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12 votes
1 answer
996 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
472 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
  • 1,453
9 votes
2 answers
671 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
0 answers
409 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
1 answer
476 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
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
6 votes
0 answers
261 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 ...
John Binder's user avatar
  • 1,453
5 votes
1 answer
752 views

Some question about cupidal automorphic representation and supercuspidal representation

Question 1: Let $G$ be a connected reductive group defined over a number field $K$, and $\pi$ is an irreducible cuspidal automorphic representation of $G(\mathbb{A}_K)$. Then by a theorem of Flath, we ...
Cooler Panda's user avatar
5 votes
1 answer
781 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 ...
Will's user avatar
  • 805
4 votes
0 answers
124 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
4 votes
0 answers
135 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 ...
Zhiyu's user avatar
  • 6,622
4 votes
0 answers
313 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-Zh's user avatar
  • 960
4 votes
0 answers
219 views

A question of integral on $p$-adic fields $\mathbb{Q_p}$

We assume that $(\pi,V)$ is an admissible, irreducible and infinite-dimensional representation of $GL_2(\mathbb{Q_p})$. In the proof of existence and uniqueness of Kirillov model, the key step is that ...
JACK's user avatar
  • 421
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
2 votes
2 answers
657 views

Compact induction as a tensor product

Let $G$ be a locally profinite (i.e., locally compact Hausdorff and totally disconnected) topological group, $H \le G$ a closed subgroup, and $(W, \sigma)$ a representation of $H$ over $\mathbb{C}$ ...
Question Mark's user avatar
2 votes
0 answers
169 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
0 answers
216 views

Confusion regarding special parahoric subgroups of the unitary group

This question is to clarify some confusion about special parahoric subgroups of a unitary group $G = \mathrm U_n(F)$ in an odd number of variables, with respect to an unramified quadratic extension $E/...
Suzet's user avatar
  • 769
2 votes
0 answers
168 views

Galois representation absolutely irreducible after restricting to open subgroup of finite index

Let $E$ and $F$ be finite extensions of $\mathbb{Q}_p$. Let $\phi:\mathrm{Gal}(\overline{E}/E)\to GL_n(F)$ be an absolutely irreducible continuous representation. Assume that the restriction of $\phi$ ...
user avatar
1 vote
1 answer
162 views

Does restriction to an open subgroup preserve projective smooth representations?

Let $G$ be a locally profinite group and $K \le G$ an open subgroup. Does the restriction functor $\mathrm{Res}^G_K$ from the category of smooth $\mathbb{C}$-linear representations of $G$ to smooth $\...
Question Mark's user avatar
1 vote
1 answer
194 views

Non-split extension of representations of $\mathrm{GL}_2$ and $\mathrm{Hom}$

Let $0\to V_1\to V\to V_1\to0$ be a sequence of representations of $\mathrm{GL}_2(\mathbb{Q}_p)$ over $\overline{\mathbb{F}}_p$, where $V_1$ is irreducible, smooth and admissible. Assume that this ...
user avatar
1 vote
1 answer
109 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
  • 6,201