Kirillov's character formula for reductive Lie groups
Kirillov's character formula for reductive Lie groups
复制标题
还原李群的基里洛夫特征公式
DOI：
10.1007/bf01390244
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发表时间：
1978
影响因子：
3.1
通讯作者：
W. Rossmann
中科院分区：
文献类型：
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作者：
W. Rossmann
Inventiones math.48，2007 - 220（1978）;在线版本。Kirillov的著名公式说，李群G的不可约酉表示的特征标X应该由以下形式的方程给出：（Φ）x（exp x）= p（x）− 1 e i（λ，x）d µ（λ）其中ω = e（X）是G的对偶g ∈ F李代数中的G-轨道，µ是g上的Kirillov标准测度，p是g上的某个函数，n amely p（x）= det 1/2 {sinh（ad（x/2））/ad（x/2）}至少对于一般轨道n [10]。当然，这个公式不能太过字面理解（（Φ）中的积分通常是发散的），但必须解释为g上测试函数的某个空间上的分布方程。为了使这一点更精确，用go表示g中零的开邻域，使得exp：g → G限制于go到G的开子集上的可逆解析映射。为了我们的目的，公式（Φ）应该被解释为：对于所有在g o中具有紧支集的C ∞函数<$$>，（Φ）t r g <$（x）π（exp（x））dx =<$g e i（λ，x）<$（x）p（x）− 1} d µ <$（λ）。(Hereπ是G的特征标为χ的表示。）当然，基里洛夫的公式并不具有这种普遍性。事实上，确定表征理论的确切有效域是表征理论中的一个主要问题。本文将证明Kirillov公式对出现在Plancherel公式中的约化真实的李群的特征标成立。实际上，我们将只详细讨论离散系列特征标。然后，用熟悉的方法，将其它特征的公式简化为离散系列特征的公式。（Duflo [3]）。基里洛夫公式的离散系列是一个公式的后果有关的傅立叶变换g与傅立叶变换的嘉当子代数的紧凑型的手段不变的积分。这就是基里洛夫公式的证明形式。
Inventiones math. 48, 2007-220 (1978); on-line version. Kirillov’s famous formula says that the characters X of the irreducible unitary representations of a Lie group G should be given by an equation of the form (Φ) χ(exp x )= p(x) −1 Ω e i(λ,x) dµΩ(λ) where ω =Ω (X )i s aG-orbit in the dual g ∗ ft he Lie algebrag of G, µΩ is Kirillov’s canonical measure on Ω, and p is a certain function on g ,n amely p(x )= det 1/2 {sinh(ad(x/2)) /ad(x/2)} at least for generic orbits Ω [10]. This formula cannot be taken too literally, of course (the integral in (Φ) is usually divergent), but has to be interpreted as an equation of distributions on a certain space of test functions on g. To make this precise, denote by g o an open neighborhoodod of zero in g so that exp : g → G restricts to an invertible analytic map of g o onto an open subset of G. For our purposes, the formula (Φ) should be interpreted as saying that (Φ � )t r g ϕ(x)π(exp(x)) dx = Ω g e i(λ,x) ϕ(x) p(x) −1 } dµΩ(λ) for all C ∞ functions ϕ with compact support in g o . (Here π is the representation of G with character χ.) Of course, Kirillov’s formula does not hold in this generality. It is in fact a major problem in representation theory to determine its exact domain of validity. In this paper we shall show that Kirillov’s formula holds for the characters of a reductive real Lie group which occur in the Plancherel formula. Actually, we shall deal in detail only with the discrete series characters. The formula for the other characters can then be reduced to the formula for the discrete series characters by familiar methods. (Duflo [3]). Kirillov’s formula for the discrete series is a consequence of a formula relating the Fourier transform on g with the Fourier transform on Cartan subalgebras of compact type by means of the invariant integral. This is the form in which Kirillov’s formula will be proved.