# Questions tagged [p-adic-groups]

The p-adic-groups tag has no usage guidance.

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### P-adic representations corresponding to the same cuspidal pair

Let $G(F)$ be a reductive $p$-adic group. A result of Bernstein says that we can correspond each smooth irreducible representation to a “cuspidal pair” where it is embedded, and at most finitely many ...

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### 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, ...

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217 views

### Basic theorem on induction for representations of $p$-adic groups

I know a lot of places where the following is sparsely proved, but I remember there was some paper where I read it in basically the same form I write it, but unfortunately I can't remember where it ...

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76 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 ...

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### Volume of a double class of a parahoric subgroup

Let $F$ be a non-archimedean local field with residue field $F_q$. Let $G$ be the group of $F$-rational points of a connected reductive group defined and split over $F$. Fix a maximal split torus $T$ ...

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### $GL_n(\Bbb Z_p)$ conjugacy classes in a $GL_n(\Bbb Q_p)$ conjugacy class

It is easy to classify conjugacy classes in $GL_n(\mathbb Q_p)$ by linear algebra. How to classify $GL_n(\Bbb Z_p)$ conjugacy classes in a $GL_n(\Bbb Q_p)$ conjugacy class? For example, for general ...

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### Low dimensional representations of $SL_n(\mathbb{Z}/p^\ell \mathbb{Z})$

When $\ell = 1$ I know that the smallest non-trivial irreducible complex representations of $SL_n(\mathbb{Z}/p\mathbb{Z})$ has dimension $\frac{p^n - 1}{p-1} - 1$ (with maybe some exceptions for ...

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43 views

### A convergence condition on tempered representation

Assume $\pi$ is a tempered representation of $GL_n(\mathbb{Q}_p)$. $N_n$ is a maximal unipotent subgroup of $GL_n$, and $\xi$ is a non-degenerate character of $N_n(\mathbb{Q}_p)$. Let $\Pi$ be the ...

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88 views

### A Question on Plancherel decomposition of p-adic groups

Assume $G$ is a $p$-adic group. If $\pi$ is an irreducible quotient of $C_c^\infty(G)$, then there is a surjection $C_c^\infty(G)\twoheadrightarrow \operatorname{Hom}_G(C_c^\infty(G),\pi)^\star\otimes\...

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245 views

### Topological dimension of $p$-adic manifolds

What is the topological dimension of a (locally analytic) $p$-adic manifold over a non Archimedean field $K$?
Is the topological dimension of $K^n$, $n$?

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55 views

### Twisted Jacquet Module

All the groups considered will be Hausdorff, totally disconnected and locally compact groups. And all representations will be smooth representations over the complex numbers.
Let $G$ be a group and $(...

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98 views

### Group schemes and Hyperspecial maximal compact subgroups

Let $F$ be a number field. For each non-archimedean place $v$ let $O_v$ denote the ring of integers. Let $G$ be a connected linear algebraic group defined over $F$. Consider the set of sequences $(K_v)...

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199 views

### Integral structures via lattices

I am looking at the paper "p-adic Groups" by Bruhat (in the Boulder Proceedings, 1965). I have a question about one of the statements. Let $k$ be the quotient field of a complete discrete valuation ...

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187 views

### 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 ...

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39 views

### A normalized embedding $\mathbb C \rightarrow \mathfrak a_M^{\ast}$ via $\tilde{\alpha}$

Let $G$ be a connected, reductive group over a field $k$. Let $S$ be a maximal $k$-split torus of $G$ with Weyl group $W$, $\Delta$ a set of simple roots of $S$ in $G$, and $P = MN$ a maximal ...

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80 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) =...

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603 views

### Classification of p-adic representations

I am studying this paper https://arxiv.org/pdf/1412.0737.pdf . The classification in theorems 1-3 is extremely elegant, but from what I understand it is implied from this paper specifically for mod $p$...

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102 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 ...

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### Naive definition of parahoric subgroup

Background
Let $F$ be a $p$-adic local field, and let $G$ be a connected reductive group over $F$. Recall that there is a rich theory of compact open subgroups of $G(F)$ which is, essentially, ...

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### Open subgroups of finite index of p-adic semisimple groups

My set up is the following: I have an affine algebraic group $G$ over a $p$-adic field $F$, we assume that $G$ is semisimple and simply connected. I have an abstract subgroup $H\leq G(F)$ of the group ...

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### Finite dimensional irreducible representations of quasisplit p-adic groups

For split groups over a $p$-adic field, every irreducible smooth (complex) representation is either infinite-dimensional or one-dimensional. Is it true for quasisplit groups that split over an ...

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### 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, $GSpin_{n}(F)$...

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### Some question about cupidal automorphic representation and supercuspidal representation

Question 1: Let $G$ be a connected reductive group defined over a number field $K$, and $\pi$ is an irreducible cuspidal automorphic representation of $G(\mathbb{A}_K)$. Then by a theorem of Flath, we ...

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99 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 $p$-...

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99 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(...

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334 views

### Hecke algebra of GL(2,F)

I was studying about the Hecke algebra from Bernstein's notes on p-adic representation theory and various other sources. First a disclaimer: everything below is fairly new to me so please feel free to ...

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238 views

### Connections between representations of $\operatorname{SL}_n$ and $\operatorname{GL}_n$

Let $G = \operatorname{GL}_n(F)$ for a $p$-adic field $F$, and let $G_D = \operatorname{SL}_n(F)$. I am wondering if there is a connection between irreducible, admissible representations of $G$ and ...

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### Jacquet Module of an Essentially Square Integrable Representation

Let $F$ be a $p$-adic field. Let $G$ be a connected reductive group and $\rho$ an irreducible admissible representation of $G(F)$. Let $P$ be a parabolic subgroup of $G$ and suppose further that $\rho$...

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199 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 $$\...

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### Orbits of $GL(n, \mathcal{O})$ on pairs of linear subspaces over non-Archimedean local fields

Let $F$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers. Let $Gr_{i,n}$ denote the Grassmannian of $i$-dimensional linear subspaces in $F^n$.
Can one describe ...

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165 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 ...

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156 views

### What's the unipotent radical of the reduction of a bad orthogonal group?

Consider a DVR $A$ with fraction field $K$ and residue field $k$. Assume $2 \in A^\times$. Let $Q: A^n \rightarrow A$ be a quadratic form defined over $A$. Then one has the (naively defined) ...

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380 views

### A question on representation theory of p-adic groups

Let $V$ be a complex vector space of infinite dimension and let $(\pi,V)$ be a representation of the $p$-adic group $G:=GL_2(\mathbb{Q}_p)$. From representation theory, we know that if the ...

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### 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 ...

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### Distributions on the mirabolic subgroup which are left invariant to the unipotent radical

I'm trying to find a reference or a proof to the following statement used by Matringe, N., 2012. Cuspidal representations of GL (n, F) distinguished by a maximal Levi subgroup, with F a non-...

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### system of generators for non-connected topological groups

It is well-known that a connected topological group can be generated by any neighborhood of the identity. There are non-connected topological groups for which this is still true, such as $\mathbb{Q}$. ...

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### 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 ...

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### 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 $G$ ...

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**1**answer

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### Whittaker functions estimates proof

I am reading the proof of the estimates of Whittaker functions from Jacquet, Hervé, Ilja Iosifovitch Piatetski-Shapiro, and Joseph Shalika. "Automorphic forms on GL (3) I." Annals of Mathematics 109.1 ...

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233 views

### Is $SL_n(\mathbb{Z}_p)$ virtually torsion free?

If so, is there a way to conclude this from Malcev's theorem?
In general, what is known about virtually torsion freeness of non-finitely generated linear groups?

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### multiplicity one for restriction of representations from $GU$ to unitary group

Let $E/F$ be a quadratic field extension of p-adic fields. Let $V$ be a (skew-)Hermitian space and $U(V)$ be the unitary group. Let $GU(V)$ be the similitude unitary group. Given an irreducible ...

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### Compact subgroups of general linear groups over affinoid algebras

Let $k$ be a finite extension of $\mathbb{Q}_p$ and let $(A,A^0)$ be a $k$-affinoid algebra, where $A^0$ is the subring of power bounded elements. Suppose given a compact subgroup, L, of $GL_n(A)$, is ...

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### the moment set of unitary representation of lie groups, analogue in the p-adic case

Let $G$ be a real Lie group with Lie algebra $\mathfrak{g}$ and $\pi$ a unitary representation of $G$ on a separable Hilbert space $\mathcal{H}_{\pi}$. Note $\mathcal{H}_{\pi}^{\infty}$ the space of $...

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### Twisted Levi of a quasi-split group that is not quasi-split

Let $F$ be, say, a non-archimedean local field. Let $G$ be a connected reductive (can be assumed simply connected) quasi-split group $G$ over $F$. Let
$X\in\operatorname{Lie}G$ be semisimple and $G_X:...

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### Why are Trace characters regular functions on the Bernstein Variety?

Given a $p$-adic reductive group $G$ with Grothendieck group $R(G)$ and $f$ an element of the Hecke Algebra $H(G)$ we can consider the function $x: R(G) \to \mathbb{C}$ given by $\pi \mapsto trace \...

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### What's the point of a Whittaker model?

Let $G$ be a quasi-split connected reductive group over a $p$-adic field $F$. Let $B$ be a Borel subgroup which is defined over $F$, with $B = TU$, $T$ defined over $F$. The choice of $T$ and $B$ ...

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### 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. ...

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235 views

### How does Jacquet's “Generic Representations” classify tempered representations?

Let $L$ be a $p$-adic field $G = GL_n(L)$. Let $P$ be a standard parabolic subgroup with Levi decomposition $P = MU$, where $M \cong G_1\times \ldots \times G_r$, for $G_i \cong GL_{n_i}(L)$.
The ...

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236 views

### Reference Request: Compact subgroups of p-adic Reductive Groups

First, I'd like to understand what the compact open subgroups of $H(\mathbb{Q}_p)$ are, where $H$ is an inner form of $GL_n$ over $\mathbb{Q}_p$.
Second, I'd like to know where I can read about this ...

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### Do the contragredient and twisted Jacquet functors commute?

Suppose that $G$ is a connected reductive group over a non-archimedean local field $F$, and let $U\subset G$ be a unipotent subgroup with a nontrivial character $\psi: U\to \mathbb{C}^\times$. Also, ...