关于SL(2,R)上速降函数的反演公式

在局部紧可分群的一般理论中，分解正则表示以及获得反演公式（或 Ｐｌａｎ－ｃｈｅｒｅｌ定理的明确表示）是调和分析的基本目标之一．ＳＬ（２，　）是最简单的非交换局部紧么模半单Ｌｉｅ群．Ｈａｒｉｓｈ－Ｃｈａｎｄｒａ在 Ｃ∞ｃ（ＳＬ（２，　））上获得了反演公式，Ｘｉａｏ和ｈｅｎｇ在文［１］中证明了Ｃ３ｃ（ＳＬ（２， ）上的反演公式．在文［２］中Ｚｈｅｎｇ引入了Ｌｉｅ群Ｇ上函数的广义微分（Ａ导数）概念．在本文中，我们利用文［２］中的微分概念来研究ＳＬ（２，　）上可微函数的Ｆｏｕｒｉｅｒ变换的阶，并获得了ＳＬ（２，　）上速降函数的反演公式．

A Note on Inversion Formula for Rapidly Decreasing Functions on SL (2, R)

In the general theory of locally compact separable groups, one of the basic objectives in harmonic analysis is to give the decomposition of regular representations and to deduce the inversion formula (or the abstract Plancherel theorem is formulated) in that context. The group SL(2, ) is the simplest of locally compact noncommutative unimdular semisimple Lie group. Harish-Chandra obtained the inversion formula for f ∈ C∞c (SL(2, )), Xiao and Zheng proved the inversion formula for f ∈ C3c (SL(2, )) in 1]. In 2] the generalized derivatives (A-derivatives) of functions on Lie group G was introduced. In this paper we will study the descending order of the Fourier transform of differential functions on SL(2, ) by using the concept of the derivatives in 2], and the inversion formula for rapidly decreasing functions on SL(2. ).