Questions tagged [p-adic-groups]

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Bernstein-Zelevinsky classification: viewing a representation as a subrepresentation or a quotient

$\DeclareMathOperator{\GL}{GL}$ $\DeclareMathOperator{\Ind}{Ind}$I have a question on the details of the Bernstein-Zelevinsky classification. This classification allows us to obtain irreducible ...
D_S's user avatar
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10 votes
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427 views

Haar measure on $PGL(2,\mathbb{Q}_p)$, the local Jacquet-Langlands correspondence, and Ihara's theorem

Goal. I would like to calculate the product of the formal dimension of a discrete series representation of $GL(2,\mathbb{Q}_p)$ with trivial central character (so, an irreducible unitary ...
L.C. Ruth's user avatar
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9 votes
0 answers
234 views

On the status of some conjectures mentioned/used in Harish Chandra's 1970 lecture notes

In van Dijk's notes of Harish Chandra's lectures on harmonic analysis, several conjectures are mentioned throughout, such as in Part 1, section 4 of van Dijk's notes Conjecture I : Let $\omega$ be ...
edgarlorp's user avatar
  • 113
9 votes
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298 views

Colimit of continuous cohomology over subgroups

Suppose $G$ is a profinite group, in fact in the applications I'm interested in it would be a $p$-adic analytic group similar to $GL_{n}(\mathbb{Z}_{p})$. Say $M$ is a profinite $G$-representation, ...
Piotr Pstrągowski's user avatar
9 votes
0 answers
391 views

The proof of Kazhdan's density theorem (And does it hold over positive characteristic?)

When proving identities about traces of functions on representations of $p$-adic groups, Kazhdan's density theorem indicates one only has to check equalities of traces on tempered representations. ...
Charles Denis's user avatar
7 votes
0 answers
355 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
7 votes
0 answers
201 views

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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7 votes
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425 views

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 ...
PrimeRibeyeDeal's user avatar
7 votes
0 answers
481 views

Intersections of anisotropic tori with split Levi subgroups

Let G be a connected reductive group defined and split over a finite field k with Frobenius morphism F. Let T be an F-stable minisotropic maximal torus. Let P be an F-stable proper parabolic ...
Monica's user avatar
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6 votes
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148 views

$SL_2(\mathbb{Z}_p)$ 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 $\text{Gal}(E/K)\cong G$? ...
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6 votes
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217 views

When is an irreducible unramified principal series representation $K$-spherical?

Let $G = \operatorname{GL}_n(\mathbb Q_p)$, $T$ the usual maximal torus of $G$, and $K = \operatorname{GL}_n(\mathbb Z_p)$. Let $\chi$ be an unramified character of $T$, with $\chi(t_1, ... , t_n) =...
D_S's user avatar
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6 votes
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206 views

Dimension of space of K-fixed vectors

If $G$ is an unramified group over an $p$-adic field $F$, the Satake isomorphism identifies the spherical Hecke algebra with respect to a special maximal compact subgroup $K$. In particular, (1) $H(G(...
Dylon Chow's user avatar
6 votes
0 answers
317 views

Bi-Whittaker functions and local Langlands compatibility

I'm trying to figure out the arithmetic analogue of a key conjecture in the geometric local Langlands correspondence. Briefly, one expects for $K=\mathbb{C}((t))$ an equivalence of dg categories $$\...
dhy's user avatar
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6 votes
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117 views

A question of Jacquet-Rallis on self-contragredient representation

In their paper "Uniqueness of linear periods", Compositio Mathematica, $\textbf{102}$ (1996), 65-123, Jacquet and Rallis asked the following question in the middle of page 67. Let $F$ be a $p$-adic ...
Q. Zhang's user avatar
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244 views

Local character expansion for discrete series representations of $GL_n(F)$

I'm interested about what, if anything, is known about the local character expansion of discrete-series representations of $GL_n(F)$, where $F$ is a $p$-adic field. First, some notation: let $G$ be a ...
John Binder's user avatar
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6 votes
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192 views

Is there a Noetherian profinite group of infinite rank?

Is there a profinite group $G$ such that any closed subgroup $H \leq G$ is finitely generated, but there is no $n \in \mathbb{N}$ such that every closed subgroup of $G$ can be generated by at most $n$ ...
Pablo's user avatar
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6 votes
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Newton Method in $p$-adic case

The Newton Method over $\mathbb{R}$ has the property that the precision is doubled (under some continuous differentiable assumption) in each iteration. For the ring $\mathbb{Z}_p$ of $p$-adic integers,...
user565739's user avatar
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5 votes
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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
5 votes
0 answers
112 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 $K$ ...
pbarron's user avatar
  • 71
5 votes
0 answers
163 views

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 $G(\mathbb{Q}_p)$-...
Pol van Hoften's user avatar
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0 answers
199 views

Does $G$ act 2-transitively on its Bruhat-Tits building?

Let $k$ be a finite extension of $\mathbb{Q}_p$ and let $G$ be a semisimple Lie group over $k$. We consider the action of $G$ on its Bruhat-Tits building $X$. Question: If $x,y,x',y'$ are vertices, ...
nikola karabatic's user avatar
5 votes
0 answers
142 views

Characters on $PGL(2)$

I am interested in computations with characters on $PGL(2, F)$ and not in $GL(2, F)$, and have some issues concerning definitions. The notion of conductor is standard for characters $\chi$ of a $p$-...
Desiderius Severus's user avatar
5 votes
0 answers
208 views

If an irreducible admissible representation is generic, so is its contragredient?

Let $G$ be a $p$-adic reductive group, and $\pi$ be an irreducible admissible representation of $G$ that is generic, do we know that the contragredient representation of $\pi$ is also generic? If $G$ ...
Q. Zhang's user avatar
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0 answers
556 views

How do you understand the Moy-Prasad filtration of G_2?

Starting on page 44 of this paper of Reeder and Yu, the authors describe the first graded piece of the Moy-Prasad filtration on $G_2$ at a certain point (in this case it's $GL_2$ of the residue field),...
user84144's user avatar
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123 views

double coset decomposition of $U_\beta\setminus GL_n(E)/ U_\beta$ over p-adic field

Let $E/F$ be a quadratic extension of local fields. Let $\beta$ be a non degenerate Hermitian matrix of rank $n$, and let $U_\beta$ be the unitary group defined by $\beta$. Do we have an explicit ...
user64433's user avatar
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221 views

Decomposition of k-split tori of p-adic reductive groups

Let $G$ be a reductive group over a $p$-adic field $k$, $S \subset G$ a maximal $k$-split torus, $\Phi(G,S)$ the relative root system and $\Delta$ a basis of $\Phi$. There is a group homomorphism : $$...
Arkandias's user avatar
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4 votes
0 answers
98 views

Hecke algebra $\mathcal{H}(K_1\backslash \mathrm{GL}_n(\mathbb{F})/K_1)$

$\DeclareMathOperator\GL{GL}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers, and let $\frak{m}$ be its maximal ideal Let $\GL_n(\mathcal{O})$ be the group ...
asv's user avatar
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4 votes
0 answers
186 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
4 votes
0 answers
112 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
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
4 votes
0 answers
191 views

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 ...
Suzet's user avatar
  • 687
4 votes
0 answers
102 views

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 ...
Lucas's user avatar
  • 289
4 votes
0 answers
120 views

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(\...
A413's user avatar
  • 433
4 votes
0 answers
93 views

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 $\mathcal{O}_{F}$...
Rajkarov's user avatar
  • 933
4 votes
0 answers
275 views

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 ...
Monty's user avatar
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4 votes
0 answers
66 views

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 $...
IMED's user avatar
  • 41
4 votes
0 answers
121 views

Finite dimensional irreps of $p$-adic groups

What are some examples of finite dimensional irreducible complex representations of $SL_2(\mathbb{Q}_p)$? One knows such a representations cannot be smooth, so probably the examples will be ...
Spinoza's user avatar
  • 81
4 votes
0 answers
134 views

Growth of the number of fixed points of a $p$-adic group under natural filtrations

Let $G$ be a $p$-adic reductive group, so by definition as a locally profinite group it's the group of $\mathbb Q_p$ points of a connective reductive group over $\mathbb Q_p$, $K$ be a parahoric ...
sawdada's user avatar
  • 6,148
4 votes
0 answers
417 views

Index of the congruence subgroups of $PGL_2(\mathbb{Z}_p)$

Let $\Gamma_n$ be the $n$-th congruence subgroup of $GL(2,\mathbb{Z}_p)$. So $\Gamma_n$ consists of matrices in $GL(2,\mathbb{Z}_p)$ which are congruent to the identity matrix modulo $p^n$. Let $Z(\...
MathStudent's user avatar
4 votes
0 answers
272 views

How to determine the unramified character corresponding to an unramified Langlands parameter?

Let $F$ be a p-adic field with ring of integers $\mathcal{O}$. Let $\textbf{G}$ be a connected split reductive algebraic group over $F$. For simplicity, we assume that $\textbf{G}$ is a Chevalley ...
Q. Zhang's user avatar
  • 960
4 votes
0 answers
445 views

Iwahori-Hecke algebra of $GL_2$

I had make an old post about this, but since seeing it now shows that I had no idea how to explain my question (because I didn't understand it) I'll try to do it normally here. So I am studying this ...
Kostas Psaromiligkos's user avatar
4 votes
0 answers
301 views

Restriction of a representation of a reductive group to the derived subgroup

Let $G$ be a reductive p-adic or Lie group and let $G'$ be its derived subgroup. The main examples I am interested at are $GL_n(F)$ and $SL_n(F)$, similitude groups and classical groups, $GSpin_{n}(F)$...
Matht111101111's user avatar
4 votes
0 answers
211 views

A question of integral on $p$-adic fields $\mathbb{Q_p}$

We assume that $(\pi,V)$ is an admissible, irreducible and infinite-dimensional representation of $GL_2(\mathbb{Q_p})$. In the proof of existence and uniqueness of Kirillov model, the key step is that ...
JACK's user avatar
  • 421
4 votes
0 answers
478 views

Parahorics and their normalizers

Let $G$ be a reductive group over a local non-arch field $F$. For convenience let's assume $G$ has anisotropic center (or even that $G$ is semisimple if preferred). Let $x$ be any point in the ...
Cheng-Chiang Tsai's user avatar
4 votes
0 answers
167 views

parametrization of irreducible finite dimensional representation of Weil group

Let $F$ be a p-adic field, with p a prime denoting the residue field characteristic. Let $\mathcal{W}_F$ be the Weil group. In the local Langlands correspondence for $GL(n,F)$, it is important to know ...
user1832's user avatar
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3 votes
0 answers
45 views

Question on the genericity of unramified representation

Let $F$ be a $p$-adic local field and $W$ be a 2n-dimensional symplectic space over $F$. Let $G_n$ be the isometry group of $W$ and $B_n$ be the Borel subgroup of $G_n$. Then the maximal torus $T_n$ ...
Andrew's user avatar
  • 865
3 votes
0 answers
45 views

One parameter subgroups of reductive algebraic groups

If I have a reductive algebraic group $G$ defined over a non-archimedean local field $F$. We can define a one-parameter subgroup to be a group homomorphism from $G_{m}$ to $G$. I was wondering, if I ...
Ekta's user avatar
  • 63
3 votes
0 answers
104 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
3 votes
0 answers
317 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
3 votes
0 answers
130 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
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