p-adic典型群上正规化结算子的极点

The harmonic analysis on p-adic linear algebraic groups plays an important role in Langlands program since many important problems raised from this area rely on it, for example, the reducibility and classification problem of representations, the functoriality and reciprocity laws, etc. However, the relationship between Plancherel measures and standard intertwining operators in p-adic harmonic analysis revealed by Harish Chandra shows that sovling the reducibility problem is equivalent to studying the poles and zeroes of the attached standard intertwining operators..When the inducing representation is unitary generic supercuspidal, the reducibility and L-function problem were determined by Shahidi, while the lifting problems on classical groups were mainly solved by Jiang and Soudry. Shahidi's result can be briefly restated as follows:.1. The standard intertwining operator has a pole at s=0 or equivalently, the induced representation is irreducible at s=0 if and only if one of two L-functions L(s,1),L(s,2) attached to the standard intertwining operator has a pole, the poles raised from these functions are exclusive. Where L(s,i), i=1,2, roughly speaking, are the L-functions related to the adjoint action of the Levi subgroup on i-th unipotent part of the unipotent radical (radical of the parabolic subgroup)..2. If L(s,i) has a pole at s=0, then the zeroes of the standard intertwining operators are exactly at s=1/i. On 0<s<1/i,the induced representations are in the complementary series. On other place, it's never in the complementary series.. When the inducing representation is general unitary supercuspidal,there is no big breaking through has been obtained so far. Langlands conjectured that there are only finitely many types of L-packets attched to automorphic forms, and Shahidi even conjectured that on every local L-packet, there is a generic one that attched to it. These conjectures imply that on concerns of L-function and reducibility, the non-generic cases is "very close" to generic ones.. Our project is designed to approach solutions to some of these problems in general cases, mainly on determining the poles at s=0, by studying the norm correspondence. We have extracted the L-functions L(s,i), i=1,2, exactly as those obtained from generic representations, and proved that these L-functions control the reducibility in the same way as in generic case. Moreover, when the pole comes from L(s,1), it also reflects similar lifting property between representations as generic case. Howcver, our answers to these questionss are mostly on necessary condition now, we still need to pay much effort towards to the sufficient direction.

p-adic线性代数群上的调和分析是Langlands纲领的主要内容之一，正规化结算子是其中的一个重要工具。它可用于确定导出表示的可约性及分类；揭示表示之间的函子性（表示的提升）；定义联系表示的L-函数并确立它们之间的函数方程。这些局部性质被广泛地应用于表示论的其它领域，如自守表示、逆定理、Langlands对应等方向的研究。..本课题主要研究由极大抛物子群的supercuspidal表示所导出的p-adic典型群上的表示在原点处的可约性、表示之间的提升及与L-函数的关系。

郎兰兹L函数是自守形式与调和分析的重要内容之一，它揭示表示的函子性及一些重要的不变性。正规化结算子是定义和研究郎兰兹L函数的重要工具，它的极点和零点刻画了这些L函数的许多重要性质，包含这深刻的代数数论的意义。而确定它的极点是定义L函数的首要步骤。. 本课题主要研究在局部典型群上，联系由极大抛物子群上的超尖点表示所导出表示的正规化结算子的极点。首先，我们将极点处的留数转化为范对应下的轨道积分，证明了这些积分只可能在正则椭圆轨道上非零；其次，我们在这些轨道积分上分离出了与两类标准L函数对应的不变积分，证明了这两种不变积分的非零性是互斥的；最后，我们证明了这两类积分具有与它们对应的L函数相似的函子性。这些结果与langlands 和Shahidi 猜想的一般局部L函数的特性相符合，虽然这些L函数还尚未被定义。