On Rossmann's character formula for discrete series
On Rossmann's character formula for discrete series
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论离散级数罗斯曼的特征公式
DOI：
10.1007/bf01391173
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发表时间：
1979
影响因子：
3.1
通讯作者：
M. Vergne
中科院分区：
文献类型：
--
作者：
M. Vergne
在该杂志最近的一篇文章中，W。Rossmann [3]证明了Kirillov给出的一个公式，它将Plancherel公式中半单李群表示的特征标与余伴随表示中轨道上测度的Fourier变换联系起来。罗斯曼公式具有相当重要的概念意义，因为它通过轨道方法建立了一般李群的表示理论和半单李群表示的cabalistic研究之间的联系。我在这里给一个简单的证明罗斯曼的基本定理，其中涉及傅立叶变换的g和嘉当子代数的紧凑型。1.设V是具有非退化对称形式B的真实的向量空间。本文考虑由（mB.f）（x）=B（x，x）f（x）给出的乘法算子m8，AB = c~176拉普拉斯~176（AB”f）（x）=（B（~--x' ~x）”f）（x）”i的谐振子i（mB-AB）为JB，[若V是一维的，则JB=i [. X2
In a recent article in this journal, W. Rossmann [3] has proved a formula conjectured by Kirillov connecting characters of the representations of semisimple Lie groups occurring in the Plancherel formula with Fourier transforms of the measures on orbits in the coadjoint representation. Rossmann's formula has considerable conceptual importance, as it established, via the orbit method, the connection between the representation theory of general Lie groups and the cabalistic study of representations of semi-simple Lie groups. I give here a simple proof of Rossmann's basic theorem, which relates the Fourier transforms on g and on a Cartan subalgebra of compact type. 1. Let V be a real vector space with a non-degenerate symmetric form B. I consider the multiplication operator m 8 given by (mB.f)(x)=B(x,x)f(x), and AB the c~176 Laplace ~176 (AB" f)(x)=(B (~--x' ~x)" f )(x)" i ing the harmonic oscillator i(mB--AB) as JB, [if V is one dimensional, JB=i [.X 2