All Questions
Tagged with automorphic-forms algebraic-groups
54 questions
2
votes
0
answers
73
views
Number of rational points of a connected reductive group in a compact subset
Let $G$ be a connected reductive $\mathbb{Q}$-group. Let $\mathbb{A}$ denote the ring of adèles of $\mathbb{Q}$. Let $B \subset G(\mathbb{A})$ be a compact, let $x \in G(\mathbb{A})$ and consider the ...
- 81
2
votes
0
answers
55
views
Question on generic A-packet
Let $G$ be a classical group and $\phi$ be a generic $A$-parameter of $G$.
I am wondering whether each automorphic representations in the $A$-packet associated to $\phi$ are locally generic at almost ...
- 1,019
1
vote
0
answers
140
views
Explicit construction of $T$-orbits of generic characters of unitary groups
Let $F$ be a $p$-adic field. Let $E$ be a quadratic extension of $F$ and $G$ be a quasi-split unitary group $U(2n)$ or $U(2n+1)$ over with respect to $E/F$. Let $N_{E/F}$ be a norm map.
Let $B=TU$ be ...
- 1,019
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 $...
11
votes
1
answer
575
views
Do people prefer working on $\mathrm{GSp}$ and $\mathrm{GU}$ rather than $\mathrm{Sp}$ and $\mathrm{U}$, and why?
I am a new learner of Iwasawa theory and currently reading the famous paper by Skinner-Urban in 2014, and the following-up works by many other people. When reading these papers, I found that some ...
- 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
1
vote
0
answers
151
views
Action of $T_p$ on automorphic forms, and error in Gelbart's "Automorphic forms on adele groups"?
Let $f\in\mathcal{S}_k(\Gamma_0(N),\chi)$ be a cuspidal modular form, and $\phi_f\in\mathcal{A}_0(\text{GL}_2(\mathbb{Q})\backslash\text{GL}_2(\mathbb{A}_\mathbb{Q}),\omega)$ be its corresponding ...
- 111
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
3
votes
1
answer
206
views
Questions on norms on Adelic group
This might be stupid question to some experts who works in the realm of automorphic form.
Let $K$ be a number field and $\mathbb{A}$ is a adele ring of $K$. Let $G$ be a connected reductive group ...
- 1,019
3
votes
0
answers
149
views
The inclusion of Siegel sets of the general linear groups
In some paper, it is written that none of Siegel set of $GL_{n}$ is not contained in any Siegel set of $GL_{n+1}$. I don't understand this because I think it should hold.
To explain my question more ...
- 1,019
4
votes
0
answers
108
views
Question on Iwasawa decomposition of unitary groups over adele ring
Let $E/F$ be a quadratic extension of number fields and $V,\langle,\rangle$ is a hermition vector space over $E$. Let $\mathbb{A}$ be the adele ring of $F$.
Assume that there is a hermitian line $e \...
- 1,019
3
votes
0
answers
250
views
Functoriality for compactifications of locally symmetric spaces
Let $X$ be a symmetric space associated to an algebraic group $G$ defined over $\mathbb{Q}$ and $G(\mathbb{R})$ acts on $X$ from the left. Let $\Gamma \subset G(\mathbb{Q})$ be an arithmetic subgroup ...
- 443
6
votes
1
answer
811
views
What is the meaning of the $L$-group?
Langlands' functoriality conjecture predicts that to a suitable homomorphism of $L$-groups
$$
\psi : {}^LG \to {}^LH
$$
there should be a transfer of automorphic representations from $G$ to $H$. For ...
- 3,799
1
vote
0
answers
257
views
Does the standard arithmetic subgroup of a closed $\mathbb{Q}$-algebraic groups have non-trivial $\mathbb{Q}$-characters?
I am trying to understand the Borel-Harish Chandra theorem about arithmetic subgroups being lattices.
Suppose $G$ is an algebraic group inside $GL_n(\mathbb{C})$ such that it is definable as a zero ...
- 885
3
votes
0
answers
142
views
Iwasawa decomposition on unitary group of anisotropic kernel
Let $E/F$ be a quadratic extension of number fields. If $V$ is a hermitian space over $E$, let $V=X+V_0+Y$ be its Witt decomposition, where $X,Y$ are maximal totally isotropic subspaces and $V_0$ is ...
- 1,759
2
votes
0
answers
95
views
Metaplectic group $Mp(2n)(\mathbb{A}_F)$ splits over $Sp(2n)(F)$?
My question is the title.
In some literature, authors seem to use this without assumption.
Is it ture in general?
- 1,759
8
votes
1
answer
310
views
Restriction of irreducible representations from $G(\mathbb Q_p)$ to $[G, G](\mathbb Q_p)$
Let $\pi_p$ be a smooth irreducible representation of $G(\mathbb Q_p)$, where $G$ is a connected reductive group over $\mathbb Q_p$. Consider the restriction of $\pi_p$ to $[G, G](\mathbb Q_p)$, how ...
- 6,622
0
votes
1
answer
862
views
A question on standard parabolic subgroup
Let $G$ be a connected reductive group over a number field $F$ and $P_0$ its minimal parabolic subgroup. Then we call a parabolic subgroup $P$ of $G$ is standard if $P_0 \subset P$.
Let $K$ be a ...
- 1,759
1
vote
1
answer
350
views
Small questions in studying Arthur 's book 'Introduction to the Trace formula'
I am reading Arthur's book "Introductionto the trace formula".
In reading the book, two small question has arised and so I would like to ask it.
Let $G$ be a connected reductive group over $\mathbb{...
- 1,759
9
votes
1
answer
972
views
Why is the Langlands dual group always taken over $\mathbb{C}$?
Whenever I read a statement of the Langlands conjectures for a reductive group $G$, they are formulated in terms of the Langlands dual group, which is essentially the reductive group over $\mathbb{C}$ ...
5
votes
0
answers
229
views
The number of rational semisimple conjugacy class/the Arthur-Selberg trace formula
I was trying to understand a statement in Theorem 1.5 of this where the author seems to imply that if $G$ is a reductive group over $\mathbb{Q}$ such that $G/Z(G)$ is anisotropic, then for any ...
0
votes
0
answers
79
views
What is meant by "roots in $Lie(N)$" in root space decomposition of Lie algebras?
Let $G = GL_n$ and $T$ the invertible diagonal matrices and $N$ the upper triangular matrices with only $1$'s on the diagonal. Then the Lie algebra $\mathcal{G}$ has the roots space decomposition
$$
\...
- 3,625
3
votes
0
answers
63
views
What is meant by singular hyperplane of $c(w, \cdot)$? (global intertwining operator related to Eisenstein series)
Let $P_0$ be a minimal $\mathbb{Q}$-parabolic subgroup of $G$, a semisimple linear algebraic group over $\mathbb{Q}$. Then $P_0 = M_0 N_0$ where $M_0$ is a Levi subgroup of $P_0$. Let $E^G_{P_0}$ be ...
- 3,625
5
votes
1
answer
616
views
Question on an application of Langlands' result on the constant term of Eisenstein series (Is this a typo?)
I would like to understand an argument in https://link.springer.com/content/pdf/10.1007/BF01393904.pdf, which uses Langlands' result on the constant term of Eisenstein series, but I'm not getting it ...
- 3,625
2
votes
0
answers
196
views
Trying to understand why Eisenstein series is well defined
I am struggling to see why Eisenstein series is well defined, and I would greatly appreciate clarification.
Let
$$
E(x, \lambda) = \sum_{\delta \in P(\mathbb{Q}) \backslash G(\mathbb{Q}) }
e^{\...
- 3,625
2
votes
1
answer
311
views
Decomposition of parabolic subgroup in reductive group
Let $G$ be a reductive group over $\mathbb{Q}$ and let $P_0$ minimal parabolic subgroup.
If $P_1=M_{1}N_{1} \subset P_2=M_2N_2$ are standard parabolic subgroups of $G$, then can we decompose $P_1=(...
- 1,759
0
votes
1
answer
181
views
Does every character from group factor through largest central subgroup?
Let $G$ be a coonected reductive algebraic group over $\mathbb{Q}$ and $A_G$ its largest $\mathbb{Q}$-split central torus over $\mathbb{Q}$.
Let $X(G)_{\mathbb{Q}}$ be the addtive group of ...
- 1,759
1
vote
0
answers
124
views
Does Hom functor preserve restricted tensor product?
Let $\pi$ is an automorphic representation of reductive group $G$.
Then we can decompose $\pi = \otimes \pi_v$ as restricted tensor product by Flath theorem.
I am wondering whether $\text{Hom}_G(\pi,...
- 1,759
5
votes
2
answers
737
views
What condition makes unitary reductive group unramified?
I am a little bit confused with the definition of an unramified unitary group.
Let $F$ be a local field of characteristic zero whose residue field is finite field of characteristic $p$.
Then for a ...
- 1,759
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), ...
- 6,180
9
votes
1
answer
546
views
The space of Whittaker functionals is at most one-dimensional
Let $\mathbf G$ be a connected, reductive group over a local field $F$, and let $(\pi,V)$ be a smooth, irreducible, admissible representation of $G = \mathbf G(F)$. Assume there exists a Borel ...
- 6,180
2
votes
0
answers
418
views
How can this argument calculating the Haar measure on a parabolic subgroup be generalized to the non-split case?
Let $\mathbf G$ be a connected, reductive group over a local field $F$. Assume there is a maximal torus $\mathbf T$ which is split over $F$. Let $\mathbf P$ be a parabolic subgroup of $\mathbf G$ ...
- 6,180
17
votes
2
answers
3k
views
What's the point of a Whittaker model?
Let $G$ be a quasi-split connected reductive group over a $p$-adic field $F$. Let $B$ be a Borel subgroup which is defined over $F$, with $B = TU$, $T$ defined over $F$. The choice of $T$ and $B$ ...
- 6,180
3
votes
2
answers
638
views
Compactness of the automorphic quotient
Let $F$ be a (totally real) number field, and $E$ a (totally imaginary) quadratic extension of $F$. We consider $U$ a unitary group (with respect to a given hermitian form over $E$). The question is:
...
- 5,424
8
votes
2
answers
976
views
Relation between representations of p-adic groups and affine Hecke algebras
Let $R_n$ be the category of complex-valued smooth finite-length representations of the group $GL_n(F)$, where $F$ is a local field.
By the result of Borel, the subcategory of $R_n$ consisting of ...
- 6,201
8
votes
1
answer
615
views
Bernstein–Zelevinsky classification for classical groups
Bernstein and Zelevinsky classifies the irreducible complex smooth representations of a general linear group over a local field in terms of cuspidal representations. The irreducible modules are ...
- 6,201
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....
- 6,201
9
votes
1
answer
717
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,
$\...
- 2,516
2
votes
1
answer
344
views
On the reductive group [closed]
I know that the automorphic representation can be defined only for reductive algebraic group.
What property of algebraic group makes it hinder to define for all algebraic group and what nice property ...
- 1,759
3
votes
1
answer
487
views
What is the current status of representations of $GL_n(F)$ (and other algebraic groups)?
What is the current status of representations of $GL_n(F)$ (and other algebraic groups)?
When $F$ is a local field, the representations of $GL_n(F)$ are classified by Bernstein and Zelevinsky in ...
- 6,201
11
votes
1
answer
565
views
What can the theory of automorphic forms for $SL(n,\mathbb{Z})$ say about $SL(n,\mathbb{Z})$?
While reading "Automorphic Forms and L-functions for the Group $GL(n,R)$" by D. Goldfeld, I've got a feeling that linear groups over $\mathbb{R}$ and $\mathbb{Z}$ are considered only as technical ...
- 3,186
3
votes
2
answers
403
views
Gelfand pair and double coset decomposition
Let $F$ be a non-Archimedean local field with ring of integers $O$, $\pi$ be a uniformizer. Let $\tilde{G}$ be a connected algebraic group over $F$ and splits over $F$, fix a split maximal torus $\...
- 1,692
3
votes
1
answer
611
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 unitary/...
- 11.2k
3
votes
1
answer
552
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 ...
- 263
3
votes
0
answers
183
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 ...
- 407
1
vote
0
answers
132
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 ...
- 63
5
votes
0
answers
266
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.
...
- 3,472
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-...
- 809
7
votes
1
answer
1k
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 $F$ , $H=GL_{n-1}$ ...
- 809
3
votes
0
answers
803
views
Tamagawa number for functional fields
Let $G$ be a split semi-simple simply connected group over a global field $F$ and let
$\omega$ be a top-degree differential form on $G$ without zeroes (defined over $F$). It is well
known that $\omega$...
- 7,037