# Tagged Questions

The p-adic-groups tag has no wiki summary.

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142 views

### When are toral orbits in buildings the difference of fixed-sets?

Let $L$ be a $p$-adic field, let $G$ be a reductive group over $L$ (I'm even okay assuming semisimplicity for now). Let $T$ be a maximal torus of $G$. Let $B$ be the building for $G(L)$. (Edit 1: ...

**1**

vote

**2**answers

346 views

+50

### orbits of linear algebraic group $G({\Bbb Q}_p)$ acting on subgroups of ${\Bbb Q}_p^n$

Let $G\subseteq GL(n)$ be a linear algebraic group, and let $G({\Bbb Q}_p)\subseteq GL(V)$ act on a ${\Bbb Q}_p$-vector space V of finite dimension.
Consider the action of $G$ on abelian subgroups ...

**3**

votes

**0**answers

76 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 $n$ ...

**2**

votes

**2**answers

101 views

### admissible characters for $PGL_{2}(F)$

What are the irreducible admissible representations of $PGL_{2}(F)$ for $F$ a local nonarchimedean field and do we have formulas for their characters?

**4**

votes

**3**answers

308 views

### What is the difference between p-adic Lie groups and linear algebraic groups over p-adic fields?

I thought they were the same, just different names. Let me make question more precise:
Let $G$ be any linear algebraic group over a p-adic field $\mathbb{Q}_p$, is $G$ a p-adic Lie group w.r.t. the ...

**1**

vote

**0**answers

60 views

### Are A and A^T unitarily equivalent over a p-adic field?

Let $E/F$ be a quadratic extension of p-adic field. Let $U=\{u\in GL_2(E): uu^*=1\}$ be the unitary group of rank 2.
My question is: given a matrix $A\in GL_2(E)$ can we find $u_1,u_2\in U$ such ...

**2**

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**0**answers

55 views

### Mellin transform of Plancherel measure

Let $G$ be a reductive p-adic group with a chosen Haar measure $dg$. The Plancherel measure is the measure $\mu$ on the set of (tempered) irreducible representations of $G$ such that for any locally ...

**3**

votes

**1**answer

181 views

### Abelian varieties as analytic manifolds

Assume we have an Abelian varieties over the p-adic numbers, namely $
k=\mathbb{Q}_p$. Then the question is whether $A(k)$, the rational points over $k$, will form a p-adic analytic manifold.
I am ...

**4**

votes

**0**answers

60 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 ...

**3**

votes

**1**answer

225 views

### Abelian varieties over $p$-adic fields

Theorem : Let $A$ be an abelian varieties of dimension $d$ over a field $k$, non-archimedian valued complete, i.e. $\mathbb{Q}_p$, then $A(k)$ contains a subgroup of finite index analytically ...

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**0**answers

78 views

### Buildings associated to generalized $BN$ pairs

I'll begin by asking a general question, and then specializing to the situation I really care about.
Let $G$ be a group and let $(B, N)$ be a $BN$-pair in $G$ (see, for instance, page 39 of Tits' ...

**2**

votes

**1**answer

123 views

### Why are compactly induced representations projective in the category of admissible representations?

I am reading part of Dipendra Prasad's paper found here: http://arxiv.org/pdf/1306.2729v1.pdf.
In it (in the middle of page 8) he writes that compactly induced representations are projective. Why is ...

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vote

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51 views

### Exhausting a free pro-p group

Recall that for a profinite group $G$ we define the subgroup rank to be $$\sup \{d(H): H \leq_c G\}$$ where $d(H)$ stands for the minimal cardinality of a set of topological generators of $H$.
Let ...

**3**

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**0**answers

91 views

### Newvectors in tensor product representations

Let $\pi_p$ and $\pi_p^\prime$ two smooth admissible irreducible complex representations of ${\rm GL}_2(F)$ where $F$ is a non archimedean local field of residual characteristic $p$ of central ...

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**0**answers

51 views

### Integrating new vectors of GL(n,F)

I'd be interested in the following: Let $\pi$ be an irreducible admissible generic representation of $GL(2n,F)$, $F$ a p-adic field. Assume that $\pi$ is ramified and let $W$ be a (non-trivial) new ...

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108 views

### Thin profinite groups - nonabelian analogues of p-adic integers

Let $p$ be a prime number, $S = C_p$ a cyclic group of order $p$, $G = \mathbb{Z}_p$ the profinite additive group of $p$-adic integers. It is well known that all the closed nontrivial subgroups of $G$ ...

**1**

vote

**1**answer

71 views

### Does restriction to an open subgroup preserve projective smooth representations?

Let $G$ be a locally profinite group and $K \le G$ an open subgroup. Does the restriction functor $\mathrm{Res}^G_K$ from the category of smooth $\mathbb{C}$-linear representations of $G$ to smooth ...

**2**

votes

**2**answers

199 views

### Compact induction as a tensor product

Let $G$ be a locally profinite (i.e., locally compact Hausdorff and totally disconnected) topological group, $H \le G$ a closed subgroup, and $(W, \sigma)$ a representation of $H$ over $\mathbb{C}$ ...

**7**

votes

**1**answer

215 views

### On unramified p-adic groups

Let G be a reductive group over a local field F. Let O be the ring of integers of F.
The following are equivalent (and groups satisfying these conditions are called unramified):
(a) G is quasisplit ...

**2**

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**0**answers

81 views

### A question on Plancherel measure for $p$-adic group

Let $G$ be a reductive group over a $p$-adic field. My understanding is that the Plancherel measure on $G$ is a measure on the unitary dual $\hat{G}$.
But at the same time, for example, in his famous ...

**1**

vote

**0**answers

92 views

### Symmetric spaces which are compact modulo the unipotent radical are compact

Is the following true?
Let $X = G/H$ be a symmetric space of a reductive group over a p-adic field $F$.
Let $X^0$ be an open orbit w.r.t. the action of the minimal parabolic $B$ of $G$.
Let $U$ be ...

**3**

votes

**1**answer

82 views

### Density of $\Gamma(N)$ in $\mathrm{Sp}_{2g}(\mathbb{Z}_{\ell})$ where $\ell \not | N$

Let $\mathrm{Sp}_{2g}$ denote the symplectic group of $2g \times 2g$ matrices for some $g \geq 1$, and let $\Gamma(N)$ be the level-$N$ principal congruence subgroup of $\mathrm{Sp}_{2g}(\mathbb{Z})$. ...

**1**

vote

**1**answer

115 views

### $p$-adic analogues of $SO(3)$

I read in the paper " From Laplace to Langlands via representations of orthogonal groups" by Benedict Gross and Mark Reeder that there are, up to isomorphism, two orthogonal groups of the ...

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36 views

### Compatability of depth for elements in p-adic groups under base change

Suppose $F$ is a non-archimedean local field and $E$ is a tame extension of it. Let $G$ be any connected reductive group over $F$. For any $r>0$, let $G(E)_r$ be the set of elements with depth $\ge ...

**1**

vote

**1**answer

169 views

### Orbits of an action of maximal compact subgroups of p-adic orthogonal groups

Let $Q$ be a non-degenerate indefinite quadratic form on ${\mathbb R}^n$ and write $G=SO(Q)$ for the associated special orthogonal group. Let $K$ be a maximal compact subgroup of $G$ and consider the ...

**1**

vote

**1**answer

142 views

### classification of irreducible finite dimensional representation of affine hecke algebra of type A

Let $H_{n}$ be the affine Hecke algebra with parameter q, where q is not root of unity.
The classification of irreducible finite dimensional representations has been given by Kazhdan-Lusztig in terms ...

**16**

votes

**2**answers

666 views

### Are there any simple, interesting consequences to motivate the local Langlands correspondence?

Let's pretend that we know local Langlands at a fairly high level of generality... i.e. we know something along the lines of:
Let $G=\mathbf{G}(F)$ be the group of $F$-points of a connected ...

**5**

votes

**3**answers

193 views

### Questions on constructions of supercuspidal representations

To my knowledge, usually there are two ways to construct supercuspidal representations over p-adic fields. The first is via theory of types (for GL(n) and classical groups), notably by Bushnell, ...

**6**

votes

**2**answers

147 views

### Spherical functions for sl(2,Q_p)

I kindly would like to ask you the following- I am refering to page
175 in the Book by Gelfand, Graev, Shapiro, etc, on "Automorphic forms
..."
My question to which I would kindly ask you to answer ...

**4**

votes

**0**answers

162 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 ...

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90 views

### Are the only discrete groups with nontrivial p-adic Haar measure finite?

Let K be a complete non-archimedean field (say $\mathbb{C_p}$) and let G be a discrete group. Since {e} is an open p-compatible compact subgroup of G, G admits a (left) K-valued Haar measure $\mu$. ...

**3**

votes

**1**answer

172 views

### Weyl group action on continuous characters of the group of $\mathbf{Q}_p$-points of the torus

Let $G$ be a split reductive group over $\mathbf{Q}_p$ and assume $G$ has connected center.
Let $T$ be a maximal split subtorus of $G$ and $R$ be the roots of $(G,T)$.
Let $\chi : T(\mathbf{Q}_p) \to ...

**1**

vote

**1**answer

70 views

### Strictly contracting elements in the center of a Levi subgroup

Let $G$ be a connected reductive group over a non archimedean local field $k$.
Let $P \subset G$ be a parabolic subgroup with Levi decomposition $P=MN$, $Z_M \subset L$ be the center of $M$ and $S_M ...

**1**

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133 views

### Clarification about Tits' article in the Corvallis

I am studying Tits' article in the Corvallis wherein he defines the apartment in the general case (not necessarily split). I wish to know what he means about the filtration of the groups $U_a(K)$ ...

**2**

votes

**2**answers

254 views

### Density of characters

Did Harish-Chandra prove that characters of irreducible representations of a $p$-adic reductive group $G$ span a dense subspace of the space of conjugation-invariant distributions on $G$? What is the ...

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154 views

### Algebraic characters and quasi-characters of reductive algebraic group over non-archimedean local field

Let $G$ be a reductive algebraic group over $F$, where $F$ is a non-archimedean local field.
Then $G(F)$ is a p-adic group.
Let $\Psi(G)$ be the lattice of algebraic characters.
Let $\Lambda_G$ be the ...

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vote

**0**answers

64 views

### Centralizer of a maximal split torus

Can you help me find a reference for the following fact?
"If $G$ is a quasi-split $p$-adic group and $T$ is a maximal split torus in $G$, then the centralizer of $T$ is a maximal torus in $G$."
Or ...

**6**

votes

**5**answers

521 views

### Is a unitary representation always semisimple?

I have been reading the online lecture notes by Fiona Murnaghan
http://www.math.toronto.edu/murnaghan/courses/mat1197/notes.pdf
The first lemma in p.35 says that every unitary representation of ...

**5**

votes

**1**answer

185 views

### When does the derived subgroup of $G(F)$ contains the $F$-points of unipotent subgroups of $G$

Let $F$ be a local field of characteristic $0$ and $G$ a connected split reductive group over $F$.
Let's look at the derived groups. We have $(G(F),G(F)) \subset (G,G)(F)$ and this inclusion is of ...

**4**

votes

**0**answers

112 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 ...

**2**

votes

**0**answers

158 views

### Correct definition of locally algebraic parabolic induction of a locally algebraic character

Let $L$ be a finite extension of $\mathbf{Q}_p$ and $G$ the group of $L$-points of a split connected reductive group $\mathbf{G}$ over $L$, $T$ the $L$-points of a split maximal torus in $\mathbf{G}$, ...

**2**

votes

**1**answer

334 views

### On a result due to Zelevinskii

In his paper on the p-adic analogue of the Kazhdan-Lusztig hypothesis (Functional Analysis and Its Applications 15.2 (1981): 83-92), Zelevinskii proves a combinatorial proposition (outlined in Section ...

**3**

votes

**3**answers

190 views

### Modules for an idempotented algebra

Recall that an associative algebra $A$ is called idempotented provided that is the filtered union of subalgebras $eAe$ for $e \in A$ idempotent. I think sometimes people say that $A$ has approximate ...

**8**

votes

**2**answers

389 views

### Proving that some principal series representations of SL(2,F) are irreducible

I am sorry in advance if this question is not "research level".
Let $F$ be a p-adic field.
I saw, in Bumps book, a proof which I liked, showing which principal series representations of $GL(2,F)$ ...

**0**

votes

**0**answers

99 views

### Postnikov system for a tree

The Postnikov system of a path-connected space X is a tower of spaces $...\longrightarrow X_{n+1}\longrightarrow X_{n}\longrightarrow ...\longrightarrow X_{1}\longrightarrow X_{0}$ with the following ...

**1**

vote

**0**answers

99 views

### Restriction and then induction of the Steinberg representation of GL(n)

Let $G_{n}=GL(n,F)$, where $F$ a locally compact non-Archimedean field, $St_{G_{n}}$ the Steinberg representation of $G_{n}$, and $B$ the standard Borel subgroup of $G_{n}$.
We denote $\pi_{n}$ the ...

**2**

votes

**2**answers

209 views

### fixed vector of a generic representation of GL(n,F)

Let $F$ be a locally compact non-archimedean field and $G_{n}$ the locally profinite group $GL(n,F)$.
Let $\Gamma_{n,k}$ be the subgroup of $G_{n}$ whose elements are the matrices of the form
$$ ...

**3**

votes

**0**answers

134 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 :
...

**4**

votes

**1**answer

239 views

### p-adic Lie group vs Lie algebra cohomology with mod p coefficients

My question concerns the cohomology of a compact $p$-adic Lie group $G$ (wich is pro-$p$).
Let $M$ be a finite dimensional $\mathbb{Q}_p$-vector space with continuous linear $G$-action.
Lazard ...

**9**

votes

**2**answers

434 views

### Upper bound on order of finite subgroups of GL_n(Z_p)?

Fix a prime $p$ and integer $n>1$, along with the ring $R$ of integers in a finite extension of the field $\mathbb{Q}_p$ (for example $R = \mathbb{Z}_p$).
Is there an upper bound $C(n,p)$ on ...