Questions tagged [p-adic-groups]

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28 votes
3 answers
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What is a tamely-ramified Weil-Deligne representation?

Let $W_F$ denote the Weil group of a finite extension of $\mathbb{Q}_p$. Let $I$ denote the inertia subgroup and $I^{>0}$ the (pro-$p$) subgroup of wild inertia. (I hope I've got my notation right.....
Geordie Williamson's user avatar
22 votes
2 answers
1k 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 ...
sawdada's user avatar
  • 6,148
19 votes
2 answers
1k 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 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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16 votes
3 answers
5k 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 ...
m07kl's user avatar
  • 1,672
13 votes
3 answers
2k views

Naïve 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, ...
SomeGuy's user avatar
  • 833
13 votes
0 answers
379 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 ...
D_S's user avatar
  • 6,100
12 votes
1 answer
917 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 ...
JACK's user avatar
  • 421
12 votes
1 answer
441 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 $\pi$ ...
John Binder'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
11 votes
2 answers
505 views

Does local Langlands functoriality preserve genericity?

Let $G=Sp_4$ over a p-adic field $F$. After Gan-Takeda's work, the local Langlands correspondence for $G$ is known. Thus we have the local functoriality from $G$ to $GL_5$. Do we know that this local ...
user100750's user avatar
10 votes
1 answer
2k 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 ...
Peter McNamara's user avatar
10 votes
1 answer
742 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 $K^n$, $n$?
user avatar
10 votes
2 answers
262 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 ...
asv's user avatar
  • 21.1k
10 votes
1 answer
241 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 ...
Nate's user avatar
  • 1,992
10 votes
0 answers
427 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 ...
L.C. Ruth's user avatar
  • 229
9 votes
2 answers
1k views

Is it true that irreducible generic representations of $G_2(F)$ are self-dual?

Let $G_2$ be the split exceptional group of type $G_2$ and $F$ be a p-adic field. Is it true that every irreducible smooth representation of $G_2(F)$ is self-contragredient? If the answer is Yes, can ...
Q. Zhang's user avatar
  • 960
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
  • 21.1k
9 votes
2 answers
639 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 ...
John Binder's user avatar
  • 1,453
9 votes
2 answers
1k 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 ...
Kostas Psaromiligkos'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
  • 143
9 votes
1 answer
348 views

The maximum order of torsion elements in ${\rm GL}_n(\mathbb{Z}_p)$ or ${\rm GL}_n(\mathbb{F}_p[[T]])$

This question is inspired by Upper bound on order of finite subgroups of GL_n(Z_p)?. It's showed that the supremum of orders of finite subgroups of ${\rm GL}_n(\mathbb{Z}_p)$ is finite and can be ...
Nobody's user avatar
  • 795
9 votes
0 answers
234 views

On the status of some conjectures mentioned/used in Harish Chandra's 1970 lecture notes

In van Dijk's notes of Harish Chandra's lectures on harmonic analysis, several conjectures are mentioned throughout, such as in Part 1, section 4 of van Dijk's notes Conjecture I : Let $\omega$ be ...
edgarlorp's user avatar
  • 113
9 votes
0 answers
298 views

Colimit of continuous cohomology over subgroups

Suppose $G$ is a profinite group, in fact in the applications I'm interested in it would be a $p$-adic analytic group similar to $GL_{n}(\mathbb{Z}_{p})$. Say $M$ is a profinite $G$-representation, ...
Piotr Pstrągowski's user avatar
9 votes
0 answers
391 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. ...
Charles Denis's user avatar
8 votes
2 answers
1k 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 ...
MathStudent's user avatar
8 votes
2 answers
730 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 $GL(2,F)$ ...
Sasha's user avatar
  • 5,492
8 votes
1 answer
311 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 ...
D_S's user avatar
  • 6,100
8 votes
1 answer
453 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 ...
John Binder's user avatar
  • 1,453
8 votes
1 answer
363 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 ...
Alexander Braverman's user avatar
8 votes
1 answer
288 views

Tempered Iwahori-spherical representations

Consider a local field $F$ of characteristic 0 and $G=GL_n(F)$. It is well known (for example Cartiers article in Corvallis) that an admissible, irreducible representation $\pi$ of $G$ has a ...
Julien's user avatar
  • 153
7 votes
3 answers
538 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, ...
user1832's user avatar
  • 2,679
7 votes
5 answers
1k 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 ...
Windi's user avatar
  • 833
7 votes
2 answers
546 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 ...
user64433's user avatar
  • 181
7 votes
1 answer
480 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 ...
Monica's user avatar
  • 211
7 votes
1 answer
247 views

When a pro-$p$ group of finite rank can be embedded into the first congruence subgroup of ${\rm GL}_{N}(\mathbb{Z}_{p})$

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$Let $p$ be a odd prime. We say that a pro-$p$ group has finite rank if it is isomorphic to a closed subgroup of $\GL_d(\mathbb{Z}_p)$ for some ...
stupid boy's user avatar
7 votes
0 answers
355 views

Subgroups of ${\rm GL}_{2}(\mathbb{Z}/2^{n} \mathbb{Z})$

Assume that $n \geq 3$. Is there a subgroup $H \leq {\rm GL}_{2}(\mathbb{Z}/2^{n} \mathbb{Z})$ whose order is a power of $2$ so that $\bullet$ $\det : H \to (\mathbb{Z}/2^{n} \mathbb{Z})^{\times}$ is ...
Jeremy Rouse's user avatar
7 votes
0 answers
201 views

Limit of the Casselman–Shalika Formula for the Spherical Whittaker Function

$\DeclareMathOperator{\GL}{GL}$Consider $G = \GL_{r+1}(F)$, where $F$ is a local non-archimedian field with the ring of integers $\mathcal{O}_F$ and the maximal ideal $\mathfrak{p}$, and let $q = \...
user avatar
7 votes
0 answers
425 views

intuition for lattices in p-adic (or other non-Archimedean) vector spaces?

I could use some help to jumpstart my intuition for lattices in vector spaces over non-Archimedean fields, like $\mathbb{Q}_p$ and $\mathbb{F}_q((t))$. I have some intuition for $\mathbb{Z}$-lattices ...
PrimeRibeyeDeal's user avatar
7 votes
0 answers
481 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 ...
Monica's user avatar
  • 211
6 votes
2 answers
265 views

Is every compact simply-connected reductive p-adic group perfect?

Let $k$ be a nonarchimedean local field and $G$ a reductive $k$-group, which we assume to be semisimple and simply-connected. Recall that an abstract group $H$ is perfect if it is generated by ...
David Schwein's user avatar
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
  • 223
6 votes
2 answers
438 views

When is compact induction cuspidal?

Let $G=GL_2(\mathbb{Q}_p)$, and let $K$ be a compact-modulo-center subgroup of $G$, $\rho$ an irreducible smooth representation of $K$. Question 1: Is $\mathrm{ind}_K^G \rho$ cuspidal? Here ...
nikola karabatic's user avatar
6 votes
1 answer
323 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: "...
John Binder's user avatar
  • 1,453
6 votes
2 answers
403 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 ...
Florin Radulescu's user avatar
6 votes
1 answer
578 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 $U_a(K)$ (...
Abhishek Parab's user avatar
6 votes
1 answer
342 views

On the Artin-Rees Lemma for non-commutative rings

Consider a commutative noetherian ring $A$ with an ideal $I\subset A$. The Artin-Rees lemma implies that for f.g. modules $N\subset M$, the $I$-adic topology on $N$ agrees with the subspace topology ...
Fernando Peña Vázquez's user avatar
6 votes
1 answer
533 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 ...
user avatar
6 votes
1 answer
380 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' "...
John Binder's user avatar
  • 1,453
6 votes
1 answer
265 views

Realization of the $p$-adic Steinberg representation as a subrepresentation

Let $G = \mathrm{GL}_n(F)$ where $F$ = non-archimedean local field. The Langlands Classification tells one that all irreducible admissible reps of $\mathrm{GL}_n(F)$ can be realized as (the unique ...
Maty Mangoo's user avatar

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