Questions tagged [p-adic-groups]
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270
questions
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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 ...
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 $\Theta_{\pi}$ ...
3
votes
0
answers
98
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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 ...
1
vote
0
answers
220
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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 ...
3
votes
0
answers
159
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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 ...
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 ...
1
vote
0
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234
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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) ...
3
votes
0
answers
202
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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 ...
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 ...
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 ...
3
votes
1
answer
320
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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 ...
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 ...
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_\...
9
votes
0
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234
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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 ...
4
votes
1
answer
141
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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 ...
4
votes
0
answers
122
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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 ...
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-...
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 ...
5
votes
1
answer
478
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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.
...
28
votes
3
answers
2k
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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.....
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 ...
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 $X$...
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 ...
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-...
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 ...
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_\...
3
votes
0
answers
127
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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 ...
2
votes
0
answers
55
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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 ...
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, ...
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(\...
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 $U_{E/F}(2)$? ...
5
votes
1
answer
1k
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$Hom_G(C_c^{\infty}(G),\pi)\cong Hom_{\mathbb{C}}(\pi^{\vee},\mathbb{C}) ?$
$G$ 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 ...
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 ...
6
votes
1
answer
165
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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 ...
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, ...
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 ...
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 ...
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 $T$ ...
22
votes
2
answers
1k
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$GL_n(\Bbb Z_p)$ 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 ...
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 ...
1
vote
0
answers
61
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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 ...
3
votes
0
answers
113
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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\...
10
votes
1
answer
745
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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 $K^n$, $n$?
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 $(...
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)...
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 ...
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 ...
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 ...
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) =...
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 $p$...