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3
votes
0answers
41 views

Intertwining Operators Associated to Simple Reflections

Let $G$ be a quasi-split reductive group, over a local field, with a Borel subgroup $B=T\cdot N$ and the associated Weyl group $W$. Given a family of induced representations $\pi_s = Ind_B^G \chi\cdot ...
7
votes
1answer
476 views

Which L-functions are not “Langlands-Shahidi L-functions”?

The Langlands-Shahidi method, among other things, obtains certain L-functions from the constant term of Eisenstein series attached to so-called $(G,M)$ pairs, where $G$ is a reductive group, $M$ a ...
5
votes
1answer
192 views

Complete L-function and FE of Rankin-Selberg on GL(2)?

Let $f$ be a Maass cusp form of $\Gamma_0(N)$ on the upper half plane with character $\chi$ mod $N$ and eigenvalue $1/4+\mu^2$. What is the complete $L$-function of the Rankin-Selberg product ...
5
votes
2answers
229 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 ...
1
vote
0answers
74 views

Sign of the functional equation of L function and Shimura lift

I would like to know what happens to the root number of a half integral weight automorphic form (holomorphic or not), i.e. the sign of the functional equation of its L-function when we apply the ...
2
votes
0answers
173 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 ...
1
vote
0answers
135 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 ...
15
votes
3answers
650 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 ...
4
votes
1answer
212 views

Half integral weight Hecke operators

I would like to find a source giving the exact formula for the product of two Hecke operators $T_{\kappa}(n^2)$ and $T_\kappa(m^2)$ of half integral weight. That is, $\kappa \in \frac 12 \mathbb{Z} - ...
1
vote
1answer
246 views

Is there a non-tempered representation of U(2)?

I am wondering why the first well known example of non-tempered irreducible admissible representation of $p$-adic group $U(n)$ should be $U(3)$. Because, Gelbart and Rogawski suggested the ...
7
votes
1answer
218 views

Geometric interpretation of Cusps for general groups?

Let $\mathrm{G}$ be a reductive group over a number field $F$, but for simplicity we can think about $\mathrm{G}=\mathrm{GL_n}$ for $n>2$ and $F =\mathbb{Q}$. Then for an automorphic form, ...
13
votes
2answers
656 views

What's the status of Arthur's announced classification for GSp(4)?

In "Automorphic representations of GSp(4)" (2004) (see http://www.math.toronto.edu/arthur/), James Arthur announces a classification of discrete automorphic representations of GSp(4). There are no ...
0
votes
0answers
67 views

On continuous part of the L^2 spectrum

Suppose $G$ is a real reductive Lie group and $\Gamma$ is a lattice in $G$ (of finite co-volume). I am reading Langlands's paper " On the functional equation satisfied by the Eisenstein series". I ...
24
votes
1answer
1k views

Underlying idea for (automorphic) L-function?

Edit: So with a few more months of math under my belt, I recognize some of the issues with this question. I still hope for an answer, so let me say a few things. Within the Langlands philosophy, ...
5
votes
0answers
140 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 ...
4
votes
0answers
55 views

Global theta series on GL

Let $E/F$ be a quadratic extension of number field and let $V$ be a Hermitian space over $E$. Then we have Weil representations for the dual pair $U(n,n)\times U(V)$, and we can consider the theta ...
2
votes
1answer
164 views

Rankin-Selberg convolution and product of degrees

As I'm kinda obsessed with the Selberg class and because of the general converse conjecture, I'm still trying to get a rough idea of what automorphic representations and their L-functions as well as ...
1
vote
0answers
76 views

About Theorem $3.1.3$ in Kubota's book: Elementary theory of eisenstein series

My question is about the proof of Theorem $3.1.3$ given in kubota's book, which shows how the function $\varphi(s)$ appearing in the Fourier expansion of eisenstein series can be continued ...
4
votes
1answer
157 views

Is the restriction of a representation semisimple?

Let $F$ be local field of characteristic zero and $\pi$ be a irreducible admissible representation of $GL_n(F)$. Let us consider its restriction to $GL_{n-1}(F)$. Then I want to know whether ...
4
votes
1answer
1k views

Corvallis 1979 proceedings

These proceedings have long been freely available on the AMS website, but now it seems we can't even find them anymore (e.g. http://www.ams.org/publications/online-books/pspum331-index and ...
8
votes
1answer
310 views

Self-dual automorphic forms on $GL(4)$

As is known among experts, all self-dual automorphic forms on $GL(3)$ come from symmetric square lifts from $GL(2)$. You can find this in Ramakrishnan ...
50
votes
7answers
4k views

What are the local Langlands conjectures nowadays, for connected reductive groups over a $p$-adic field?

Let me stress that I am only interested in $p$-adic fields in this question, for reasons that will become clear later. Let me also stress that in some sense I am basically assuming that the reader ...
3
votes
0answers
214 views

On the local root number(or local $\epsilon$-factor)

I want to ask some question related to the local root number. Let $E/F$ be a quadratic extension of p-adic local fields and $\psi:E \to \mathbb{C}$ is an additive character of $E$. Let $\phi:WD(E) ...
0
votes
0answers
72 views

faithful action of Hecke algebra

Let $G$ be a connected reductive group split over a number field $F$, $\mathbb{A}$ the adeles. Let $v$ be a finite place and $\mathcal{H}_{v}$ the spherical hecke algebra at palce $v$. ...
3
votes
0answers
92 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 ...
2
votes
0answers
94 views

“simulteneous eigenvectors” under the full set of weighted Laplacians on a $g$-fold product of the Poincare half plane

This question is closely related to the following MO question Characterizing the real analytic Eisenstein series Let $\mathfrak{h}=\{z=x+iy\in\mathbf{C}\}$ be the Poincare upper half plane endowed ...
10
votes
1answer
403 views

Reference for: CM Hilbert Modular forms arise from Hecke characters

For classical modular forms, the correspondence between the form having CM by an imaginary quadratic field $K$ and it being induced from a Hecke character on $K$ is well-known. (Ribet's paper is a ...
6
votes
2answers
474 views

Characterizing the real analytic Eisenstein series

Consider the classical real analytic Eisenstein series $$ E(z,s)=\left(\pi^{-s}\Gamma(s)\frac{1}{2}\right)\sum_{(m,n)\neq(0,0)}\frac{y^s}{|mz+n|^{2s}}, $$ where $z=x+iy$. We think of $E(z,s)$ as a ...
2
votes
0answers
117 views

Functoriality for non-split orthogonal groups

I am trying to understand the functoriality conjectures of Langlands. We know that the functoriality conjectures imply that automorphic $L$-functions of a connected reductive group are equal to ...
4
votes
1answer
171 views

Voronoi formula and twists by additive characters

I was wondering if there are any references for the error term in the problem $$\sum_{n\leq x} r(n) \exp(2\pi i\frac{a}{q}n)$$ where $r(n)$ is the number of representations of $n$ as a sum of two ...
3
votes
0answers
86 views

Oscillatory integral moments of $L(\frac{1}{2} + it, f \times f)$

Understanding moments and subconvexity bounds for $L$-functions is a big topic with a lot of activity. I'm currently looking at a related problem, bounding $$ \int_0^T L\left(\tfrac{1}{2} + it, f ...
7
votes
0answers
275 views

Lindelof Hypothesis implying Selberg Eigenvalue Conjecture?

(General) Lindelof Hypothesis which says for any $L$-function we have $$L(1/2+it)\ll Q(t)^{\epsilon}$$ for any $\epsilon>0$ where $Q(t)$ is the conductor of $L(s)$ at $t$. For a Maass form $\phi$ ...
8
votes
1answer
302 views

Regularity assumption in the simple trace formula

In the simple trace formula of Deligne Kazhdan one assumes that the test function is supported at the elliptic regular elements at one place and is a supercusp form at another place. Why can't one ...
6
votes
1answer
423 views

Generalization of Watson's triple product

In Watson's thesis (page 51) we can find his beautiful triple product formula. My question is that does there exist a generalization of this formula? By generalization, I mean: If $\phi_n$'s are ...
4
votes
1answer
175 views

Asymptotic behaviour of $K$-Bessel function in transition range

It is known that the famous mistake of Iwaniec-Sarnak in their paper of $L^\infty$ norm of eigenfucntion of non-cocompact arithmetic surfaces in lemma (A1) is because of they did not consider the bump ...
1
vote
1answer
180 views

About Section 4.2 in 'introduction to the spectral theory of automorphic forms' by Iwaniec

In Section 4.2 in 'introduction to the spectral theory of automorphic forms' by Iwaniec, the author said that the kernel of the invariant integral operator $$ (Lf)(z)=\int_{\mathbb{H}}k(z,w)f(w)d\mu w ...
5
votes
1answer
173 views

The sixth power integral moment of automorphic L-function attached to Maass Forms

It is known that the sixth power integral moment of automorphic L-function attached to Cusp Forms has been proved by M. Jutila, that is $\int_{0}^{T}|L(1/2+it,f)|^{6}dt \ll T^{2+\varepsilon}$. And ...
1
vote
1answer
111 views

On the Saito Kurokawa representation

I know Saito-Kurokawa(SK) representation is the famous non-tempered representation of $SO(5)$. But since the tempered or non-tempered terms are concerned with local phenomenon, I am wondering that ...
3
votes
1answer
131 views

Lower bound of first moment of $L$-function on $\mathrm{GL}(3)$

Let $\pi$ be an automorphic form on $\mathrm{GL}(3,\mathbb{A}_{\mathbb{Q}})$. Do we know any case that \[\int_0^{T} \left|L(\frac{1}{2} + it, \pi)\right| dt \gg T\] holds unconditionally? I know the ...
2
votes
1answer
194 views

Lower bound of Hecke eigenvalues of Maass form

If $f$ is a Maass form and $p$-Hecke eigenvalue (i.e. Hecke eigenvalue of usual Hecke operator $T_p$) of $f$ is $\lambda_f(p)$, do we know anything about lower bound of the sum$$S(x) = \sum_{x\le p\le ...
9
votes
0answers
141 views

Weyl law for Maass forms with nontrivial character

The classical Weyl law for $\Gamma = \mathrm{SL}_2(\mathbb{Z})$ counts the number of Maass cusp forms on $\Gamma \backslash \mathbb{H}$ with Laplace eigenvalue less than $T$. This is originally due to ...
1
vote
0answers
171 views

on the Rankin-Selberg L-function

Let $n,m$ be two different positive integers. I heard that for cuspidal tempered automorphic representations $\pi_{n}$ and $\pi_m$ of $GL_n$ and $GL_m$, the Rankin-Selberg L-function $L(s,\pi_n ...
1
vote
1answer
130 views

On the reductive group [closed]

I know that the automorphic representation can be defined only for reductive algebraic group. What property of algebriac group makes it hinder to define for all algebraic group and what nice property ...
10
votes
1answer
372 views

Holomorphic cusp forms and cohomology of GL(2,Z)

Let $V_{k}$ denote the complex representation of $\mathrm{GL}(2)$ given by $\mathrm{Sym}^k(V)$, where $V$ is the defining 2-dimensional representation. Assume that $k$ is even. I would like to compute ...
1
vote
0answers
50 views

limit of regular hyperbolic integrals is a unipotent integral (GL2 calculation)

In developing a simple trace formula for $G$=GL$_2$ over a number field $F$ one encounters the following identity of local integrals (for example, in Gelbart-Jacquet, 1979): $$\int_{Z_v N_v\backslash ...
0
votes
0answers
51 views

Definition of Degenerate hyper-Kloosterman Sum

If $(a,q)=1$, we all know the usual hyper-Kloosterman sum $$Kl_n(a,q)= \sum_{x_1...x_n\equiv a \mod q} e(\frac{x_1+\dots+x_n}{q}).$$ I wonder what would be the correct definition when $(a,q)\neq 1$. ...
3
votes
0answers
141 views

multiplicity of automorphic representation of unitary similitude group

Let $G$ be a unitary similitude group over $\mathbb{Q}$ (as in the book of Harris-Taylor), $\pi$ an irreducible automorphic representation of $G(\mathbb{A})$. I'm looking for some results on its ...
18
votes
2answers
1k 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 ...
7
votes
1answer
946 views

An interesting double coset in the theory of automorphic forms

Does anyone have some idea to describe the double coset $P(F)\backslash G(F)/H(F)$ , say using Weyl group elements ? Here $G=GL_n\times GL_{n-1}$ is defined over a number field $F$ , $H=GL_{n-1}$ ...
1
vote
1answer
127 views

On the pole of local L-function

Let $F$ be a number field and $v$ a finite place of $F$. Let $\chi_v$ be a unramified unitary character of $F_v$. Then we define local L-function $L_v(s,\chi_v):=\frac{1}{1-\chi_v(\omega)q^{-s}}$ ...