Unramified Whittaker functions for GL(3,R)
Unramified Whittaker functions for GL(3,R)
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DOI：
10.1007/bf02788764
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发表时间：
1995-01-01
影响因子：
1
通讯作者：
Huntley, J
中科院分区：
文献类型：
--
作者：
Bump, D;Huntley, J
本文将给出GL（3，~）Whittaker函数的一部分理论，它可以用复变函数论的初等方法得到。本文研究GL（3，R）上球Whittaker函数在无穷远点的渐近展开式.这些惠特克函数是两个变量的偏微分方程的超定系统的解，有六个解，这是合流超几何方程的秩二模拟。我们将获得所有六个解决方案的渐近展开，使我们能够表征一个特定的解决方案，其中发生在傅立叶展开的自守形式的条件下，适度增长在无穷大。（这种类型的结果也可以在Wallach [11]和Shalika [7]中找到。我们还计算了高阶项的渐近展开，这可能有自守形式的应用。Wallach [11]用表示论的方法研究了任意约化李群上Whittaker函数的渐近展开。瓦拉赫的结果与我们的结果重叠，是确定的。然而，我们能够获得非常精确的信息，这是新的，通过限制到GL（3，R）的特殊例子。我们将在第二节讨论我们的工作与Wallach的工作之间的关系。我们发现（在所考虑的区域内）Whittaker函数的一阶渐近性与泛包络代数的指标特征无关。然而，高阶渐近依赖于索引字符。鉴于亨特利[4]的方法，高阶渐近的这种依赖性似乎对自守形式有影响。
In this paper, we will present a portion of the theory of GL (3,~) Whittaker functions which is obtainable using elementary methods of the theory of complex variables. We will study the asymptotic expansions at infinity for spherical Whittaker functions on GL (3, R). These Whittaker functions are solutions to an overdetermined system of partial differential equations in two variables, having six solutions, which is a rank two analog of the confluent hypergeometric equation. We will obtain asymptotic expansions for all six solutions, allowing us to characterize the one particular solution which occurs in the Fourier expansions of automorphic forms by a condition of moderate growth at infinity.(Results of this type may also be found in Wallach [11] and Shalika [7].) We also compute the higher order terms in the asymptotic expansions, which may have applications to automorphic forms. Asymptotic expansions of Whittaker functions on an arbitrary reductive Lie group have been considered by Wallach [11] using methods of representation theory. Wallach's results, which overlap with ours, are definitive. However, we are able to obtain extremely precise information, which is new, by restricting to the particular example of GL (3, R). We will discuss the relationship between our work and Wallach's in Section 2.We find that (in the region under consideration) the first order asymptotics of the Whittaker functions are independent of the indexing character of the universal enveloping algebra. However, the higher order asymptotics depend upon the indexing character. In view of the methods of Huntley [4], it seems likely that this dependence of the higher order asymptotics will have implications for automorphic forms.