103 views

Continuity of central character [closed]

Let $G$ be a $p$-adic reductive group and $Z\subseteq G$ the center. Let $\pi$ be an irreducible admissible representation of $G$. By Schur's lemma, it is easy to show that there is a group ...
Windi's user avatar
  • 833
2 votes
0 answers
193 views

Classification of generic representations of $\mathrm{GL}(n)$ over non-archimedean fields

Let $k$ be a local field and $n \in \mathbb{N}$. Question: I would like to know precisely, which irreducible (admissible) representations of $G := \textrm{GL}_n(k)$ are generic, i.e. which admit ...
Maty Mangoo's user avatar
4 votes
0 answers
111 views

Two definitions of intertwining operators and Harish-Chandra's Plancherel measure

I guess this question is a well-known fact to experts, but I didn't find any explicit explanation in the literature. So let $F$ be a $p$-adic field. (There're parallel definitions and results in the ...
youknowwho's user avatar
2 votes
0 answers
106 views

Understanding segments in Bernstein-Zelevinsky Classification

All reps shall be admissible in what follows. Let $k$ be a non-arch. field, $n = a\cdot b$ natural numbers and $P = M \cdot N \subset \mathrm{GL}_n(k)$ the standard parabolic subgroup with $$ M = \...
Maty Mangoo's user avatar
2 votes
2 answers
154 views

Structure theorem for Iwasawa modules over $p$-adic rings of integers

Let $K/\mathbb Q_p$ be a finite extension, and $\mathcal O_K$ the ring of integers of $K$. I am asking for a reference for a structure theorem of finitely generated modules over the completed group ...
Adelhart's user avatar
  • 207
2 votes
0 answers
105 views

Admissible representations of an $\ell$-group are a (neutral) Tannakian category?

Let $G$ be an $\ell$-group in the sense of Bernstein/Zelevinsky (sometimes also called td-group), i.e. $G$ is a Hausdorff locally compact totally disconnected topological group. Prominent examples ...
Maty Mangoo's user avatar
1 vote
0 answers
86 views

A closed subgroup of $p$-adic analytic group having same dimension is open?

Let $G$ be $p$-adic analytic pro-$p$ group and $H$ a closed subgroup of $G$. Suppose that $G$ and $H$ have the same dimension as $p$-adic analytic groups. Question: Is it true that $H$ is an open ...
trivialquestions's user avatar
2 votes
0 answers
110 views

A simple question on representation on $p$-adic groups

Let $G$ be a $p$-adic group (for example classical groups) and $\pi$ is a smooth finite length representation of $G$. Let $U$ be a unipotent subgroup of $G$ and $\psi$ is a non-trivial character of $U$...
Andrew's user avatar
  • 835
3 votes
0 answers
311 views

References on $p$-adic Langlands

As a student with a background in Algebraic Geometry (up to chapter 3 of Hartshorne) and basic Algebraic Number Theory, where should I begin learning about the $p$-adic Langlands program? What are the ...
Luiz Felipe Garcia's user avatar
9 votes
1 answer
343 views

The maximum order of torsion elements in ${\rm GL}_n(\mathbb{Z}_p)$ or ${\rm GL}_n(\mathbb{F}_p[[T]])$

This question is inspired by Upper bound on order of finite subgroups of GL_n(Z_p)?. It's showed that the supremum of orders of finite subgroups of ${\rm GL}_n(\mathbb{Z}_p)$ is finite and can be ...
Nobody's user avatar
  • 765
2 votes
0 answers
168 views

When is an infinite pro-$p$ group generated by its torsions

Let $p$ be a prime and $\mathcal{O}=\mathbf{Z}_p$ or $\mathbf{F}_p[[T]]$, i.e. the ring of $p$-adic integers or the ring of formal power series over a finite field $\mathbf{F}_p$ of order $p$. Let $G\...
stupid boy's user avatar
3 votes
0 answers
129 views

Just-infinite quotients of pro-$p$ groups that are linear over a complete Noetherian local ring

This question is a sequel to Quotients of pro-p groups linear over a complete Noetherian local ring. Recall that an infinite pro-$p$ group is called just-infinite if it has no proper, infinite ...
Nobody's user avatar
  • 765
0 votes
0 answers
79 views

An identity for lattices in vector spaces over non-Archimedean local field

Let $V$ be a finite dimensional vector space of a non-Archimedean local field $\mathbb{F}$. Let $\Lambda\subset V$ be a lattice, i.e. an open compact $\mathcal{O}$-submodule. Let $W_1,W_2\subset V$ be ...
asv's user avatar
  • 20.9k
3 votes
0 answers
238 views

Is insoluble $p$-adic analytic just-infinite pro-$p$ group torsion-free?

Recall that an infinite pro-$p$ group $G$ is called just-infinite if all non-trivial closed normal subgroup of $G$ have finite index. Question: Let $G$ be an insoluble $p$-adic analytic just-infinite ...
stupid boy's user avatar
1 vote
1 answer
88 views

Quotients of pro-$p$ groups linear over a complete Noetherian local ring

Let $R$ be complete Noetherian local ring with finite residue field $\mathbb{F}$ of characteristic $ p $. We say that a pro-$p$ group $G$ is linear over $R$ if it is isomorphic to a closed subgroup of ...
Nobody's user avatar
  • 765
4 votes
0 answers
161 views

Reference for Iwahori-Hecke algebras

I recently came across the notion of an Iwahori-Hecke algebra. I would like to learn the basics about this type of algebras (mainly to get an intuition about them, as they seem to be related to some ...
Fernando Peña Vázquez's user avatar
3 votes
1 answer
413 views

$L$-parameters and parabolic induction

I apologize in advance if the answer to this question is well-known to experts. So let $F$ be a $p$-adic field, and $G$ a reductive group over $F$. Let $P$ be an $F$-parabolic subgroup of $G$ and $M$ ...
youknowwho's user avatar
2 votes
1 answer
194 views

Finite quotients of $p$-adic congruence subgroups of $\operatorname{SL}_2$

$\DeclareMathOperator\SL{SL}$Let $p$ be an odd prime and let $ \SL^1_2(\mathbb{Z}_p)$ denote the kernel of the natrual surjective morphism $\SL_2(\mathbb{Z}_p)\rightarrow \SL_2(\mathbb{Z}_p/p\mathbb{...
Nobody's user avatar
  • 765
3 votes
1 answer
161 views

Does $\mathrm{Ext}^i_G(\pi,\pi')$ vanish if $\pi$ and $\pi'$ are smooth irreducible representations of $G$ with different central characters?

Let $G$ be a $p$-adic group, ie. the group of $F$-points of some connected reductive group over $F$, where $F$ is a $p$-adic field. We consider complex smooth representations of $G$. Any irreducible ...
Suzet's user avatar
  • 687
5 votes
1 answer
329 views

A question on linear algebra over non-Archimedean local field

Let $\mathbb{F}$ be a non-Archimedean local field. Let $\{T_a\}_{a=1}^\infty$ be a sequence of linear operators $\mathbb{F}^n\to\mathbb{F}^n$ of rank $n$. After a choice of subsequence, is it ...
asv's user avatar
  • 20.9k
5 votes
1 answer
587 views

Understand the $p$-adic local Langlands correspondence with examples

Let $\rho:G_{\mathbb{Q}}\rightarrow \mathbf{Gl}_{n}(\mathbb{Q}_{p})$. I would like to understand in depth why the local Langlands correspondence for $\rho_{|\mathbb{Q}_{p}}$ must consider $p$-adic ...
Marsault Chabat's user avatar
5 votes
1 answer
174 views

Iwahori-spherical tempered representations with fixed vectors for larger parahoric subgroups

Let $F$ be a $p$-adic field and $G$ be split over $F$ plus all other usual adjectives. Let $\pi$ be an admissible tempered representation with Iwahori-fixed vectors. Let $P\supset I$ be some parahoric ...
Stefan  Dawydiak's user avatar
2 votes
1 answer
128 views

The dimension of a torsion-free $p$-adic analytic group generated by two generators

$\DeclareMathOperator\GL{GL}$Let $G$ be a $2$-generator pro-$p$-group of finite rank, i.e. it is isomorphic to a closed subgroup of $\GL_d(\mathbb{Z}_p)$ for some integer $d$. Assume that $G$ is ...
stupid boy's user avatar
7 votes
1 answer
247 views

When a pro-$p$ group of finite rank can be embedded into the first congruence subgroup of ${\rm GL}_{N}(\mathbb{Z}_{p})$

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$Let $p$ be a odd prime. We say that a pro-$p$ group has finite rank if it is isomorphic to a closed subgroup of $\GL_d(\mathbb{Z}_p)$ for some ...
stupid boy's user avatar
2 votes
1 answer
147 views

How do you construct elements in $\operatorname{Ind}_P^G\pi$?

Let $G$ be a $p$-adic reductive group, and $P=MN\subseteq G$ a parabolic subgroup. How do you know that the space of the induced representation $\operatorname{Ind}_P^G\pi$ is non-zero? Namey, how do ...
Windi's user avatar
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