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**16**

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**2**answers

749 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 ...

**10**

votes

**1**answer

162 views

### Growth of dimension of fixed spaces in $GL_n(\mathbb{Q}_p)$-representations

Let $\pi$ be a generic irreducible admissible representation of $GL_n(L)$, where $L$ is a $p$-adic field, $R$ is its ring of integers, and $\mathfrak{p}$ is its prime ideal. The conductor of $\pi$ ...

**9**

votes

**2**answers

467 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 ...

**8**

votes

**2**answers

436 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 $GL(2,F)$ ...

**8**

votes

**2**answers

236 views

### Characters of cuspidal representations

Let $\pi$ be an irreducible cuspidal representation of a semi-simple $p$-adic group $G$. It is well-known that the character of $\pi$ is concentrated in the set of compact elements in $G$.
What is ...

**7**

votes

**1**answer

342 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 ...

**7**

votes

**1**answer

312 views

### When is a Moy-Prasad filtration subgroup the stabilizer of a subset of the building (up to center)?

Let $G$ be a connected, simply connected, semi-simple algebraic group defined and split over a local non-arch field $k$ with integer ring $R$. Let $B$ be the corresponding reduced building. Fix an ...

**7**

votes

**2**answers

439 views

### On a theorem of Kazhdan

Let $G=GL_n(F)$, where $F$ is a p-adic local field, $U$ be the upper triangular maximal unipotent group, and $\theta$ a character of $U$. Then a Theorem of Kazhdan says that for any irreducible smooth ...

**7**

votes

**0**answers

244 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 ...

**6**

votes

**5**answers

608 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 ...

**6**

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**2**answers

220 views

### What are the special parahoric subgroups in unitary groups?

Let $L$ be a $p$-adic field and let $L'/L$ be a quadratic extension. Let $U_{L'/L}(n)$ be a quasi-split unitary group of $n\times n$ matrices with entries in $L'$. I'm curious about what the special ...

**6**

votes

**2**answers

190 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 ...

**6**

votes

**1**answer

241 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: ...

**6**

votes

**1**answer

155 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' ...

**6**

votes

**0**answers

119 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 ...

**5**

votes

**3**answers

271 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, ...

**5**

votes

**1**answer

267 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 ...

**5**

votes

**1**answer

219 views

### Arithmetic quotients of Bruhat-Tits buildings for groups over local fields of positive characteristic

I have been led to believe that there is a result giving a description of the quotient of a Bruhat-Tits building $\Delta(G,k)$, for a semisimple algebraic group $G$ over a non-archimedean local field ...

**5**

votes

**1**answer

317 views

### Reference request - Jacquet module and asymptotic of matrix coefficients

Hello,
I would like to know some nice references about the relation between asymptotics of matrix coefficients of representations of reductive groups over local fields, and the pairing between the ...

**4**

votes

**3**answers

745 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 ...

**4**

votes

**4**answers

677 views

### cuspidal types and Iwahori subgroup for $SL(2,F)$

Let $(J,\pi)$ be a cuspidal type in $SL(2,F)$, $F$ is a non-Arch. local field and let $I$ be the Iwahori subgroup of $SL(2,F)$.
Is there any possibility that $J\subset I$ or even a subgroup?

**4**

votes

**1**answer

130 views

### Examples to keep in mind while reading the book 'The Admissible Dual…' by Bushnell and Kutzko and the importance of Interwining of representations

I am a beginner in the field of representation theory. I was reading the book 'The Admissible Dual of $GL(N)$ Via Compact Open Subgroups' by Bushnell and Kutzko.
Let me first describe the book a ...

**4**

votes

**1**answer

123 views

### Is Howe's construction of tame supercuspidal representations independent of additive character?

Let $F$ be a $p$-adic field.
In "Tamely ramified supercuspidal representations of $Gl_n$" (Am. J. Math 73 (1977)), Howe constructs a supercuspidal representation $\pi_{\psi}$ of $GL_n(F)$ from the ...

**4**

votes

**1**answer

313 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 $U_a(K)$ ...

**4**

votes

**1**answer

472 views

### Representations of reductive groups over local fields through parahoric induction

Let me take $G$ to be a simple (connected) split reductive group over a local field $K$. One way I might go about constructing a (smooth, admissible) complex representation $\sigma$ of $G$ is as ...

**4**

votes

**1**answer

90 views

### Cartan integral formula for a p-adic group?

Let $G$ denote a reductive group over a local field $F$. Suppose that $G$ is split over $F$ and fix a maximal (split) torus $A$. Let $A^+$ denote a Weyl chamber in $A$ and let $K$ be a suitable ...

**4**

votes

**1**answer

87 views

### Intertwining Operators Associated to Simple Reflections

Let $G$ be a quasi-split reductive group, over a local field, with a Borel subgroup $B=T\cdot N$ and the associated Weyl group $W$. Given a family of induced representations $\pi_s = Ind_B^G \chi\cdot ...

**4**

votes

**1**answer

287 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 ...

**4**

votes

**2**answers

184 views

### Orbits of the maximal compact subgroup on the light cone for $p$-adic groups

It is known that if $Q$ is an indefinite non-degenerate quadratic form on $ \mathbb{R}^n$ with $n \ge 3$, then any maximal compact subgroup $K$ of the orthogonal group $SO(Q)$ acts transitively on the ...

**4**

votes

**0**answers

110 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 $n$ ...

**4**

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**0**answers

79 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 ...

**4**

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**0**answers

200 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 ...

**4**

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**0**answers

126 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 ...

**4**

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**0**answers

641 views

### Newton Method in $p$-adic case

The Newton Method over $\mathbb{R}$ has the property that the precision is doubled (under some continuous differentialbe assumption) in each iteration. For the ring $\mathbb{Z}_p$ of $p$-adic ...

**3**

votes

**2**answers

605 views

### 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

**1**answer

672 views

### Discrete Series representations for $SL_{2}$ over $p$-adic field.

I am working on the chamber homology for $SL(2,F)$, and stuck at some basic stuff on D.S. reps of $SL(2,F)$.
Let $ I=\left(
\begin{array}{cc}
\mathcal{O}_{F} & \mathcal{O}_{F} \\
...

**3**

votes

**1**answer

201 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 ...

**3**

votes

**1**answer

91 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 $\mathrm{Sp}_{2g}(\mathbb{Z})$. ...

**3**

votes

**3**answers

207 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 ...

**3**

votes

**1**answer

251 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 ...

**3**

votes

**1**answer

220 views

### Fixed points in the Bruhat-Tits building

Let $G$ be a connected reductive group over a complete discrete valuation field with perfect residue field (or just a non-arch local field). Let $\mathcal{B}$ be its reduced Bruhat-Tits building, and ...

**3**

votes

**1**answer

184 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 ...

**3**

votes

**2**answers

268 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 ...

**3**

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**0**answers

96 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 ...

**3**

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**0**answers

118 views

### Ext and representations with fixed central characters

In this paper (http://arxiv.org/pdf/1108.3668v2.pdf) Adler and Prasad compute certain Ext groups. On page 2 they write,
"Since extensions of representations of abelian groups are well understood
...

**3**

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**0**answers

98 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 ...

**3**

votes

**1**answer

186 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 ...

**3**

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**0**answers

96 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

**0**answers

145 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 :
...

**2**

votes

**3**answers

789 views

### The Weyl group of $SL(2, F)$

Let $G= SL(2, F)$, given a torus $T$, the Weyl group with respect to $T$ is defined to be $W=N(T)/Z(T)$, the quotient of the normalizer $N(T)$ of the torus by the centralizer $Z(T)$ of the torus.
My ...