Unanswered 'rt.representation-theory+automorphic-forms' Questions

13 votes

0 answers

523 views

Euler Subgroups and Automorphic L-functions

Recently, I have read about the Whittaker expansion for $\mathrm{GL}_n$ and was struck by the utility of the mirabolic subgroup, $\mathrm{P}_n\subset \mathrm{GL}_n$ of matrices with bottom row $(0\; 0 ...

Spencer Leslie

2,516

asked Feb 6, 2015 at 16:15

11 votes

0 answers

283 views

Why are there so few irreducible admissible representations of $\text{GL}(n,\mathbb{R})$ (up to infinitesimal equivalence)?

Studying Langlands's classification of irreducible admissible representations, I have been rather stunned by the following: Theorem Up to infinitesimal equivalence, all irreducible admissible ...

Daniel Miller

511

asked Oct 17 at 21:50

11 votes

0 answers

359 views

Representation theory of $\operatorname{GL}_2(\mathbb Z/n\mathbb Z)$

Is there a nice reference for the finite dimensional (characteristic 0) representation theory of $\operatorname{GL}_2(\mathbb Z/n\mathbb Z)$ and $\operatorname{PGL}_2(\mathbb Z/n\mathbb Z)$ for ...

Asvin

7,746

asked Aug 12, 2020 at 16:36

11 votes

0 answers

332 views

Fourier Transforms of Convolutions

Straightforward computations lead to the following standard property of Fourier transformation: it transforms convolutions into products, i.e. for functions $f$ and $g$ Schwartz class we have $$\...

Desiderius Severus

5,424

asked Jan 7, 2018 at 9:41

9 votes

0 answers

233 views

A detail in I. Piatetski-Shapiro and S. Rallis's "Doubling paper": computing the integral on negligible orbits

I'm currently reading the paper "L-functions for the classical groups" by I. Piatetski-Shapiro and S. Rallis, where they introduced the doubling method over classical groups. I'm confused at ...

Hetong Xu

639

asked Jan 13, 2023 at 12:29

9 votes

0 answers

210 views

Why and how is a representation "continuously decomposable"?

What I am asking may apply to a much more general setting and I am interested in the underlying level of generality of these statements, mostly with canonical references. However I state the question ...

Desiderius Severus

5,424

asked Jun 20, 2022 at 9:06

8 votes

0 answers

481 views

Formal degree of discrete series representations

Let $G$ be a locally compact unimodular group. A continuous irreducible unitary representation $\pi$ of $G$ is said to be a discrete series if its matrix coefficients (defined by $\xi^\pi_{v,w} : g \...

Desiderius Severus

5,424

asked Mar 25, 2021 at 6:55

8 votes

0 answers

265 views

$L^2$ norms of Whittaker vectors and zeros of Intertwining operators

For $\mu,\nu\in \mathbb{C}^2$ we denote $I(\mu,\nu)$ to be the principal series of $\mathrm{GL}_2(\mathbb{Q}_p)$ induced from $|.|^\mu\otimes |.|^\nu$. For $s=\mu-\nu$ one defines the standard ...

Subhajit Jana

1,692

asked Dec 7, 2016 at 8:27

7 votes

0 answers

137 views

Computing explicit matrix coefficients

I would like to understand in a more explicit way the Fell topology on unitary duals, that is to say the convergence of matrix coefficients of local representations. If I consider a local ...

Desiderius Severus

5,424

asked Oct 18, 2020 at 2:43

7 votes

0 answers

597 views

Reference for shtuka and trace formula

I really want to learn the work of Laurent Lafforgue and the joint work of Zhiwei Yun and Wei Zhang. They both involve shtuka and trace formula, which I only know the basic idea. So I would like to ...

user330928

409

asked Jul 26, 2017 at 0:08

7 votes

0 answers

140 views

Algebraic construction of the modular representation of $\mathrm{SL}_2(\hat{\mathbf Z})$

The answer to this question is probably to be found in the theory of automorphic forms, but (I don't know much about it and consequently) after some tries, I did not catch it. Thus I'd be grateful if ...

few_reps

1,980

asked Apr 8, 2014 at 16:35

6 votes

0 answers

268 views

Correspondence between motives and automorphic representations

What I know: I understand motives via its realization; in Coates' and Perrin-Riou's paper On $p$-adic L-functions Attached to Motives over $\mathbb{Q}$ (see http://doi.org/10.2969/aspm/01710023), the ...

Maty Mangoo

748

asked May 2, 2023 at 12:50

6 votes

0 answers

239 views

Direct sum decomposition of the space of cuspidal automorphic forms

$\newcommand{\G}{\mathbb{G}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\A}{\mathbb{A}} \newcommand{\Autom}{\mathcal{A}} \newcommand{\cen}{\mathcal{Z}} \newcommand{\lieg}{\...

Aurel

5,382

asked Apr 5, 2018 at 9:27

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

91

asked Feb 15, 2018 at 6:09

6 votes

0 answers

1k views

Definition of Admissible Representation

Let $G$ be a connected, reductive group over a number field $k$. Let $v$ be a place of $k$. If $v$ is finite, an admissible representation of $G(k_v)$ is defined to be an abstract representation of $...

D_S

6,180

asked May 11, 2017 at 15:46

6 votes

0 answers

320 views

Is the space of global Whittaker functions complete?

Let $f$ be a complex valued function of $GL_n(\mathbb{A})$, where $\mathbb{A}$ is the adeles of some number field. Assume $f(ug)=\psi(u)f(g)$ for any $u$ in the standard maximal unipotent subgroup $...

user1832

2,709

asked Aug 4, 2011 at 16:02

5 votes

0 answers

220 views

Explicit Satake isomorphism

Consider the group $G=GL(n)$ over a non-Archimedean field and $K$ a maximal compact subgroup. Let $(\pi, V)$ be a smooth admissible irreducible representation of $G$, say spherical. I would like to ...

Wolker

551

asked Mar 29, 2022 at 14:34

5 votes

0 answers

213 views

Truncation and weighted orbital integrals in hyperbolic term of trace formula for $\mathrm{GL}(2)$

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PGL{PGL}$I am looking at Gelbart--Jacquet's article in the first Corvallis volume (the article entitled Forms of $\GL(2)$ from an analytic point of ...

babu_babu

241

asked Mar 20, 2021 at 3:03

5 votes

0 answers

388 views

Why admissible representations?

In the theory of automorphic forms we often right away reduce the study to admissible representations, and I wonder how much everything breaks when not. If I understood well, admissible ...

Wirdspan

181

asked Nov 1, 2020 at 3:16

5 votes

0 answers

215 views

Explicit description of the Plancherel measure for $GL_n(\mathbb{R})$

Let $G:=\mathrm{GL}_n(\mathbb{R})$ and $f\in C_c^\infty(G)$. One can uniquely determine the Plancherel measure $d\mu_p$ on $\hat{G}$, the unitary (actually tempered) dual of $G$, by the equation $$f(g)...

Subhajit Jana

1,692

asked Sep 17, 2016 at 16:15

5 votes

0 answers

359 views

Examples of Rankin-Selberg L-functions from Eisenstein series

I've been digging for awhile to not much success, so I figure I would try here: I am looking for some references which compute explicitly examples of Rankin-Selberg L-functions from the constant ...

Tian An

3,799

asked Jul 23, 2015 at 21:16

5 votes

0 answers

163 views

Uniqueness of cohomological holomorphic discrete series representation

In Claus Sorenson's PhD thesis, he proves a theorem about level lifting of paramodular forms whose associated automorphic representation has component $\pi_{\infty}$ that is the "cohomological ...

Watson Ladd

2,429

asked Jun 4, 2015 at 20:09

5 votes

0 answers

681 views

Base change and Langlands' combinatorial exercise

Is it correct that Langlands' combinatorial exercise (as he terms it in his paper "Shimura varieties and the Selberg trace formula") is to establish base change identities between orbital integrals of ...

Nicolás

2,842

asked Jun 9, 2011 at 14:07

4 votes

0 answers

155 views

How to obtain the harmonic theta series via the global theta correspondence explicitly?

I am interested in the following kind of modular forms: let $(L,q)$ be an even unimodular lattice inside $V:=L\otimes \mathbb{Q}$, and $P$ is a harmonic homogeneous polynomial of degree $d$ on $\...

Erica

391

asked Jun 24, 2022 at 12:37

4 votes

0 answers

149 views

Global Vogan A-packet is infinite set?

For an cuspidal automorphic representation of general linear group, we can attach its global Vogan A-packet. Though I thought that it is finite set, in some paper, it is written that there are ...

Andrew

1,019

asked Nov 27, 2021 at 7:55

4 votes

0 answers

135 views

Values at 1 of symmetric power L-functions of Maass cusp forms

I have a blur that whether one has $L(1,\text{sym}^2f)\ll \log^A q$ for some $A>0$? Here $f$ is assumed to be a Maass cusp form of square-free level $q$. If any experts here know something about ...

hofnumber

563

asked Aug 23, 2021 at 5:42

4 votes

0 answers

130 views

System of eigenvalues of partial Laplacians and the Jacquet Langlands correspondence

I'm looking for a precise reference. I could not dig it up in the original paper of Jacquet-Langlands. Let $F$ be a totally real number field with $[F:\mathbb{Q}]=r$, let $B_1=M_2(F)$ be the split ...

Hugo Chapdelaine

7,609

asked Oct 1, 2019 at 19:45

4 votes

0 answers

215 views

L-functions of tempered automorphic representations

Let $G$ be a reductive group over the adele ring $\mathbb A_F$ of a number field $F$. Let also $r$ be a complex finite dimensional representation of the $L$-group $r:{^LG}\to GL(V)$. It is generally ...

Tian An

3,799

asked Jan 9, 2019 at 20:51

4 votes

0 answers

398 views

Why Whittaker functions are useful?

Whittaker functions appears in Langlands program. Recently, it is shown that some Whittaker functions can be obtained by integrating a function related to decoration over a geometric crystal in http://...

Jianrong Li

6,201

asked Mar 19, 2014 at 12:10

4 votes

0 answers

169 views

parametrization of irreducible finite dimensional representation of Weil group

Let $F$ be a p-adic field, with p a prime denoting the residue field characteristic. Let $\mathcal{W}_F$ be the Weil group. In the local Langlands correspondence for $GL(n,F)$, it is important to know ...

user1832

2,709

asked Oct 18, 2013 at 9:09

4 votes

0 answers

218 views

Restriction of representations from $SO_{2n}$ to $SO_{2n-1}$ and $K$-fixed vectors

Let $F$ be a $p$-adic field, $G=SO_{2n}(F)$ the split special orthogonal group and $H=SO_{2n-1}(F)$, taken as a subgroup of $G$. Assume that we have an irreducible, admissible representation $\pi$ of $...

Vita Kala

179

asked Oct 14, 2012 at 15:23

3 votes

0 answers

91 views

Question on the genericity of unramified representation

Let $F$ be a $p$-adic local field and $W$ be a 2n-dimensional symplectic space over $F$. Let $G_n$ be the isometry group of $W$ and $B_n$ be the Borel subgroup of $G_n$. Then the maximal torus $T_n$ ...

Andrew

1,019

asked Mar 11 at 8:45

3 votes

0 answers

242 views

Explicit expression of automorphic representations as automorphic forms

Let‘s take $G=GL_n$ over a number field $F$ for example. It's already known that every irreducible automorphic representation $\pi$ is a irreducible component of a induced representation $I(G,P;\...

Adjoint Functor

111

asked Nov 24, 2022 at 21:49

3 votes

0 answers

166 views

Relation between automorphic representations and Laplacian eigenvalues

Let $G=SL(2, \mathbb{R})$. I would like to understand how and why automorphic forms correspond to automorphic representations, and how general is this fact (for other groups). Let $\Gamma$ be a ...

Wirdspan

181

asked Sep 22, 2020 at 3:26

3 votes

0 answers

95 views

Principal series representation of $SL(2,\mathbb{R})$ restricted to principal congruence subgroup

Given a principal congruence subgroup $\Gamma(N)$ of $SL(2,\mathbb{R})$, since $\Gamma(N)$ is free, consider a probability distribution $\mu_1$ of a simple random walk on $\Gamma(N)$ and consider its ...

user158970

131

asked Sep 15, 2020 at 1:30

3 votes

0 answers

156 views

Question on the proper sub-representation of induced representation

$\DeclareMathOperator\Ind{Ind}$Let $G$ be a reductive group over a $p$-adic local field $F$, and $P=MN$ a parabolic subgroup. Let $\sigma$ be an irreducible representation of $M(F)$ and consider its ...

Monty

1,759

asked May 22, 2020 at 13:53

3 votes

0 answers

153 views

Global Arthur packet consist of only globally generic representations?

I would like to ask very stupid two questions to experts. I am wondering whether every globally generic automorphic representation of unitary groups are contained some global Arthur packet associated ...

Monty

1,759

asked Mar 23, 2020 at 20:18

3 votes

0 answers

166 views

Automorphy Factor from Vector Bundles on Compact Dual

So I'm coming from an algebraic geometry perspective and I'm trying to carefully piece together the story of interpreting automorphic forms as sections of vector bundles on Shimura varieties. I think ...

Benighted

1,701

asked Mar 15, 2019 at 17:35

3 votes

0 answers

255 views

Some basic question on the parabolic induction

I would like to ask some basic question about parabolic induction. Let $F$ be a local field and $G=GL_n(F)$ and $P=MN$ its parabolic subgroup whose Levis subgroup $M=GL_{n_1}(F) \times GL_{n_2}(F)$ ...

Monty

1,759

asked Jul 14, 2018 at 6:59

3 votes

0 answers

188 views

Non-vanishing of left- vs. right-averages over lattices in $SL(2,\mathbb{R})$

I asked the same question on MSE one week ago, but it has not received any answers. Background. Let $G=SL(2,\mathbb{R})$, let $K=SO(2)$, and let $\Gamma$ be a lattice in $G$, e.g. $SL(2,\mathbb{Z})$...

L.C. Ruth

229

asked Aug 13, 2017 at 17:49

3 votes

0 answers

272 views

Spectral decomposition on GL(n)

If $\Delta_1, \ldots, \Delta_{n-1}$ are a basis of the ring of commuting bi-$SL(n,R)$-invariant differential operators, $L_0^2=L_0^2(SL(n,Z)\backslash SL(n,R))$ is the space of cuspidal automorphic ...

Jack Buttcane

151

asked Jul 13, 2016 at 3:23

3 votes

0 answers

132 views

Newvectors in tensor product representations

Let $\pi_p$ and $\pi_p^\prime$ two smooth admissible irreducible complex representations of ${\rm GL}_2(F)$ where $F$ is a non archimedean local field of residual characteristic $p$ of central ...

AdLibitum

221

asked Nov 27, 2014 at 19:31

3 votes

0 answers

651 views

Constant terms of Eisenstein series and Gindikin-Karpelevich formula

Let $G$ be a split reductive group over $\mathbb{Q}$ and $P=NM$ be a standard parabolic subgroup of G. Let $\chi$ be an unramified character of $M$ and $f_\chi$ be the spherical section of the ...

John

273

asked May 9, 2014 at 13:37

3 votes

0 answers

173 views

Bounding global matrix coefficient for PGL_2

I'm trying to find a reference that gives a bound for the adelic matrix coefficient for $\text{PGL}_2$ using the bound towards Ramanujan conjecture. More specifically: Let $G=\text{PGL}_2$. Let $F$ ...

Yueke Hu

31

asked Feb 4, 2014 at 18:05

3 votes

0 answers

158 views

Bessel function for $GL_3(\mathfrak{R})$?

In the $GL_2(\mathfrak{R})$ case, assume that $\pi$ is an irreducible unitary representation and $W_{\pi}(g)$ is the Whittaker functional associates with $\pi$. Then there is a Bessel function $j_{\pi}...

Aalogon

31

asked Aug 22, 2013 at 5:43

3 votes

0 answers

740 views

The operator \boxtimes and \boxplus in automorphic representations

Given two automorphic representations $\pi_1, \pi_2$ of $GL_2(\mathbb A_Q)$ and $GL_3(\mathbb A_Q)$ respectively. Let $\pi_i =\otimes_v \pi_{i, v}$. Now, for each $v$, let $\pi_{1, v}\boxtimes \pi_{...

boxtimes

31

asked Feb 11, 2013 at 3:11

3 votes

0 answers

522 views

On the theta lift and its L-function

I am wondering how the relation is between of the automorphic L-function and its lift's. More precisely, Let $E/F$ be a quadratic extension of number fields. Let $W$ be a hermitian space over E of ...

Jude

263

asked Jan 19, 2013 at 6:51

2 votes

0 answers

99 views

Factorization of global Waldspurger's integrals and connection to central L-values

Let $\pi$ be the irreducible cuspidal automorphic representation of $\mathrm{GL}_2$. Let $E/F$ be a quadratic extension with given embedding $E^{\times} \to \mathrm{GL}_2(F)$. For $f_1 \in \pi$, $f_2 \...

Alvin

21

asked Sep 5 at 20:02

2 votes

0 answers

76 views

Simple question on the genericity of induced representation

$\DeclareMathOperator\GL{GL} \DeclareMathOperator\Sp{Sp} \DeclareMathOperator\Ind{Ind}$ Let $F$ be a $p$-adic field and $\Sp(2n)$ symplectic group over 2n dimensional symplectic space over $F$. Let $B=...

Andrew

1,019

asked Feb 20 at 17:10

2 votes

0 answers

155 views

Meaning of the meromorphic continuation of intertwining operators

I am trying to make sure the meaning of the meromorphic continuation of the intertwining operators. Assume we deal with a non-Archimedean field $F$ and just consider $G= SL_2$, for simplicity. We fix ...

Qingzhi Li

111

asked Feb 2, 2023 at 9:17

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