# Questions tagged [p-adic-groups]

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208
questions

6
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1
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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 ...

8
votes

2
answers

157
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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 $...

2
votes

1
answer

106
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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 $...

1
vote

1
answer

116
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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 $...

3
votes

1
answer

113
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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 $F$...

2
votes

0
answers

61
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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} ...

1
vote

1
answer

109
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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 $...

1
vote

0
answers

109
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### What can the $p$-component of an automorphic representation be once we have fixed its $\infty$-component?

My question concerns Fargues' 2004 Astérique paper "Cohomologie des espaces de modules de groupes $p$-divisibles", available here. I will use the same notations below. Before stating it, I ...

1
vote

0
answers

92
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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/...

2
votes

1
answer

220
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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 ...

3
votes

0
answers

81
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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 ...

3
votes

0
answers

117
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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 $\mathcal{G}$. ...

1
vote

0
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133
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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 ...

1
vote

0
answers

67
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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 ...

5
votes

0
answers

93
views

### 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 $K$ ...

1
vote

0
answers

42
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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 $P\supset B \supset T$. ...

2
votes

1
answer

161
views

### 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 ...

4
votes

0
answers

86
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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 ...

7
votes

0
answers

152
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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 = \...

7
votes

0
answers

141
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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 ...

4
votes

0
answers

88
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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(\...

2
votes

0
answers

274
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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(...

1
vote

0
answers

81
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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 \...

3
votes

1
answer

152
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### Globalising tori and weak approximation

Let $G$ be a semi-simple and simply connected reductive group over $\mathbb{Q}$ and let $T \subset G_{\mathbb{Q}_p}$ be a maximal torus. A classical result of Harder tells us that we can find a ...

0
votes

1
answer

83
views

### The stabiliser group of an isotropic quadratic form over $\mathbb{Q}_p$ is non-compact?

Let $\mathbb{Q}_p$ denote the $p$-adic integers. Let $V$ be a $\mathbb{Q}_p$-vector space and $Q : V \rightarrow \mathbb{Q}_p$ be a non-degenerate integral quadratic form. We say that the pair $(Q,V)$ ...

2
votes

0
answers

118
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### Galois representation absolutely irreducible after restricting to open subgroup of finite index

Let $E$ and $F$ be finite extensions of $\mathbb{Q}_p$. Let $\phi:\mathrm{Gal}(\overline{E}/E)\to GL_n(F)$ be an absolutely irreducible continuous representation. Assume that the restriction of $\phi$ ...

3
votes

0
answers

189
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### Tits Reductive Groups over Local Fields Example 1.15 (Quasi-split special unitary groups in odd dimension)

I hope this question about Tits's paper "Reductive groups over local fields" in Algebraic groups and discontinuous subgroups ends up having an easy answer, but I'm a little stuck on the ...

5
votes

0
answers

135
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### When is $G(\mathbb{Z}_p)$ topologically generated by maximal tori?

Let $G/\mathbb{Z}_p$ be a connected reductive group, and let $T_1, \cdots, T_n \subset G_{\mathbb{Q}_p}$ be maximal tori. Suppose that we have precisely one maximal torus in each $G(\mathbb{Q}_p)$-...

4
votes

0
answers

62
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### Compactly induced representations and the Whittaker model

Let $G$ be the group $GL_n(F)$, where $F$ is a non-archimedean locally compact field. Let us fix a character $\psi$ of $(F,+)$ which is trivial on $\mathcal{P}_{F}$ and non-trivial on $\mathcal{O}_{F}$...

4
votes

0
answers

180
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### Frobenius reciprocity law in the $p$-adic group represenation

Let $G$ be a $p$-adic classical group and let $P_0$ be a minimal parabolic subgroup of $G$. Let $P=MN$ be a
standard parabolic subgroup containing $P_0$. Let $\text{Ind}$ and $ |_M$ be the normalized ...

2
votes

0
answers

162
views

### Question on external tensor product

Let $G_1,G_2$ be $p$-adic groups. Let $\rho_1,\pi_1$ be smooth representations of $G_1$ and $\rho_2,\pi_2$ be smooth representations of $G_2$. Assume that $\pi_2$ is admissible.
I am wondering that if ...

1
vote

0
answers

54
views

### Supercuspidals of GL(n) and the induced representation $Ind_{T}^{G}1_{T}$

Let $G$ be the group $GL(n,F)$, where $F$ is a non-archimedean local field. Is it true that for any smooth irreducible supercuspidal representation (with trivial central character) $(\pi,V)$ of $G$ we ...

2
votes

0
answers

90
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### Parabolic inductions for p-adic reductive groups

So I wish to ask for articles/comments surveying conjectures and theorems about parabolic induction for p-adic (non-archimedean case) reductive groups, and how local Langlands behaves under such. That ...

1
vote

0
answers

103
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### Question on induction of unramified representations

$\def\anonabs{\lvert\cdot\rvert}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Ind{Ind}\DeclareMathOperator\SO{SO}
$Let $F$ be a $p$-adic local field of characteristic zero.
Let $\chi$ be an ...

6
votes

0
answers

139
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### $SL_2(\mathbb{Z}_p)$ extension of a local field

Let $G$ be an arbitrary open group of $SL_2(\mathbb{Z}_p)$ and $K$ be a finite extension of $\mathbb{Q}_p$. Can we construct a Galois extension field $E$ of $K$ such that $\text{Gal}(E/K)\cong G$? ...

3
votes

0
answers

131
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### Question on the proper sub-representation of induced representation

$\DeclareMathOperator\Ind{Ind}$Let $G$ be a reductive group over a $p$-adic local field $F$, and $P=MN$ a parabolic subgroup.
Let $\sigma$ be an irreducible representation of $M(F)$ and consider its ...

3
votes

1
answer

90
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### Are polynomial sections for unramified principal series generated by the spherical vector?

Let $F$ be a non-archimedean local field, $G = \operatorname{GL}_n\left(F\right)$, $B$ the standard Borel subgroup, $K = \operatorname{GL}_n\left( \mathcal{O}_F \right)$. We have the Iwasawa ...

4
votes

0
answers

57
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### The weak restriction of the Jacquet module

Let $P= MN$ be a parabolic subgroup of a reductive p-adic group $G$, and $(\pi, V)$ is an irreducible, admissible representation of $G$. The Jacquet module is the representation $(\pi_N, V_N)$, where $...

7
votes

0
answers

273
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### intuition for lattices in p-adic (or other non-Archimedean) vector spaces?

I could use some help to jumpstart my intuition for lattices in vector spaces over non-Archimedean fields, like $\mathbb{Q}_p$ and $\mathbb{F}_q((t))$.
I have some intuition for $\mathbb{Z}$-lattices ...

2
votes

0
answers

61
views

### $p$-adic Harish-Chandra character of a stable virtual character

Let $F$ be a $p$-adic field and let $G$ be a reductive group over $F$. Associated to an irreducible admissible representation of $\pi$ of $G(F)$, we have a distribution character $\Theta_{\pi}$ ...

3
votes

0
answers

76
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### p-adic analogue of classification of irreducible Riemannian symmetric spaces?

For Riemannian symmetric spaces, we have a nice result that the simply-connected ones are products of irreducible symmetric spaces, of which we have a list of ten infinite families and some ...

1
vote

0
answers

131
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### compact $p$-adic Lie group can be embedded into $O_K^n$ or $\text{GL}_n(K)$?

Let $K$ be a local field of charecteristic $0$ and $G$ be a compact $p$-adic Lie group of dimension $n$, then can $G$ be embedded into $O_K^n$ or $\text{GL}_n(K)$ as a closed subgroup? This is a dual ...

3
votes

0
answers

144
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### Orbit representatives for the action of the maximal compact subgroup

Let $F$ be a non-Archimedean local field and $O$ be the ring of integers in $F$. Take $G=GL(2,F)$ and $K=GL(2,O)$. Consider the action of $K$ on $G$ by conjugation. Is it possible to explicitly write ...

9
votes

2
answers

307
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### Is the representation of $GL_n(\mathcal{O})$ in functions on Grassmannian multiplicity free?

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

1
vote

0
answers

186
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### J. Tate's article on $p$-divisible groups

I have a question about a conclusion that J. Tate arrived at in his paper $p$-Divisible Groups (in: Springer T.A. (eds) Proceedings of a Conference on Local Fields. Springer, Berlin, Heidelberg (1967) ...

3
votes

0
answers

174
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### 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 ...

4
votes

1
answer

235
views

### Unitary representations of lattices

Let $G$ be a simple linear group over a non-archimedean local field $F$.
Assume that the split-rank over $F$ is at least 2.
Let $\Gamma$ be a lattice in $G$. Then $\Gamma$ is a finitely generated ...

2
votes

0
answers

110
views

### Classical Hecke operators and Hecke algebra of type $A_1$

What's the relation between the classical Hecke operators (as defined in J. P. Serre's A course in arithmetic chapter 6) and the Hecke algebra of type $A_1$, i.e. the algebra generated by the vertices ...

3
votes

1
answer

268
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### branching laws for $p$-adic representations of reductive groups

There are many papers studying branching laws of irreducible admissible complex representations of classical groups over local fields, are there some analogues for $p$-adic representations?
For ...

3
votes

1
answer

100
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### Image of a finite dimensional complex representations of $GL_n(\mathcal{O})$

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