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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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Two basic questions on parabolic induction
I want to ask some basic two questions on the parabolic induction.
Let $F$ be a local fields.
Let $\chi_1,\chi_2$ be two characters of $GL_1(F)$ and $GL_1 \times GL_1$ be the Levi part of the parabo …
CommunityBot
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2
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Jacquet module of irreducible principal series
Let $F$ be a local field of characteristic zero and $G=\operatorname{GL}_n(F)$.
Let $B=UT$ be a Borel subgroup of $G$ and $\chi=(\chi_1,\cdots,\chi_n)$ is an unramified character of $B$.
Consider un …
- 3,054
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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 …
- 1,759
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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}^ …
- 4,219
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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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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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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
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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 …
CommunityBot
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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 …
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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 …
- 12.9k
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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
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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
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Order of some $L$-function at $s=0$
Sorry, I asked this two days ago, but this time I modified it to be easily read and added more specific explanation. I hope to get your illuminating comment on whether my approach is right.
I am comp …
- 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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answer
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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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