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4
votes
1answer
252 views

How to translate the representation theory of semisimple to reductive groups?

I am aware of the following question: Definitions of Reductive and Semisimple Groups So let me phrase a precise question: Is there a standard technique by which one can translate the ...
5
votes
0answers
349 views

a generalization of a formula of Shimura

Let $\phi$ be a $GL(2)$ automorphic form with Fourier coefficients $a(n)$ and $a(1)=1$. Obviously we have $L(s,\phi)=\sum \frac{a(n)}{n^s}$. Shimura have the following formula $L(s, Ad\; ...
3
votes
2answers
204 views

Restriction map between spaces of automorphic forms

Hello, Let $H \subset G$ be reductive groups defined over $\mathbb{Q}$. I consider the spaces of automorphic forms of $G$ and $H$. One has a restriction map from the space of automorphic forms of $G$ ...
1
vote
0answers
86 views

Theta lift to 1-dimensional vector space.

Hi! My question is very simple and it is about the theta lift of unitary group in global situation. Let $E/F$ be a quadratic number fields and $V,W$ be an $n$-dimensional and $1$-dimensional ...
0
votes
0answers
79 views

Global theta liftings with seesaw reductive dual pairs

Hello, I have the following problem : Let $(G,G')$ and $(H,H')$ be a pair of dual reductive pairs in a symplectic space $Sp(W)$ (in a global setting) forming a seesaw pair (with $H \subset G$ and $G' ...
0
votes
0answers
52 views

stablization of cuspidal terms

Kottwitz, building on the work of Langlands and Shelstad, gave a stabilization of the elliptic terms on the geometric side of the trace formula. This stabilization depended on the endoscopic transfer ...
5
votes
1answer
274 views

Cuspidal automorphic representations as the space of $K$-finite vectors in a unitary cuspidal automorphic representation.

The definition I know of for a cuspidal automorphic representation of, say, $G=\mathrm{GL}_2$ over a number field $F$ (relative to a choice of compact open subgroup $K_f$ of $G(\mathbf{A}_F^\infty)$ ...
2
votes
1answer
165 views

On the Cartan decomposition of unitary group

Hello. I have some question on Cartan decomposition of unitary group, especially $U(2)$. I am interested in local situation, that is p-adic or archimedian. Let $F$ be a local field and $E$ be its ...
2
votes
0answers
124 views

Conceptual reason behind Shimura lifts

Shimura lifts are correspondence between integer weight and half-integral weight automorphic forms. Half integral weight things are associated to representation of a double cover of $G ...
0
votes
0answers
280 views

Local Langlands conjecture for GL(2)

Let $F_v$ be a local field. Let $\sigma_v$ be a two-dimensional representation of $Gal(\overline{F}_v:F_v) \rtimes W_{F_v}$. Now, there exists an infinite-dimensional representation $\pi_v$ of ...
4
votes
2answers
172 views

No Exceptional Eigenvalues of Weight 1/2 Maass Forms on $\Gamma_0(4)$?

Some colleagues and I were wondering if there is a citation out there which shows there are no exceptional eigenvalues, $\lambda$, of classical weight 1/2 Maass forms on $\Gamma_0(4)$, which is to say ...
9
votes
1answer
363 views

Borel's Paris Lectures

I am trying to read Harish-Chandra's book on automorphic forms on Semisimple Lie groups, and he keeps referring to Borel's Paris lecture notes. Does anyone have an online version of these notes or ...
2
votes
0answers
86 views

Integral conjugacy vs. Rational conjugacy

Let $G$ be an algebraic group over a field $F$. Let $g\in G(F)$, and write $C(g)$ for the centralizer of $g$ in $G$. Conjugacy over $F$ is of course not necessarily the same thing as conjugacy over an ...
8
votes
2answers
302 views

central/critical/special values of L-functions terminology

I have a question about the terminology for special values of L-functions. Is the following a correct description of standard usage: Suppose L(s) is an L-function which satisfies a functional ...
1
vote
0answers
107 views

Off critical line zeros for half integer weight $L$-functions

Let $f(z) = \sum_{n=1}^\infty A(n)n^{\frac{k-1}{2}}e(nz)$ be a modular form of weight $k$ for a half integer $k$. Put $$L(s,f) = \sum_{n=1}^\infty \frac{A(n)}{n^s} $$ to be the $L$-function. Further ...
2
votes
1answer
131 views

On the absolute convergence of the local-zeta integral.

Though I am in a situation considering only local-zeta integral, to explain my question briefly, let me ask it in quite general form. Let $f(s,g)$ be a two variable smooth good (in a suitable sense) ...
1
vote
0answers
70 views

Non vanishing and cuspidality of the theta lift of trivial representation.

Hi! Let E/F be a quadratic number field extension. Then we make some hermitian and skew hermition vector spaces and define unitary group on it.(namely U(1) and U(3)) Then, I am wondering whether the ...
9
votes
2answers
1k views

New Geometric Methods in Number Theory and Automorphic Forms

The MSRI is organising a programme with the above title from Aug 11, 2014 to Dec 12, 2014. Here is a short description from their website : The branches of number theory most directly related ...
0
votes
0answers
91 views

On the explicit formula of the height function occuring on the doubled Weil representation.

Hi! I am wondering the exact formula of height function of $GL(n)$ which occurs in the doubling Weil representation. To be more precise, let me introduce the basic setting for this. Let $F$ be the ...
3
votes
1answer
325 views

Is Eisenstein series not identically zero

How does one prove that an Eisenstein series (adelically formulated as in the book of Moeglin-Waldspurger) is not identically zero? Namely how does one prove that the sum $\sum_{\gamma\in ...
1
vote
2answers
230 views

A question on twisted L-function

Hi! This is my first question here. In studying automorphic form, I am wondering the relation of critical L-values of some representation and its twisted representation by a character. For example, ...
26
votes
3answers
2k views

What is the difference between an automorphic form and a modular form?

This is more of a question about terminology than about math. The term "automorphic form" is clearly a generalization of the term "modular form." What is not clear is exactly which generalization it ...
3
votes
1answer
215 views

Volume of PGL(2,F) \ PGL(2, A)

Let $F$ be a global field. What is the measure of $PGL_2(F) \backslash PGL_2(\mathbb{A})$? This depends of course on the normalizations of the Haar measures on $PGL_2(F)$ and $PGL_2(\mathbb{A})$. ...
2
votes
1answer
159 views

Hecke eigenvalue at p and at p^k

I am interested in the relationship between the Hecke eigenvalue at $p$ and at $p^k$ for $k \geq 2$ in the unramified and ramified situation for modular/Maass forms. More precisely, I know from a ...
1
vote
1answer
91 views

Maass-Hecke construction

I heard this name that it can construct GL(2) automorphic forms or L-functions from GL(1)? I did not find it anywhere. Or does it have another name which we are familiar with?
2
votes
1answer
210 views

Arthur-Clozel Prop 3.1 for Function Fields?

The subject says it all. I would like to know if Proposition 3.1 in Arthur-Clozel's book on the trace formula holds for local fields of positive characteristic. Thanks! EDIT: Here is Prop 3.1 of ...
8
votes
2answers
817 views

Most understandable notes on Jacquet-Langlands?

I am particularly interested in the comparison of the trace formula part of Jacquet-Langlands. But I found the original text hard to read.
8
votes
1answer
341 views

Request for errata for Automorphic Forms on GL(2)

Hello everyone, I'm currently proofreading Hervé Jacquet's and Robert Langlands's Automorphic Forms on GL(2) for future republication. I was wondering if some of you had noticed mistakes in it that ...
3
votes
0answers
176 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 ...
2
votes
1answer
131 views

Refining the moderate growth condition on automorphic forms

Let $G$ be a simple algebraic group defined over $\mathbb Q$. In their Corvallis article (automorphic forms and automorphic representations), Borel and Jacquet define an automorphic form to be a ...
6
votes
1answer
513 views

A stupid question on automorphic l-function

This may be a silly question for experts in this area. But I am really suffering for not being able to compute local-L function of some automorphic representation. So, I post it hoping some ...
12
votes
1answer
921 views

Understanding the “idea” behind Langlands

Apologies in advance if this is a bit too simple to ask here, but I think I'm probably more likely to get an answer here than at stackexchange. I've been trying to learn the basics of the Langlands ...
1
vote
0answers
185 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 ...
4
votes
1answer
225 views

On the restriction of cusp. irr. representation and period.

Let's consider only global case. Let $G_n$ be classical algebraic group over global # field (eg, $GL(n),SO(n), U(n)$...) and let $\pi_n$ be its irr. cusp. reps of $G_n$. Then we can define the ...
1
vote
0answers
232 views

on the fundamental lemma

I consider the fundamental lemma for the spherical Hecke algebra. Let $G$ a connected reductive quasisplit group on $F$, a local field of equal characteristic $p$. and $H$ an endoscopic group. Can ...
1
vote
1answer
153 views

Global square integrability ensures local sq. integrability?

This might be a stupid question for expert in this area. I am considering automorphic representation of algebraic group. In studying it, local tempered, local square integrable representations ...
2
votes
2answers
312 views

a question about the proof of analytic continuation of Eisenstein series for GL(2)

I'm reading Gelbart and Jacquet's paper 'forms of GL(2) from the analytic point of view', and was confused at a point in the proof of analytic continuation of Eisenstein series. On the top of page ...
3
votes
1answer
215 views

Extending cuspidal representation to more bigger group.

I am thinking of extending an irreducible cuspidal representation to more bigger group. My question is almost same with the earlier one posted by Neal Harris except the only one. Let me first ...
13
votes
1answer
443 views

Is a reductive adelic group a Type I group?

I foresee that to experts of automorphic forms this question will sound unimportant or useless or even not worthy of an answer; but none of these are going to stop me from asking it! The question is ...
1
vote
1answer
323 views

On the Weil representation of unitary groups.

I suppose I am the first one who asked about Weil representation here. In studying Weil representation, I fell into a slough and so determined to ask you for shedding a light. I think your responses ...
2
votes
0answers
201 views

On the L-function of unique subrepresentation of induced representation.

In studying the L-functions of induced representation, it is not easily come up with me the papers or books dealing the L-function of irreducible subrepresentation of induced representation, while the ...
6
votes
1answer
275 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 $\pi$, ...
5
votes
0answers
134 views

On Langlands Pairing and transfer factors

In the paper "On the definition of transfer factors" Langlands and Shelstad define a certain number of factors $\Delta_{I}$, $\Delta_{II}$,$\Delta_{III,1}$,$\Delta_{III,2}$, which are roots of unity. ...
1
vote
0answers
155 views

On the Weil representation of U(1) and U(3).

Let $E/F$ quadratic extention number fields. Let $V$ be the $m$-dimension hermition vector space over $E$. Let $W$ be the $2n$-dimension skew hermitian vector space over $E$ and $Y_n + Y_n^*=W$ be ...
0
votes
1answer
202 views

Reference on Casselman-Shalika formula for GL(n) and PGL(n)?

I am looking for reference on Casselman-Shalika formula for GL(n) and PGL(n) at finite place p.
13
votes
3answers
552 views

Why only half-integral weight automorphic forms?

Why is that the theory of automorphic forms concentrates on the case of half-integral weight? I read in Borel's book "Automorphic forms on $SL_2$" (Section 18.5) that by considering the finite or ...
5
votes
2answers
634 views

Explicit examples of algebraic Hecke characters with infinite image?

Jerry Shurman has a lovely set of notes explaining the classical definition of Hecke characters, the idelic definition of Hecke characters, their relationship, and the classification of algebraic ...
10
votes
0answers
998 views

Why doesn't functoriality immediately imply the modularity theorem?

Let $E/\mathbb{Q}$ be an elliptic curve. By the modularity theorem, the prime indexed coefficients of its $L$-function agree with those of a weight $2$ cusp eigenform $f$ with integer coefficients. ...
1
vote
1answer
236 views

Does FE of Selberg Zeta function imply Trace formula?

Does the functional equation of the Selberg Zeta function imply the Selberg trace formula? BTW, the trace formula implies the functional equation.
4
votes
0answers
147 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 ...