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Results tagged with automorphic-forms
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An automorphic form is a well-behaved function from a topological group $G$ to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup $\Gamma \subset G$ of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups.
3
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Theta lifting over function fields
Let $F$ be a number field and $\mathbb{A}$ its adele ring.
For a dual reductive pair $G$ and $H$, let $\pi$ be a cuspidal irreducible representation of $G(\mathbb{A})$. Let $\Theta(\pi)$ be the global …
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4
votes
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122
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$L^2$-spectrum versus automorphic discrete spectrum
Let $G$ be a classical group defined over a number field $F$.
In his monumental book, (https://www.ams.org/books/coll/061/coll061-endmatter.pdf) Arthur described a spectral decomposition of $L_{disc}^ …
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6
votes
1
answer
364
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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 …
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0
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65
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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!
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1
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116
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Is it possible $L(\frac{1}{2},\phi \times \phi')=0$ for all $\phi'$?
Let $\phi$ be an irreducible cuspidal automorphic representation of $GL_n(\mathbb{A})$ of symplectic type, that is, the exterior square $L$-function $L(s,\phi,\Lambda^2)$ has a pole at $s=1$.
Then I a …
- 1,759
2
votes
1
answer
292
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Question on the residual representation
Let $G=SO_n$ and fix a borel subgroup $P_0$ of $G$. Let $P=MN$ be a standard maximal parabolic subgroup $G$ and $\sigma$ a cuspidal representation of $M$
Consider the normalized parabolic induced rep …
- 1,759
1
vote
0
answers
149
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Question on induction of unramified representations
$\def\anonabs{\lvert\cdot\rvert}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Ind{Ind}\DeclareMathOperator\SO{SO}
$Let $F$ be a $p$-adic local field of characteristic zero.
Let $\chi$ be an unramif …
- 1,759
3
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answers
151
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Question on the proper sub-representation of induced representation
$\DeclareMathOperator\Ind{Ind}$Let $G$ be a reductive group over a $p$-adic local field $F$, and $P=MN$ a parabolic subgroup.
Let $\sigma$ be an irreducible representation of $M(F)$ and consider its …
- 1,759
1
vote
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292
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Intertwining operator is not an isomorphism?
Let $F$ be a number field and $G$ a symplectic group over $F$.
Let $P=MN$ is a maximal parabolic subgroup of $G$ and $W_M=N_G(M)/M$. Since $P$ is maximal, $W_M \simeq S_2$. Let $w$ be a non-trivial e …
- 1,759
0
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2
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238
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Left translation of automorphic form satisfies $K$-finiteness?
Does a left translation of an automorphic form satisfy left $K$-finiteness?
Let $F$ be a number field and $G$ is an algebraic group. Let $\phi$ is an automorphic form on $G$. Let $K$ be a maximal com …
- 1,759
7
votes
1
answer
276
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Restriction of product of automorphic forms
Let $W \subset V$ be quadratic spaces over a number field $F$.
Let $G_n=SO(V)$ and $G_m=SO(W)$ and we consider $G_m$ as a subgoup of $G_n$ via a diagonal embedding.
Let $f$ be an automorphic form of …
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3
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149
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Global Arthur packet consist of only globally generic representations?
I would like to ask very stupid two questions to experts.
I am wondering whether every globally generic automorphic representation of unitary groups are contained some global Arthur packet associated …
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2
votes
1
answer
209
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Semidirect product of metaplectic group and Heisenberg group
I know that Symplectic group has an action on Heisenberg group.
I am wondering how to extend this to non-trivial two fold metaplectic covering?
Thanks in advance!
- 1,759
3
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answers
134
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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 a …
- 1,759
2
votes
0
answers
95
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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?
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