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17 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
  • 6,180
13 votes
0 answers
406 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
  • 6,180
9 votes
1 answer
331 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
  • 163
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
1 answer
320 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
  • 6,180
6 votes
2 answers
486 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
0 answers
236 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
  • 6,180
6 votes
0 answers
217 views

Dimension of space of K-fixed vectors

If $G$ is an unramified group over an $p$-adic field $F$, the Satake isomorphism identifies the spherical Hecke algebra with respect to a special maximal compact subgroup $K$. In particular, (1) $H(G(...
Dylon Chow'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
5 votes
1 answer
280 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
5 votes
1 answer
279 views

Asymptotic behavior of matrix coefficients

I'm reading Casselman's notes "Introduction to the theory of admissible representations of p-adic reductive groups". In chapter 4 "The asymptotic behavior of matrix coefficients", ...
Zahi Hazan's user avatar
5 votes
0 answers
122 views

Problem with affine root subgroups of $SU_3$ ramified, residue characteristic $p=2$

Let $L/K$ be ramified quadratic extension of local fields, and let characteristic of the residue field of $K$ be $2$. Let $\mathbb{G}=SU_3$, $G=\mathbb{G}(K)$. Let $\text{val}$ be a valuation on $K$ ...
pbarron's user avatar
  • 71
4 votes
1 answer
130 views

Do parabolic inductions share a composition factor if and only if the inducing data are associate?

Let $F$ be a local field of characteristic zero and $G$ a connected reductive group over $F$. Let us call an inducing datum a triple $(P,M,\sigma)$, where $P$ is a parabolic subgroup of $G$, $M$ is a ...
user449595's user avatar
4 votes
1 answer
355 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
4 votes
0 answers
213 views

Gelfand-Kazhdan criterion, exposition by Paul Garrett

Here is Paul Garrett's exposition on the Gelfand-Kazhdan Criterion. In page 4 of the exposition, he showed the following lemma. Lemma (Page 4): Let $B, t, S$ be as above and for $\alpha, \beta$ in $...
L-JS's user avatar
  • 43
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
510 views

Parahorics and their normalizers

Let $G$ be a reductive group over a local non-arch field $F$. For convenience let's assume $G$ has anisotropic center (or even that $G$ is semisimple if preferred). Let $x$ be any point in the ...
Cheng-Chiang Tsai's user avatar
3 votes
1 answer
327 views

Pro-unipotent radical (in Bruhat-Tits) vs. unipotent radical (and reference request)

I'm a beginner in Bruhat-Tits theory, and the following phenomenon makes me puzzled. So let $F$ be a $p$-adic field, with $\mathfrak{o}\supset \mathfrak{p}$ its ring of integers and maximal ideal, and ...
youknowwho's user avatar
3 votes
1 answer
170 views

Factoring out an element of a root subgroup to make a conjugation integral

Fix a nonarchimedean local field $L/\mathbb{Q}_p$ with ring of integers $\mathcal{O}$, uniformizer $\varpi$, and residue field $k$. If I take a matrix $$\begin{pmatrix} a & \varpi b \\ c & d \...
Ashwin Iyengar's user avatar
3 votes
1 answer
181 views

Does $\mathrm{Ext}^i_G(\pi,\pi')$ vanish if $\pi$ and $\pi'$ are smooth irreducible representations of $G$ with different central characters?

Let $G$ be a $p$-adic group, ie. the group of $F$-points of some connected reductive group over $F$, where $F$ is a $p$-adic field. We consider complex smooth representations of $G$. Any irreducible ...
Suzet's user avatar
  • 769
3 votes
1 answer
339 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 ...
Zhiyu's user avatar
  • 6,622
3 votes
1 answer
408 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
2 votes
1 answer
203 views

Orbital integrals of $\operatorname{SL}_2$ and the fundamental lemma

$\DeclareMathOperator\SL{SL}$When I was checking some orbital integral computations of Sally-Shalika's The Plancherel Formula for $\SL_2$ over a Local Field, Proceedings of the National Academy of ...
youknowwho's user avatar
2 votes
0 answers
44 views

Nonvanishing of Jacquet modules of principal series subquotients

Let $G$ be a connected reductive group over a $p$-adic field $F$. For simplicity, we assume $G$ to be split. Fix a Borel $B=TU$ with its Levi decomposition ($U$ unipotent radical, $T$ maximal torus). ...
youknowwho's user avatar
2 votes
0 answers
76 views

Simplest way to classify reducibility of principal series for $p$-adic $\mathrm{SL}_2$

Let $F$ be a $p$-adic field and $G_1=\mathrm{SL}_2(F)\subset \mathrm{GL}_2(F)=G$. For simplicity, we assume $p>2$. Denote by $|\cdot|$ the normalized absolute value on $F$. Here I shall focus on ...
youknowwho's user avatar
2 votes
0 answers
415 views

Algebraic characters and quasi-characters of reductive algebraic group over non-archimedean local field

Let $G$ be a reductive algebraic group over $F$, where $F$ is a non-archimedean local field. Then $G(F)$ is a p-adic group. Let $\Psi(G)$ be the lattice of algebraic characters. Let $\Lambda_G$ be the ...
JJH's user avatar
  • 1,457
1 vote
1 answer
252 views

Can we compare $K$-spherical representations of $p$-adic groups for varying special maximal subgroups $K$?

In this question, I'm borrowing the notations from Minguez' paper on unramified representations of unitary groups. Let $F$ be a $p$-adic field and let $G$ be a connected reductive group over $F$. Let $...
Suzet's user avatar
  • 769
1 vote
1 answer
112 views

Continuity of central character [closed]

Let $G$ be a $p$-adic reductive group and $Z\subseteq G$ the center. Let $\pi$ be an irreducible admissible representation of $G$. By Schur's lemma, it is easy to show that there is a group ...
Windi's user avatar
  • 833
1 vote
0 answers
77 views

Asymptotic behavior of Shalika germs near non-regular elements

Let $G$ be a connected reductive group over a $p$-adic field $F$. Let $T\subset G$ be a maximal torus. Fix a special maximal compact subgroup $K$ of $G(F)$ and for any closed subgroup $H\subset G(F)$ ...
youknowwho's user avatar
1 vote
0 answers
59 views

Distributivity property for smooth parabolic induction

Let $G$ be a reductive group over a local field $k$ of characteristic zero with maximal split torus $T$, Weyl group $W$, Borel $ B$ and a parabolic subgroup $P$ such that $P\supset B \supset T$. ...
KKD's user avatar
  • 473
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
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
  • 6,180