All Questions
Tagged with rt.representation-theory automorphic-forms
248 questions
5
votes
1
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230
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Explicit Jacquet-Langlands correspondence for real reductive groups
Let $G$ be a connected reductive group over $\mathbb R$. Let $G'$ over $\mathbb R$ be an inner form of $G$ with ${}^LG={}^LG'$. By local Langlands correspondence over $\mathbb R$, if a $L$-packet of $...
- 6,622
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 ...
- 511
1
vote
0
answers
125
views
When is a vector bundle on a Shimura variety an automorphic vector bundle?
Let $(G, X)$ be a Shimura datum, let $K \subset G(\mathbb{A}_f)$ be an open compact subgroup, and denote by $\text{Sh}_K(G,X)$ the Shimura variety whose complex points are given by $G(\mathbb{Q})\...
- 159
1
vote
0
answers
77
views
Distinguishedness of discrete series induction
Let $D_k$ denote a discrete series representation of $\text{GL}_2(\mathbb{R})$ of weight $k\geq 2$. Consider the parabolically induced representation $D_k \times D_k$, which is a representation of $\...
- 145
3
votes
1
answer
228
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On the local factor of Rankin-Selberg L-functions
I have a puzzle on the local factors of Rankin-Selberg $L$-functions. Consider two newforms on $\text{GL}_2$. Let $f$ be a newform of square-free level $N$, and $g$ a newform of trivial level. As ...
- 353
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 \...
- 21
0
votes
0
answers
98
views
Morphisms on L^2(G) induced by morphisms of LCA groups
I am looking for a good reference to understand the space $L^2(G)$ for a locally compact abelian (LCA) group $G$.
In particular, I would like to understand when $L^2(-)$ is functorial, so that if $\...
- 71
9
votes
1
answer
680
views
Roadmap to Carayol-Deligne-Langlands
Having begun self-study of Fermat's Last Theorem a few years ago, I have only recently begun to understand and appreciate the theorem of Carayol-Deligne-Langlands on local-global compatibility for ...
- 283
5
votes
1
answer
187
views
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 $\mathrm{GL}_{n+1}(F)$ ...
- 639
1
vote
0
answers
48
views
Genericity of local representation with a non-generic local A-parameter
Let $\pi$ be an irreducible smooth representation of a classical $p$-adic group. Suppose that $\pi$ has a local L-parameter associated to some non-generic local A-parameter $\psi$. Then I am wondering ...
- 1,019
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 $T_n$ ...
- 1,019
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 ...
- 423
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=...
- 1,019
1
vote
0
answers
99
views
Question on the unramified representation
$\DeclareMathOperator\GL{GL}$Let $F$ be a $p$-adic field and $\chi$ be an unramified character of $\GL_1(F)$.
Consider an induced representation $\pi$ of $\GL_2(F)$ induced from the character $\chi|\...
- 1,019
8
votes
1
answer
567
views
Symmetric power lift of modular forms
Let $f_1$ and $f_2$ be two cuspforms of weights $k_1$ and $k_2$ and nebentypus $\epsilon_1$ and $\epsilon_2$ respectively such that $f_1 \neq f_2 \otimes \chi$ for some Dirichlet character $\chi$ of ...
- 424
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 ...
- 639
6
votes
1
answer
536
views
How to see that Eisenstein series are eigenfunctions of the laplacian?
Let $\Gamma$ be a discrete subgroup of $PSL_2(\mathbb{R})$ of finite type. Let $c_1,\ldots,c_h\in\mathbb{R}\cup\{\infty\}$ be a set of representatives of the $\Gamma$-equivalence classes of cusps. For ...
- 7,527
6
votes
1
answer
574
views
Automorphic representation of GL(1)
These might be very silly questions, but somehow I am not able to understand it or I might have misunderstood something.
I am reading automorphic forms from this book.
What I have understood till now:
...
- 424
5
votes
1
answer
306
views
Explicit description for action of Weyl element in Whittaker model for GL2
Let $F$ be a non-archimedean local field and let $\pi =\mathscr{B}(\chi, \chi^{-1})$ be a principal series representation of $\mathrm{PGL}_2(F)$ induced from a character $\chi$ of $F^\times$. Let $w = ...
- 53
1
vote
1
answer
187
views
Question of pole and zeros of symmetric or exterior global $L$-function of $\mathrm{GL}_n(\mathbb{A})$
Let $\pi$ be a unitary cuspidal representation of $\mathrm{GL}_n(\mathbb{A})$.
It is written is some paper that using the results towards the generalized Ramanujan conjecture in the paper "On the ...
- 1,019
1
vote
0
answers
124
views
Question on the Rankin-Selberg epsilon function
Let $\pi$ and $\pi'$ be unitary cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A})$ and $\mathrm{GL}_m(\mathbb{A})$, respectively.
It is well known that the complete Rankin-Selberg $L$-...
- 1,019
1
vote
0
answers
186
views
Consult a question about subconvexity bounds for symmetric-square L-functions in an Arxiv-eprint due to P. D. Nelson
Sorry to disturb, the experts here. Recently, I read a paper of Nelson ("Subconvex equidistribution of cusp forms: reduction to Eisenstein observables--"https://arxiv.org/pdf/1702.02908.pdf)....
- 563
4
votes
1
answer
299
views
Can any pair of associate parabolics be related by opposite parabolics?
Let $G$ be a reductive group, say over an algebraically closed field of characteristic zero.
We have the following definitions for a pair of parabolic subgroups $P_1$ and $P_2$ with Levi quotients $...
3
votes
1
answer
211
views
Question on the relation of global theta lifting and local theta lift
Let $F$ be a number field and $G$ (resp. $H$) an odd orthogonal (resp. metaplectic group) over $F$.
Let $v$ be a finite place of $F$ and $\sigma_v$ a supercuspidal representation of $G_v(F_v)$. Let $\...
- 1,019
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 ...
- 748
1
vote
0
answers
139
views
Zeroes of certain $L$-functions on the critical line and GGP conjectures
Global Gan-Gross-Prasad conjecture (on various groups) says that nonvanishing of certain automorphic $L$-function $L(s, \pi)$ (of cuspidal representation $\pi$ of some reductive group $G$) at $s = 1/2$...
- 2,215
2
votes
1
answer
160
views
Basic results concerning the intertwining operator in the $\mathrm{SL}_2$ case
I am reading [Ikeda, Tamotsu, On the location of poles of the triple L-functions]. On page 194, the author recalled some known results concerning $\operatorname{SL}_2$. I would like to know any ...
- 111
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 ...
- 111
6
votes
1
answer
404
views
Local component of cuspidal automorphic representation
Let $F$ be a number field and $\mathbb{A}$ its adele ring. $G$ be a classical group and $
\pi$ be a unitary cuspidal automorphic representation of $G(\mathbb{A})$.
Then I am wondering whether there is ...
- 1,759
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 ...
- 639
1
vote
0
answers
121
views
Spectral decomposition of the automorphic space for a unipotent group
Let $k$ be a global field of positive characteristic, $\mathbb{A}$ its adele ring. Let $U$ be a unipotent algebraic group over $k$, of dimension sufficiently small relative to ${\rm char} (k)$. Is ...
- 5,562
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;\...
- 111
0
votes
0
answers
65
views
Is there a generic representation for non-quasi split $p$-adic group?
It seems that generic representation only occurs for quasi-split groups.
For non-quasi split groups, is it expected that generic representation doesn’t exist?
Thank you in advance!
- 1,759
1
vote
1
answer
110
views
Automorphic representations on non-cyclic covering groups
The theory of theta functions can be interpreted as automorphic representations on metaplectic groups (2-fold covering groups of $\mathrm{Sp}_{2}$, or $\mathrm{GL}_2$), and there's also a notion of $n$...
- 2,215
2
votes
0
answers
139
views
Local Rankin-Selberg Zeta-function and Coates' p-adic L-Functions
$\DeclareMathOperator\Kern{Kern}\DeclareMathOperator\diag{diag}$
Let $F$ be a non-archimedean local field, $\mathcal{O}$ its ring of integers, $\mathfrak{p}$ its maximal ideal and $\pi$ a uniformizer.
...
- 748
4
votes
1
answer
291
views
The Langlands parameters of the symmetric cube lifts of cusp forms
I have a question concerning the Langlands parameters of the symmetric cube lifts. Lets $f$ be a $GL_2$-cusp form, and $\operatorname{sym}^2f$ the symmetric-square lift of it. Assume that $\alpha_1,\...
- 563
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 $\...
- 391
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 ...
- 5,424
2
votes
0
answers
172
views
Irreducibility criterion unramified principal series
Let $G$ be a reductive algebraic group defined over $\mathbb{Q}_p$ with maximal split torus $T_s$, Borel subgroup $B = TN$ and Weyl group $W(G,T_s)$. Let us consider the $\mathbb{Q}_p-$points of $G$ ...
- 103
4
votes
1
answer
817
views
Supercuspidal, spherical and discrete series representation
Let $G$ be an algebraic group defined over $\mathbb{Q}$ with maximal unipotent radical $N$. Let $\pi$ be an admissible representation of $G(\mathbb{Q}_p)$, we say that this representation is ...
- 103
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 ...
- 551
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,...
- 11
1
vote
0
answers
109
views
Whether or not the Maass form for $\Gamma _0(N)$ on $GL(3)$ covers the classical symmetric lift of a newform on $GL(2)$?
I have a blur which needs some help from the experts here, and may look naive for some experts. Recently I read Zhou's paper "The Voronoi formula on $GL{(3)}$ with ramification" (https://...
- 563
4
votes
1
answer
287
views
$\DeclareMathOperator\SL{SL}$Multiplicities of irreducible representations in discrete part of $L^2(\SL(2,\mathbb{Z})\backslash{\SL(2,\mathbb R)})$
$\DeclareMathOperator\SL{SL}$It is well-known that the cuspidal (or discrete) part of $L^2(\SL(2,\mathbb{Z})\backslash{\SL(2,\mathbb{R})})$ decomposes into irreducible representations of $\SL(2,\...
- 391
1
vote
0
answers
102
views
How to check the non-emptyness of the A-packet of non-split $\operatorname{SO}(2n+1)$?
Let $F$ be a number field and $G_n=\operatorname{SO}(2n+1)$ be the split group over $F$. Let $G_n^{\times}$ be a non-split group over $F$.
Let $\tau$ be an irreducible cuspidal automorphic ...
- 1,019
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 ...
- 1,019
4
votes
1
answer
344
views
Tannakian fundamental group of automorphic representations
Let $\mathcal{C}_{\mathrm{aut}}(G, F)$ be the category of automorphic representations of a connected reductive group $G$ over a number field $F$.
If this is a Tannakian category, it has an associated ...
- 87
4
votes
1
answer
405
views
Schur lemma and Whittaker functions
$\DeclareMathOperator\GL{GL}$Let $(\pi,V)$ be an infinite dimensional irreducible admissible representation of $\GL_2(\mathbb{Q}_p)$. Let us fix an element $v_0\in V$ and define a vector space
$$V_{...
- 103
1
vote
0
answers
153
views
Global tempered A-parameter is the same with generic A-parameter?
It seems there are two relevant terminologies on global Arthur parameter. One is generic A-parameter and the other one is tempered A-parameter.
I thought that these two terminology are same. But is ...
- 1,019
4
votes
1
answer
274
views
Non-vanishing criterion of the Hom space of induced representation of p-adic groups?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\GL{GL}
$Let $F$ be local field of characteristic zero and $(W,\langle,\rangle)$ be a $2n$-dimensional symplectic space over $...
- 1,019