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Reference Request: Test vectors for local Rankin-Selberg L-factors in ramified cases

Let $F$ be a global number field, i.e. a finite extension of the field of rational numbers. Let $\sigma$, $\pi$ be automorphic representations of $\mathrm{GL}_n(F)$ and $\mathrm{GL}_{n+1}(F)$ ...
Hetong Xu's user avatar
  • 639
1 vote
0 answers
158 views

What is the "weight" of an automorphic form for $\mathrm{PGL}_2$?

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PGL{PGL}$I'm trying to understand what the notion of "weight" is for automorphic forms over $\GL_2(F)$ where $F$ is some number field, in ...
HASouza's user avatar
  • 423
1 vote
1 answer
155 views

Iwahori action on the $p$-ordinary line of a principal series representation

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\diag{diag}\DeclareMathOperator\Ind{Ind}\newcommand\Iw{\mathrm{Iw}}\DeclareMathOperator\ord{ord}$Let $F$ be a $p$-adic local field, i.e. a finite ...
Hetong Xu's user avatar
  • 639
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's user avatar
1 vote
0 answers
91 views

Explanation about Lapid-Rallis iductive argument (doubling method)

I am reading Lapid-Rallis "On the local factors of representations of classical groups" and I am completely stuck with the proof of Proposition 3. In the case $\mathcal V$ is not anisotropic,...
ahw's user avatar
  • 11
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's user avatar
8 votes
1 answer
452 views

Characterization of automorphic discrete spectra

I recently learned about automorphic spectral decomposition from the book "Spectral decomposition and Eisenstein series" by Moeglin and Waldspurger. (Let me call it M-W) I have a question ...
user avatar
23 votes
2 answers
859 views

Moments of Plücker coordinates on complex Grassmannian

Consider the Grassmannian $Gr(k,N)\simeq U(N)/(U(k)\times U(N-k))$ which parametrizes $k$-dimensional subspaces of $\mathbb{C}^N$. Let us put on it the $U(N)$-invariant probability measure. Let $\...
Abdelmalek Abdesselam's user avatar
2 votes
2 answers
474 views

For what automorphic representations is Ramanujan-Petersson known?

I had in mind that Ramanujan-Petersson conjecture was essentially unknown in the case of number fields. I however recently heard that If an automorphic representation on $GL(2)$ is ramified at a ...
Gory's user avatar
  • 609
6 votes
1 answer
361 views

Decay of matrix coefficients of non-tempered representation

A theorem of Cowling--Haagerup--Howe gives an effective decay rate of the matrix coefficients of a tempered representation $\pi$ of a semi-simple algebraic $G$ in terms of Harish-Chandra $\Xi$ ...
Subhajit Jana's user avatar
6 votes
2 answers
453 views

Reduction to Lie algebra version of fundamental lemma?

Ngo famously proved the Langlands-Shelstad fundamental lemma for Lie algebras using the geometry of the Hitchin fibration. For the purposes of the trace formula, one actually needs the fundamental ...
Spencer Leslie's user avatar
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's user avatar
  • 3,799
3 votes
1 answer
270 views

Eisenstein series for discrete subgroups of SL(2,C)?

I am looking for a reference for Eisenstein series for discrete subgroups of $SL(2,\mathbb C)$, in particular, finite index subgroups of $SL(2,\mathcal O_K)$ where $K$ is an imaginary quadratic field. ...
Tian An's user avatar
  • 3,799
18 votes
2 answers
1k views

What is the archimedean Hecke algebra?

Let $\mathbf G$ be a connected, reductive group over $\mathbb Q$. For each nonarchimedean place $v$, let $K_v$ be a maximal compact subgroup of $\mathbf G(\mathbb Q_v)$. The space $\mathscr H(\mathbf ...
D_S's user avatar
  • 6,180
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's user avatar
  • 5,382
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's user avatar
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's user avatar
  • 6,180
4 votes
2 answers
829 views

Reference Request: Definition of Induced Representation for reductive groups over a local field

Let $G$ be a connected, reductive group over a local field $F$ of characteristic zero, and $H$ a closed subgroup of $G$ which is defined over $F$. Let $\mu_H, \mu_G$ be right Haar measures on $H(F), ...
D_S's user avatar
  • 6,180
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's user avatar
4 votes
1 answer
421 views

Birch's conjecture from Representation Theory

Birch has a conjecture about which automorphic forms on $PGL(2)$ are the lifts from nonsplit $O(3)$. Temporarily ignore global issues, and focus on the local nonarchimedian picture. The automorphic ...
Watson Ladd's user avatar
  • 2,429
4 votes
1 answer
260 views

Reference request: normalization of intertwining operators for GL(2, C)

Take $F$ a local field and $\chi_1, \chi_2$ two characters, write $M(\chi_1, \chi_2)$ for the standard intertwining integral $$M(\chi_1. \chi_2).f(g) := \int_{F} f\left( \begin{pmatrix} 0&-1\\ 1&...
Joseph Hundley's user avatar
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's user avatar
  • 3,799
2 votes
0 answers
245 views

Reference request: proofs of the theorems in the paper "On the representation of the group GL(n, K) where K is a local field"

In the paper On the representation of the group $GL(n, K)$ where $K$ is a local field by Gelfand and Kazhdan, it is said that the proofs of the theorems in the paper are published in some other papers....
Jianrong Li's user avatar
  • 6,201
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's user avatar
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's user avatar
  • 1,980
7 votes
1 answer
788 views

What is the support of the Whittaker function of a new vector on GL(2)?

Let $W$ be the normalized Whittaker function associated to a new vector in an irreducible generic representation $\pi$ of $G=GL_2(k)$, where $k$ is a $p$-adic field. Let $c$ be the conductor of $\pi$, ...
B R's user avatar
  • 3,183
2 votes
1 answer
632 views

Pseudo coefficients and orbital integrals

I am looking for a reference/idea, how this passage from Labesse's Snowbird Lecture "Introduction to endoscopy" pg.5 can be explained: "We shall denote by $f_\pi$ a pseudo-coefficient for $\pi$, ...
Marc Palm's user avatar
  • 11.2k
26 votes
3 answers
5k views

Questions about the Bernstein center of a $p$-adic reductive group

Dear all, The "Bernstein center" of a $p$-adic reductive group appears frequently in the literature of automorphic forms, often without a precise definition. For example, in page 233 of Moeglin-...
user4245's user avatar
  • 809
14 votes
2 answers
3k views

What is the non-motivic motivation behind automorphic representations?

In one of my last questions: What is the "reason" for modularity results? it was pointed out to me that "the notion of automorphic representation developed independently of any concern with ...
James D. Taylor's user avatar
51 votes
7 answers
11k views

How is representation theory used in modular/automorphic forms?

There is certainly an abundance of advanced books on Galois representations and automorphic forms. What I'm wondering is more simple: What is the basic connection between modular forms and ...
David Corwin's user avatar
  • 15.4k
5 votes
1 answer
855 views

Rallis inner product formula for U(2,2) and U(3)

Victor Tan has a couple of papers on a regularized Siegel-Weil formula for U(2,2) and U(3). The papers I'm talking about are: "A Regularized Siegel-Weil Formula on U(2,2) and U(3)", Duke, 1998. "An ...
Neal Harris's user avatar