An inequality for local unitary Theta correspondence

Z. Gong[1]; L. Grenié[2]

  • [1] Lycée annexe à l’Université Fudan, N.383 Rue Guo Quan, Shanghai, Chine
  • [2] Università degli Studi di Bergamo, viale Marconi 5, 24044 Dalmine (BG), Italy

Annales de la faculté des sciences de Toulouse Mathématiques (2011)

  • Volume: 20, Issue: 1, page 167-202
  • ISSN: 0240-2963

Abstract

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Given a representation π of a local unitary group G and another local unitary group H , either the Theta correspondence provides a representation θ H ( π ) of H or we set θ H ( π ) = 0 . If G is fixed and H varies in a Witt tower, a natural question is: for which H is θ H ( π ) ≠ 0 ? For given dimension m there are exactly two isometry classes of unitary spaces that we denote H m ± . For ε ∈ { 0 , 1 } let us denote m ε ± ( π ) the minimal m of the same parity of ε such that θ H m ± ( π ) ≠ 0 , then we prove that m ε + ( π ) + m ε - ( π ) ≥ 2 n + 2 where n is the dimension of π .

How to cite

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Gong, Z., and Grenié, L.. "An inequality for local unitary Theta correspondence." Annales de la faculté des sciences de Toulouse Mathématiques 20.1 (2011): 167-202. <http://eudml.org/doc/219835>.

@article{Gong2011,
abstract = {Given a representation $\pi $ of a local unitary group $G$ and another local unitary group $H$, either the Theta correspondence provides a representation $\theta _H(\pi )$ of $H$ or we set $\theta _H(\pi )=0$. If $G$ is fixed and $H$ varies in a Witt tower, a natural question is: for which $H$ is $\theta _H(\pi )\ne 0$ ? For given dimension $m$ there are exactly two isometry classes of unitary spaces that we denote $H_m^\pm $. For $\varepsilon \in \lbrace 0,1\rbrace $ let us denote $m_\varepsilon ^\pm (\pi )$ the minimal $m$ of the same parity of $\varepsilon $ such that $\theta _\{H_m^\pm \}(\pi )\ne 0$, then we prove that $m_\varepsilon ^+(\pi )+m_\varepsilon ^-(\pi )\ge 2n+2$ where $n$ is the dimension of $\pi $.},
affiliation = {Lycée annexe à l’Université Fudan, N.383 Rue Guo Quan, Shanghai, Chine; Università degli Studi di Bergamo, viale Marconi 5, 24044 Dalmine (BG), Italy},
author = {Gong, Z., Grenié, L.},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
language = {eng},
month = {1},
number = {1},
pages = {167-202},
publisher = {Université Paul Sabatier, Toulouse},
title = {An inequality for local unitary Theta correspondence},
url = {http://eudml.org/doc/219835},
volume = {20},
year = {2011},
}

TY - JOUR
AU - Gong, Z.
AU - Grenié, L.
TI - An inequality for local unitary Theta correspondence
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2011/1//
PB - Université Paul Sabatier, Toulouse
VL - 20
IS - 1
SP - 167
EP - 202
AB - Given a representation $\pi $ of a local unitary group $G$ and another local unitary group $H$, either the Theta correspondence provides a representation $\theta _H(\pi )$ of $H$ or we set $\theta _H(\pi )=0$. If $G$ is fixed and $H$ varies in a Witt tower, a natural question is: for which $H$ is $\theta _H(\pi )\ne 0$ ? For given dimension $m$ there are exactly two isometry classes of unitary spaces that we denote $H_m^\pm $. For $\varepsilon \in \lbrace 0,1\rbrace $ let us denote $m_\varepsilon ^\pm (\pi )$ the minimal $m$ of the same parity of $\varepsilon $ such that $\theta _{H_m^\pm }(\pi )\ne 0$, then we prove that $m_\varepsilon ^+(\pi )+m_\varepsilon ^-(\pi )\ge 2n+2$ where $n$ is the dimension of $\pi $.
LA - eng
UR - http://eudml.org/doc/219835
ER -

References

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