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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.
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Product of two automorphic form is still automorphic form?
I am wondering whether the automorphic forms in some automorphic representation is closed under product. I think it is true by definition of automorphic form. Am I right?
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Subquotient of principal series
Let $F$ be a local field of characteristic 0.
I am wondering whether an unramified principal series representation of $\operatorname{GL}_n(F)$ can have 1-dimensional quotient when $n>1$.
In some pap …
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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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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 …
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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 whe …
- 1,759
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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 non-tempere …
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Does the global theta lift performed twice yield the identity when it doesn't vanish?
Let $E/F$ be a quadratic extension of number fields and $V,W$ are hermitian and skew hermitian vector space over $E$ whose dimension is $n,m$ respectively.
Let $\pi$ be a irreducible tempered cuspida …
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Is constant map from automorphic form surjective?
Let $G$ be a connected reductive group over $\mathbb{Q}$ and $P=NM$ be a standard parabolic subgroup of $G$ and and $K$ a 'good' maximal compact subgroup of $G$. (For precise definitions of these term …
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307
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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
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306
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Central character of some conjugate self dual representation
I want to ask some question on conjugate self dual representation.
Let $E/F$ be a quadratic extension of number fields and $c:GL_n(\mathbb{A}_E) \to GL_n(\mathbb{A}_E)$ be a automorphism induced by c …
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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 …
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833
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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 fixe …
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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 …
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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
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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 …
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