Holomorphic extensions of representations: (I) automorphic functions
Holomorphic extensions of representations: (I) automorphic functions
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DOI：
10.4007/annals.2004.159.641
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发表时间：
2002-10
影响因子：
4.9
通讯作者：
B. Kroetz;R. Stanton
中科院分区：
文献类型：
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作者：
B. Kroetz;R. Stanton
设G是一个包含在其复化GC中的连通的真实的半单李群，K是G的一个极大紧子群.本文在GC中构造了一个KC-G双陪集域，证明了G在G的任一不可约酉表示的K-有限向量上的作用在这个域上有一个全纯扩张。对于由此产生的K-有限矩阵系数的全纯扩展，我们得到的边界处的奇异性的估计，以及沿边界的主要/次要估计沿着。我们得到了G/K上全纯扩张自守函数在Sobolev范数意义下的L ∞界，并利用这些界估计了自守函数组合的Fourier系数，例如Maas形式的三重积.
Let G be a connected, real, semisimple Lie group contained in its complexification GC, and let K be a maximal compact subgroup of G. We construct a KC-G double coset domain in GC, and we show that the action of G on the K-finite vectors of any irreducible unitary representation of G has a holomorphic extension to this domain. For the resultant holomorphic extension of K-finite matrix coefficients we obtain estimates of the singularities at the boundary, as well as majorant/minorant estimates along the boundary. We obtain L ∞ bounds on holomorphically extended automorphic functions on G/K in terms of Sobolev norms, and we use these to estimate the Fourier coefficients of combinations of automorphic functions in a number of cases, e.g. of triple products of Maas forms.