The local Gan-Gross-Prasad conjecture for U(3)×U(2): The non-generic case

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• 作者：Jaeho Haan ^{jaehohaan@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Gan&ndash ; Gross&ndash ; Prasad conjecture ; Non-tempered Arthur packet ; Unitary groups ; Local theta correspondence ; ϵ-factor
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：August 2016
• 年：2016
• 卷：165
• 期：Complete
• 页码：324-354
• 全文大小：494 K

文摘

In this paper, we investigate the local Gan–Gross–Prasad conjecture for some pair of representations of 16000706&_mathId=si1.gif&_user=111111111&_pii=S0022314X16000706&_rdoc=1&_issn=0022314X&md5=1637ff21ccc54de63bd0296b2e0e4179" title="Click to view the MathML source">U(3)×U(2)$U(3)\times U(2)$ involving a non-generic representation. For a pair of generic L  -parameters of 16000706&_mathId=si2.gif&_user=111111111&_pii=S0022314X16000706&_rdoc=1&_issn=0022314X&md5=fd216a0d62add6c8cfe46bc8d2340194" title="Click to view the MathML source">(U(n),U(n−1))$(U(n),U(n-1))$, it is known that there is a unique pair of representations in their associated Vogan L-packets which produces the unique Bessel model of these L-parameters. We showed that this is not true for some pair of L-parameters involving a non-generic one.

On the other hand, we give the precise local theta correspondence for 16000706&_mathId=si3.gif&_user=111111111&_pii=S0022314X16000706&_rdoc=1&_issn=0022314X&md5=e3cfb32f46f3291599fefa96f0ee36de" title="Click to view the MathML source">(U(1),U(3))$(U(1),U(3))$ not at the level of L-parameters but of individual representations in the framework of the local Langlands correspondence for unitary group. As an application of these results, we prove an analog of Ichino–Ikeda conjecture for some non-tempered case. The main tools in this work are the see-saw identity, local theta correspondence for (almost) equal rank cases and recent results on the local Gan–Gross–Prasad conjecture both on the Fourier–Jacobi and the Bessel case.