Automorphic forms constructed from Whittaker vectors
Automorphic forms constructed from Whittaker vectors
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从 Whittaker 向量构建的自同构形式
DOI：
10.1016/0022-1236(89)90059-1
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发表时间：
1989
影响因子：
1.7
通讯作者：
N. Wallach
中科院分区：
文献类型：
--
作者：
R. Miatello;N. Wallach
设G是半单的裂秩1李群，Γ是G的上有限体积离散子群.如果P是G的一个具有幂幺根N的抛物型函数，且χ是N的一个非平凡酉特征标，使得χ（Γ <$N）= 1，则构造了一个满足自守形式定义中除中等增长条件外的所有条件的G/G上的亚纯函数族M（v）.证明了M（v）在Rev 0 ≠ 0的极点处的主部是平方可积的，并且利用M-级数的主部可以构造出“基本上”所有χ-Fourier系数不为零的平方可积自守形式.对于在极大紧子群下固定的平方可积自守形式，可以去掉“本质上”的限制条件。计算了M级数的傅里叶系数。在χ-傅立叶系数中的一个特定的项被示出来确定M-级数的奇异性的结构。这个术语与Selberg的“Kloosterman-Zeta函数”有关。导出了M级数的函数方程。对于SL（2，R）的情形，结果更加明确，并构造了一个完整的平方可积自守形式族.本文还提出了一个猜想：对于分裂秩> 1且不可约的半单李群，自守型定义中的适度增长条件是多余的。证明了这一猜想的证据是关于数域上的SO（n，1）。
LetGbe a semi-simple Lie group of split rank 1 and Γ a discrete subgroup ofGof cofinite volume. IfPis a percuspidal parabolic ofGwith unipotent radicalNand if χ is a non-trivial unitary character ofNsuch thatχ(Γ∩N) = 1 then a meromorphic family of functions M(v) ongG/Gthat satisfy all of the conditions in the definition of automorphic form except for the condition of moderate growth is constructed. It is shown that the principal part of M(v) at a polev0with Rev0⩾ 0 is square integrable and that “essentially” all square integrable automorphic forms with non-zero χ-Fourier coefficient can be constructed using the principal parts of theM-series. For square integrable automorphic forms that are fixed under a maximal compact subgroup the proviso “essentially” can be dropped. The Fourier coefficients of theM-series are computed. A specific term in the χ-Fourier coefficient is shown to determine the structure of the singularities of theM-series. This term is related to Selberg's “Kloosterman-Zeta function.” A functional equation for theM-series is derived. For the case ofSL(2, R) the results are made more explicit and a complete family of square integrable automorphic forms is constructed. Also the paper introduces the conjecture that for semi-simple Lie groups of split rank > 1 and irreducible Γ the condition of moderate growth in the definition of automorphic form is redundant. Evidence for this conjecture is given forSO(n, 1) over a number field.
Whittaker 向量的空间不可约性
DOI：
--
发表时间：
2007
期刊：
影响因子：
--
作者：
Ikehata;M and Ohe;T;松本久義
通讯作者：
松本久義