All Questions
Tagged with p-adic-groups automorphic-forms
21 questions
1
vote
0
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48
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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
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91
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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
2
votes
0
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76
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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
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0
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98
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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
1
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0
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140
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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
5
votes
1
answer
832
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Understand the $p$-adic local Langlands correspondence with examples
Let $\rho:G_{\mathbb{Q}}\rightarrow \mathbf{Gl}_{n}(\mathbb{Q}_{p})$. I would like to understand in depth why the local Langlands correspondence for $\rho_{|\mathbb{Q}_{p}}$ must consider $p$-adic ...
- 1,270
4
votes
1
answer
274
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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
2
votes
0
answers
78
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Unramified constituent of Weil representation of $U(2)$
Let $E/F$ be a quadratic extension of local field of characteristic zero.
Let $\omega$ be the quadratic character of $F^{\times}$ associated to $E/F$ by local class field theory and $\gamma:E^{\times} ...
- 1,019
1
vote
1
answer
242
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Part of some generic representation is also generic?
$\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
2
votes
1
answer
302
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Jacquet module and Frobenius reciprocity
Let $F$ be a local field of characteristic zero and $G$ be a classical group over $F$.
Let $P=MN$ be a parabolic subgroup of $G$ and $\pi$ a irreducible smooth representation of $M$.
Let $\sigma$ be ...
- 1,019
1
vote
0
answers
152
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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 ...
- 1,759
3
votes
0
answers
156
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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
6
votes
0
answers
217
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Dimension of space of K-fixed vectors
If $G$ is an unramified group over an $p$-adic field $F$, the Satake isomorphism identifies the spherical Hecke algebra with respect to a special maximal compact subgroup $K$. In particular,
(1) $H(G(...
- 91
12
votes
1
answer
993
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A question on representation theory of p-adic groups
Let $V$ be a complex vector space of infinite dimension and let $(\pi,V)$ be a representation of the $p$-adic group $G:=GL_2(\mathbb{Q}_p)$. From representation theory, we know that if the ...
- 421
4
votes
1
answer
255
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Whittaker functions estimates proof
I am reading the proof of the estimates of Whittaker functions from Jacquet, Hervé, Ilja Iosifovitch Piatetski-Shapiro, and Joseph Shalika. "Automorphic forms on GL (3) I." Annals of Mathematics 109.1 ...
- 730
17
votes
2
answers
3k
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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 $B$ ...
- 6,170
3
votes
2
answers
581
views
Generic irreducibility of parabolic induction
In J.Bernstein's notes: REPRESENTATION OF P-ADIC GROUPS, he remarked the following result(see P.88):
Let $G$ be a reductive group defined over nonarchimedean local field $F$, $P$ parabolic subgroup of ...
- 301
4
votes
1
answer
203
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Intertwining Operators Associated to Simple Reflections
Let $G$ be a quasi-split reductive group, over a local field, with a Borel subgroup $B=T\cdot N$ and the associated Weyl group $W$. Given a family of induced representations $\pi_s = Ind_B^G \chi\cdot ...
- 141
3
votes
0
answers
132
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Newvectors in tensor product representations
Let $\pi_p$ and $\pi_p^\prime$ two smooth admissible irreducible complex representations of ${\rm GL}_2(F)$ where $F$ is a non archimedean local field of residual characteristic $p$ of central ...
- 221
7
votes
5
answers
1k
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Is a unitary representation always semisimple?
I have been reading the online lecture notes by Fiona Murnaghan
http://www.math.toronto.edu/murnaghan/courses/mat1197/notes.pdf
The first lemma in p.35 says that every unitary representation of ...
- 833
4
votes
0
answers
169
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parametrization of irreducible finite dimensional representation of Weil group
Let $F$ be a p-adic field, with p a prime denoting the residue field characteristic. Let $\mathcal{W}_F$ be the Weil group. In the local Langlands correspondence for $GL(n,F)$, it is important to know ...
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