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Questions tagged [p-adic-groups]

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6
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1answer
96 views

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 ...
5
votes
0answers
106 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, ...
5
votes
2answers
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 ...
4
votes
0answers
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 ...
2
votes
0answers
53 views

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 $T$ ...
20
votes
2answers
437 views

$GL_n(\Bbb Z_p)$ 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 ...
10
votes
1answer
176 views

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 ...
1
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0answers
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 ...
3
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0answers
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\...
9
votes
1answer
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 $K^n$, $n$?
2
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0answers
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 $(...
1
vote
0answers
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)...
5
votes
1answer
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 ...
11
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0answers
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 ...
1
vote
0answers
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 ...
5
votes
0answers
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) =...
1
vote
1answer
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 $p$...
2
votes
0answers
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 ...
5
votes
2answers
282 views

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, ...
1
vote
1answer
116 views

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 ...
4
votes
2answers
267 views

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 ...
4
votes
0answers
152 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, $GSpin_{n}(F)$...
3
votes
1answer
194 views

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 ...
5
votes
0answers
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 $p$-...
4
votes
0answers
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(...
6
votes
2answers
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 ...
8
votes
1answer
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 ...
4
votes
1answer
96 views

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 $\rho$...
6
votes
0answers
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 $$\...
10
votes
2answers
214 views

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 ...
4
votes
0answers
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 ...
2
votes
0answers
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) ...
10
votes
1answer
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 ...
6
votes
0answers
99 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 ...
3
votes
1answer
71 views

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-...
0
votes
1answer
48 views

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 $\mathbb{Q}$. ...
10
votes
0answers
217 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 ...
5
votes
0answers
121 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 $G$ ...
2
votes
1answer
146 views

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 ...
2
votes
1answer
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?
6
votes
0answers
80 views

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 ...
3
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0answers
183 views

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 ...
3
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0answers
72 views

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 $...
4
votes
1answer
158 views

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:...
2
votes
1answer
109 views

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 \...
13
votes
2answers
1k views

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 $B$ ...
8
votes
0answers
272 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. ...
8
votes
1answer
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 ...
6
votes
1answer
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 ...
2
votes
0answers
124 views

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, ...