A New Proof of Harish-Chandra’s Integral Formula
A New Proof of Harish-Chandra’s Integral Formula
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DOI：
10.1007/s00220-018-3259-9
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发表时间：
2017-12
影响因子：
2.4
通讯作者：
Colin S. McSwiggen
中科院分区：
文献类型：
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作者：
Colin S. McSwiggen
我们提出了 Harish-Chandra 公式的新证明（Harish-Chandra in Am J Math 79:87–120, 1957） Π ( h 1 ) Π ( h 2 ) ∫ G e ⟨ Ad g h 1 , h 2 ⟩ d g = [ [ Π , Π ] ] | 西 | Σ w ∈ W ϵ ( w ) e ⟨ w ( h 1 ) , h 2 ⟩ ，其中G是紧连通半单李群，dgi是归一化哈尔测度，h1和h2位于复李代数的Cartan子代数中，是判别式，是Killing形式，是将Killing形式扩展到多项式的内积，Wis a韦尔集团，and 是 的标志。本文的证明源自半单李代数上的热流和嘉当子代数上的热流之间的关系，扩展了 Itzykson 和 Zuber 开发的方法（J Math Phys 21:411–421, 1980），适用于酉群 U(N) 上的积分情况。热流证明允许采用系统方法来研究各种群上轨道积分的渐近性。
We present a new proof of Harish-Chandra’s formula (Harish-Chandra in Am J Math 79:87–120, 1957) Π ( h 1 ) Π ( h 2 ) ∫ G e ⟨ Ad g h 1 , h 2 ⟩ d g = [ [ Π , Π ] ] | W | ∑ w ∈ W ϵ ( w ) e ⟨ w ( h 1 ) , h 2 ⟩ , whereGis a compact, connected, semisimple Lie group,dgis normalized Haar measure,h1andh2lie in a Cartan subalgebra of the complexified Lie algebra,is the discriminant,is the Killing form,is an inner product that extends the Killing form to polynomials,Wis a Weyl group, andis the sign of. The proof in this paper follows from a relationship between heat flow on a semisimple Lie algebra and heat flow on a Cartan subalgebra, extending methods developed by Itzykson and Zuber (J Math Phys 21:411–421, 1980) for the case of an integral over the unitary groupU(N). The heat-flow proof allows a systematic approach to studying the asymptotics of orbital integrals over a wide class of groups.