Correlation functions and matrix coefficients

2014 Fiscal Year Final Research Report

• Project/Area Number: 24654038
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Global analysis
• Research Institution: The University of Tokyo
• Principal Investigator: KANAI Masawhiko 東京大学, 数理(科)学研究科(研究院), 教授 (70183035)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: 相関関数 / 行列係数 / ワイル領域流 / 剛性定理

Outline of Final Research Achievements

Correlation functions in the theory of dynamical systems and matrix coefficents in unitary representation theory are quite similar notions. The aim of the present research plan was to uncover common principles behind them. For these purposes, I took look at what is called Weyl chamber flows, which are typical examples of Anosov actions of abelian groups. In the proof of a rigidity theorem of those flows, which was done by Katok and Spatzier, the exponential decay of the correlation function of some unitary representation took the most serious part. I wanted to generalize such a decay estimate to `less symmetric' flows. As the first step, I tried to make a new understanding of the Weyl chamber flows from a geometric view point. Although there still remain a few gaps, I am about to obtain a new proof of the rigidity theorem of Katok and Spatzier. Hopefully, I will publish a new paper on it soon.

Free Research Field

幾何学