On the nonexistence of automorphic eigenfunctions of exponential growth on SL(3,Z)∖SL(3,R)/SO(3,R)
On the nonexistence of automorphic eigenfunctions of exponential growth on SL(3,Z)∖SL(3,R)/SO(3,R)
复制标题
关于 SL(3,Z) 上指数增长的自同构本征函数不存在 — SL(3,R)/SO(3,R)
DOI：
10.1007/s40993-019-0168-8
复制
发表时间：
2019
影响因子：
0.8
通讯作者：
Trinh, Tien D.
中科院分区：
文献类型：
--
作者：
Miller, Stephen D.;Trinh, Tien D.
众所周知，存在具有指数增长并且具有指数增长的傅里叶系数(例如，负的幂或Ani-Bessel函数)的自同构特征函数。我们证明了这种现象不会发生在一般位置的商和本征值上(一个可去掉的技术假设)。更准确地说，如果这种自同构本征函数至多有指数增长，那么它在傅立叶展开中不可能有非衰变的惠特克函数。这证实了Miatello和Wallach的猜想的一部分，他们断言这个商上的所有自同构本征函数(在其他秩例中)总是有适度增长的。在某些自然假设下，例如本征函数的傅立叶展开式的绝对收敛，我们进一步证实了他们的猜想。
It is well-known that there are automorphic eigenfunctions on—such as the classicalj-function—that have exponential growth and have exponentially growing Fourier coefficients (e.g., negative powers of, or anI-Bessel function). We show that this phenomenon does not occur on the quotientand eigenvalues in general position (a removable technical assumption). More precisely, if such an automorphic eigenfunction has at most exponential growth, it cannot have non-decaying Whittaker functions in its Fourier expansion. This confirms part of a conjecture of Miatello and Wallach, who assert all automorphic eigenfunctions on this quotient (among other rankexamples) always have moderate growth. We additionally confirm their conjecture under certain natural hypotheses, such as the absolute convergence of the eigenfunction’s Fourier expansion.