# Questions tagged [p-adic-groups]

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235
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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $U$...

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### 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 ...

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### 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\...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### $L$-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 $M$ ...

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### 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{...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### On a theorem of Bernstein-Zelevinsky regarding supercuspidal resentations

Let $G$ be a $p$-adic reductive group and $\pi$ an irreducible non-supercuspidal representation. Then there exist a parbaolic subgroup $P=MN$ and a supercuspidal representation of $M$ such that $\pi$ ...

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### Abelianization of unit quaternions over a p-adic field

Suppose $p$ is a prime, that $F$ is a finite extension of the field $\mathbb{Q}_p$, $D$ is the division quaternion algebra over $F$ and $\mathcal{O}_D$ is the valuation ring of $D$. What is the ...

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### Realization of the $p$-adic Steinberg representation as a subrepresentation

Let $G = \mathrm{GL}_n(F)$ where $F$ = non-archimedean local field. The Langlands Classification tells one that all irreducible admissible reps of $\mathrm{GL}_n(F)$ can be realized as (the unique ...

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### Global irreducible admissible representations analogue

Let ${A}_{\mathbb{Q}}$ denote the adele over the rational numbers. Then it is known that cuspidal modular forms of level $N$ correspond to some unitary automorphic representation of $\operatorname{GL}...

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### 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 ...

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### What are the good maximal compact subgroups in $p$-adic unitary groups?

Let $E/\mathbb Q_{p}$ be a quadratic extension and let $V$ be an $n$-dimensional $E$-hermitian space. Denote the hermitian form by $(\cdot,\cdot):V\times V \rightarrow E$. Let $G := \mathrm{U}(V)$ be ...

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### Asymptotic behavior of matrix coefficients

I'm reading Casselman's notes "Introduction to the theory of admissible representations of p-adic reductive groups". In chapter 4 "The asymptotic behavior of matrix coefficients", ...

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### Affine Bruhat-Tits building associated to $\mathrm{SU}_3(\mathbb Q_p)$

I saw the following results on affine Bruhat-Tits building associated to $\mathrm{SU}_3(\mathbb Q_p)$ without giving any references, where $\mathrm{SU}_3$ is the quasi-split inner form of special ...

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### On the Artin-Rees Lemma for non-commutative rings

Consider a commutative noetherian ring $A$ with an ideal $I\subset A$. The Artin-Rees lemma implies that for f.g. modules $N\subset M$, the $I$-adic topology on $N$ agrees with the subspace topology ...

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### What is the quotient group $D^*/{F^*(1+P_D)}$ for a quaternion division algebra $D$ over a local field $F$?

Let $F$ be the non-archimedean local field $\mathbb{Q}_p$ for some prime $p$ and $D$ be a quaternion division algebra over $F$. Let $\mathcal{O}_D$ and $\mathcal{P}_D$ denote the ring of integers of $...

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### Non-vanishing criterion of the Hom space of induced representation of p-adic groups?

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\GL{GL}
$Let $F$ be local field of characteristic zero and $(W,\langle,\rangle)$ be a $2n$-dimensional symplectic space over $...

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1
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### Can we compare $K$-spherical representations of $p$-adic groups for varying special maximal subgroups $K$?

In this question, I'm borrowing the notations from Minguez' paper on unramified representations of unitary groups. Let $F$ be a $p$-adic field and let $G$ be a connected reductive group over $F$. Let $...

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### Is it possible to detect when a maximal parahoric subgroup is (hyper)special from its finite reductive quotient?

Let $F$ be a $p$-adic field with residue field $k$ and let $G$ be a connected reductive group over $F$. Let us assume that $G$ is simply connected as an algebraic group over an algebraic closure of $F$...

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### Unramified constituent of Weil representation of $U(2)$

Let $E/F$ be a quadratic extension of local field of characteristic zero.
Let $\omega$ be the quadratic character of $F^{\times}$ associated to $E/F$ by local class field theory and $\gamma:E^{\times} ...

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### Part of some generic representation is also generic?

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\GL{GL}
$Let $F$ be local field of characteristic zero and $(W,\langle,\rangle)$ be a $2n$-dimensional symplectic space over $...

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### Confusion regarding special parahoric subgroups of the unitary group

This question is to clarify some confusion about special parahoric subgroups of a unitary group $G = \mathrm U_n(F)$ in an odd number of variables, with respect to an unramified quadratic extension $E/...

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### Jacquet module and Frobenius reciprocity

Let $F$ be a local field of characteristic zero and $G$ be a classical group over $F$.
Let $P=MN$ be a parabolic subgroup of $G$ and $\pi$ a irreducible smooth representation of $M$.
Let $\sigma$ be ...

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### Projective limit of copies of same group w.r.t. some fixed endomorphism

In our study of automorphism groups of transcendental field extensions, we have encountered the situation where we have a group $F$ together with an endomorphism $\alpha \colon F \to F$, resulting in ...

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### density of unipotent flows in algebraic groups

Let $\mathcal{G}$ be a reductive algebraic group over $\mathbb{Q}$ with a model $G$ over $\mathbb{Z}$ such that $G(\mathbb{R})$ is compact modulo centre. Let $T$ be a maximal torus of $\mathcal{G}$. ...

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### A p-adic analogue of a result due to Kirillov

Let $k$ be a non-Archimedean local field with char$(k)=0$.
Let $N$ be the group of $k-$rational points of a unipotent algebraic group defined over $k$.
It is known that $N$ is a locally compact and ...

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### Maximal compact subgroups of p-adic orthogonal groups

Let F be a non-archimedean local field, and G be split orthogonal groups of odd degrees $\geq$ 3.
In this setting, my question is;
Is there explicit descriptions of maximal compact subgroups of G?
I ...

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### Problem with affine root subgroups of $SU_3$ ramified, residue characteristic $p=2$

Let $L/K$ be ramified quadratic extension of local fields, and let characteristic of the residue field of $K$ be $2$. Let $\mathbb{G}=SU_3$, $G=\mathbb{G}(K)$. Let $\text{val}$ be a valuation on $K$ ...

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### Distributivity property for smooth parabolic induction

Let $G$ be a reductive group over a local field $k$ of characteristic zero with maximal split torus $T$, Weyl group $W$, Borel $ B$ and a parabolic subgroup $P$ such that $P\supset B \supset T$. ...

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### Representation of $GL_2(\mathcal{O})$ in space of functions on projective line

Let $F$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers. Let $GL_n(\mathcal{O})$ denote the group of invertible $n\times n$ matrices with entries in $\mathcal{O}$ with the ...

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### Bound of word width in compact $p$-adic analytic group

A theorem proved by A. Jaikin-Zapirain in On the verbal width of finitely generated pro-$p$ groups says that:
If $G$ is a compact $p$-adic analytic group, then every word $w$ of a free group $F$ has ...

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### Limit of the Casselman–Shalika Formula for the Spherical Whittaker Function

$\DeclareMathOperator{\GL}{GL}$Consider $G = \GL_{r+1}(F)$, where $F$ is a local non-archimedian field with the ring of integers $\mathcal{O}_F$ and the maximal ideal $\mathfrak{p}$, and let $q = \...

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### 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 ...

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### No irreducible subrepresentations

Let $p$ be a prime number. In the paper The image of Colmez's Montreal functor, on the page 12, the author points out an interesting property: The smooth mod $p$ representation $\mathrm{c-Ind}^{GL_2(\...

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### Volume of double cosets $BwB$

In Macdonald's book "Spherical functions on a group of $p$-adic type", Prop. (3.1.7), it is stated that if $w=w_1\dots w_r$ is a reduced word for $w\in W$ (the affine Weyl group), and if $q(...

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### Haar measure decomposition using orbital integrals

Let $G$ be a unimodular locally compact group, $N,A \le G$ be unimodular closed subgroups. Suppose that $A$ normalizes $N$. Let $N_0 \le N$ be a compact open subgroup. Suppose that a function $f : N \...