Questions tagged [p-adic-groups]

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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
2 votes
0 answers
89 views

$p$-adic Harish-Chandra character of a stable virtual character

Let $F$ be a $p$-adic field and let $G$ be a reductive group over $F$. Associated to an irreducible admissible representation of $\pi$ of $G(F)$, we have a distribution character $\Theta_{\pi}$ ...
Alexander's user avatar
  • 861
3 votes
0 answers
98 views

p-adic analogue of classification of irreducible Riemannian symmetric spaces?

For Riemannian symmetric spaces, we have a nice result that the simply-connected ones are products of irreducible symmetric spaces, of which we have a list of ten infinite families and some ...
Roger Van Peski's user avatar
1 vote
0 answers
220 views

compact $p$-adic Lie group can be embedded into $O_K^n$ or $\text{GL}_n(K)$?

Let $K$ be a local field of charecteristic $0$ and $G$ be a compact $p$-adic Lie group of dimension $n$, then can $G$ be embedded into $O_K^n$ or $\text{GL}_n(K)$ as a closed subgroup? This is a dual ...
user avatar
3 votes
0 answers
159 views

Orbit representatives for the action of the maximal compact subgroup

Let $F$ be a non-Archimedean local field and $O$ be the ring of integers in $F$. Take $G=GL(2,F)$ and $K=GL(2,O)$. Consider the action of $K$ on $G$ by conjugation. Is it possible to explicitly write ...
user8974's user avatar
  • 175
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
1 vote
0 answers
234 views

J. Tate's article on $p$-divisible groups

I have a question about a conclusion that J. Tate arrived at in his paper $p$-Divisible Groups (in: Springer T.A. (eds) Proceedings of a Conference on Local Fields. Springer, Berlin, Heidelberg (1967) ...
user267839's user avatar
  • 5,948
3 votes
0 answers
202 views

Miraculous Parahorics

Let $G$ be a connected simple group over a local field $k$. Let $I\subset G(\mathcal{O})$ denote an Iwahori subgroup of $G(k)$ with Lie algebra $\mathfrak{i}$. Let $P\supseteq I$ be any other ...
Dr. Evil's user avatar
  • 2,641
4 votes
1 answer
276 views

Unitary representations of lattices

Let $G$ be a simple linear group over a non-archimedean local field $F$. Assume that the split-rank over $F$ is at least 2. Let $\Gamma$ be a lattice in $G$. Then $\Gamma$ is a finitely generated ...
user avatar
3 votes
0 answers
141 views

Classical Hecke operators and Hecke algebra of type $A_1$

What's the relation between the classical Hecke operators (as defined in J. P. Serre's A course in arithmetic chapter 6) and the Hecke algebra of type $A_1$, i.e. the algebra generated by the vertices ...
Student's user avatar
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3 votes
1 answer
320 views

branching laws for $p$-adic representations of reductive groups

There are many papers studying branching laws of irreducible admissible complex representations of classical groups over local fields, are there some analogues for $p$-adic representations? For ...
sawdada's user avatar
  • 6,148
3 votes
1 answer
121 views

Image of a finite dimensional complex representations of $GL_n(\mathcal{O})$

EDIT Let $\mathcal{O}$ be the ring of integers in a non-Archimedean local field. Let $GL_n(\mathcal{O})$ be the (compact) group of $n\times n$ matrices with entries in $\mathcal{O}$ such that its ...
asv's user avatar
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3 votes
1 answer
314 views

A detail in Bushnell and Henniart's book, “The local Langlands conjecture for GL(2)”

I am recently troubled with a computational detail in Bushnell and Henniart's book, "The local Langlands conjecture for Gl(2)". Let $(\mathfrak{A},n,\alpha)$ be a simple stratum, and define $K_\...
Qingzhi Li's user avatar
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
4 votes
1 answer
141 views

The reductive $p$-adic group $^{2}\!A_3''$ via Galois decent

I am running into some confusion when trying to explicitly describe the group $^{2}\!A_3''$ (using the naming convention that Tits gives in his Corvallis notes). If anyone can give me any advice, I ...
Stella Sue Gastineau's user avatar
4 votes
0 answers
122 views

Finite dimensional irreps of $p$-adic groups

What are some examples of finite dimensional irreducible complex representations of $SL_2(\mathbb{Q}_p)$? One knows such a representations cannot be smooth, so probably the examples will be ...
Spinoza's user avatar
  • 81
3 votes
0 answers
101 views

Hermitian sublattices of a given type

Consider an unramified quadratic extension $E/F$ of non-archimedean local fields, and suppose that $\langle\cdot,\cdot\rangle$ is a fixed Hermitian form on $E^d$ such that $\mathcal{O}_E^d$ is self-...
Spencer Leslie's user avatar
6 votes
2 answers
441 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
5 votes
1 answer
478 views

Is $\mathrm{SL}_n(\mathbb{Q}_p)$ virtually torsion-free?

Recall that a group is virtually torsion-free if it admits a finite index subgroup which is torsion-free. Question. Is $\mathrm{SL}_n(\mathbb{Q}_p)$ virtually torsion-free for $n > 1$? Comments. ...
Jackson Morrow's user avatar
28 votes
3 answers
2k views

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
4 votes
1 answer
323 views

Volumes of double cosets $KtK$

Let $G$ be the group of rational points of a connected reductive group $\mathbb G$ defined over a non-archimedean local field $F$. For simplicity sake, I assume that $\mathbb G$ is split over $F$. Let ...
Paul Broussous's user avatar
3 votes
0 answers
61 views

Directed galleries of the building of type $\widetilde{A}_{n}$

Let $X$ be the affine building of $GL_n(\mathbb{Q}_{p})$. We call oreinetd chamber of $X$ every sequence $\overrightarrow{C}=(s_1,...,s_n)$ of vertices such that $C=\{s_1,...,s_n\}$ is a chamber of $X$...
Rajkarov's user avatar
  • 933
2 votes
0 answers
112 views

Generic representation of PGL(3)

Let $G$ be the group $PGL(3,F)$, where $F$ is non-archimedean locally compact field, and $(\widetilde{H}_{n})_{n\in\mathbb{N}}$ the decreasing sequence of open and compact subgroups given by (image in ...
Rajkarov's user avatar
  • 933
1 vote
1 answer
102 views

Size of a multi-segment of a representation of $GL_n(F)$

Let $F$ be a p-adic field and $GL_n(F)$ the general linear group over $F$. The irreducible complex finite length smooth representations are parametrized by multi-segements in the paper. A multi-...
Jianrong Li's user avatar
  • 6,101
4 votes
0 answers
134 views

Growth of the number of fixed points of a $p$-adic group under natural filtrations

Let $G$ be a $p$-adic reductive group, so by definition as a locally profinite group it's the group of $\mathbb Q_p$ points of a connective reductive group over $\mathbb Q_p$, $K$ be a parahoric ...
sawdada's user avatar
  • 6,148
3 votes
0 answers
145 views

Iwasawa theory and cohomological $p$-dimension of Inertia

Let $E$ be a elliptic curve without complex multiplication over a number field $F$. Let $F_n=F[E_{p^n}]$ and $F_{\infty}=F[E_{p^\infty}]$. So by a well know theorem of Serre, the Galois group $Gal(F_\...
MathStudent's user avatar
3 votes
0 answers
127 views

Galois descent for profinite groups acting on local fields

Suppose that $G$ is a profinite group acting faithfully on a field $L$ and assume moreover that the action is admissible, that is, that every $l \in L$ is stabilized by an open subgroup of $G$. If ...
Piotr Pstrągowski's user avatar
2 votes
0 answers
55 views

Admissible (unitary) spherical representation $sl(2,Q_p)$. Does dimension fixed point vector increase proportional index

I am using terminology of Cartier's Harmonic analysis on trees. Take $\pi$ be one of the irreducible principal or complementary (unitary) spehrical series of $Sl(2, Q_p)$. Let $K$ be the maximal ...
Florin Radulescu's user avatar
9 votes
0 answers
299 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
4 votes
0 answers
420 views

Index of the congruence subgroups of $PGL_2(\mathbb{Z}_p)$

Let $\Gamma_n$ be the $n$-th congruence subgroup of $GL(2,\mathbb{Z}_p)$. So $\Gamma_n$ consists of matrices in $GL(2,\mathbb{Z}_p)$ which are congruent to the identity matrix modulo $p^n$. Let $Z(\...
MathStudent's user avatar
3 votes
1 answer
157 views

Is $({F^{\times})^{ diag}}\backslash(GL_2 \times E^{\times})_{det=\mathbb{N}}$ a unitary group?

Let $F$ be an p-adic field, and $E$ be a quadratic extension of $F$, then is $({F^{\times})^{ diag}}\backslash(GL_2 \times E^{\times})_{det=\mathbb{N}}$ isomorphic to some unitary group $U_{E/F}(2)$? ...
Cooler Panda's user avatar
5 votes
1 answer
1k views

$Hom_G(C_c^{\infty}(G),\pi)\cong Hom_{\mathbb{C}}(\pi^{\vee},\mathbb{C}) ?$

$G$ is an p-adic group, and $\pi$ is an irreducible representation of $G$, then do we naturally have $Hom_G(C_c^{\infty}(G),\pi)\cong Hom_{\mathbb{C}}(\pi^{\vee},\mathbb{C})$? I think it is true, but ...
Cooler Panda's user avatar
4 votes
1 answer
298 views

Reference for Shalika germs of GL(n)

I was reading the two Repka papers where he computes the leading and subleading Shalika germs for $GL_n$ and I was wondering, where are we since then? Have these germs (and the integrals) been ...
Ioannis Zolas's user avatar
6 votes
1 answer
165 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 ...
Ioannis Zolas's user avatar
5 votes
0 answers
200 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, ...
nikola karabatic's user avatar
5 votes
2 answers
493 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 ...
Ioannis Zolas's user avatar
4 votes
0 answers
276 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 ...
Q. Zhang's user avatar
  • 960
2 votes
0 answers
136 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 $T$ ...
Paul Broussous'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
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
1 vote
0 answers
61 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 ...
Cooler Panda's user avatar
3 votes
0 answers
113 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\...
Cooler Panda's user avatar
10 votes
1 answer
745 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
2 votes
0 answers
190 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 $(...
João Dias's user avatar
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
  • 223
5 votes
1 answer
273 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 ...
Mehta's user avatar
  • 223
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
1 vote
0 answers
53 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 ...
D_S's user avatar
  • 6,100
6 votes
0 answers
218 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) =...
D_S's user avatar
  • 6,100
2 votes
1 answer
772 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 $p$...
Kostas Psaromiligkos's user avatar