Questions tagged [p-adic-groups]

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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 reductive ...
user19918273's user avatar
16 votes
2 answers
3k 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 $B$ ...
D_S's user avatar
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11 votes
2 answers
793 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 ...
Jim Humphreys's user avatar
9 votes
2 answers
291 views

What is the quotient group $D^*/{F^*(1+P_D)}$ for a quaternion division algebra $D$ over a local field $F$?

Let $F$ be the non-archimedean local field $\mathbb{Q}_p$ for some prime $p$ and $D$ be a quaternion division algebra over $F$. Let $\mathcal{O}_D$ and $\mathcal{P}_D$ denote the ring of integers of $...
BPK's user avatar
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9 votes
2 answers
399 views

Is the representation of $GL_n(\mathcal{O})$ in functions on Grassmannian multiplicity free?

Let $\mathbb{F}$ be a local non-Archimedean field. Let $\mathcal{O}\subset \mathbb{F}$ be its ring of integers. Let $GL_n(\mathcal{O})$ be the (compact) group of $n\times n$ invertible matrices with ...
asv's user avatar
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7 votes
5 answers
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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 ...
Windi's user avatar
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6 votes
2 answers
846 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 ...
Mehta's user avatar
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6 votes
1 answer
563 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 ...
Sasha's user avatar
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5 votes
2 answers
478 views

Non-trivial extension of representations have same central character

Let $\pi_1, \pi_2$ be two irreducible complex representations of $G=\mathrm{GL}_2(\mathbb{Q}_p)$ and assume that there exists a non-split extension $0\to\pi_1\to \pi\to\pi_2\to0$ of representations ...
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3 votes
1 answer
483 views

$p$-adic analogues of $\mathrm{SO}(3)$

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 (...
Keivan Karai's user avatar
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3 votes
0 answers
310 views

Tits Reductive Groups over Local Fields Example 1.15 (Quasi-split special unitary groups in odd dimension)

I hope this question about Tits's paper "Reductive groups over local fields" in Algebraic groups and discontinuous subgroups ends up having an easy answer, but I'm a little stuck on the ...
Marc Besson's user avatar
2 votes
0 answers
159 views

$\mathrm{Ext}^i(\pi_1, \pi_2)\neq0$ implies same central character

If $\pi_1$ and $\pi_2$ are two smooth admissible representations of $\operatorname{GL}_2(\mathbb{Q}_p)$ over $\overline{\mathbb{F}}_p$ with central characters. I want to prove that if $\pi_1$ has ...
user avatar
2 votes
1 answer
408 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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2 votes
2 answers
214 views

The stabiliser group of an isotropic quadratic form over $\mathbb{Q}_p$ is non-compact?

Let $\mathbb{Q}_p$ denote the $p$-adic integers. Let $V$ be a $\mathbb{Q}_p$-vector space and $Q : V \rightarrow \mathbb{Q}_p$ be a non-degenerate integral quadratic form. We say that the pair $(Q,V)$ ...
Andrew Musso's user avatar
1 vote
0 answers
119 views

Explicit construction of $T$-orbits of generic characters of unitary groups

Let $F$ be a $p$-adic field. Let $E$ be a quadratic extension of $F$ and $G$ be a quasi-split unitary group $U(2n)$ or $U(2n+1)$ over with respect to $E/F$. Let $N_{E/F}$ be a norm map. Let $B=TU$ be ...
Andrew's user avatar
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1 vote
1 answer
474 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 ...
Keivan Karai's user avatar
  • 6,064
1 vote
0 answers
230 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)...
Mehta's user avatar
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