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The singularity of orbital measures on compact Lie groups. (English) Zbl 1052.43011

Let \(G\) be a compact, connected, simple Lie group of type \(A_n, B_n, C_n\) or \(D_n\). A measure on \(G\) is said to be central if it commutes under convolution with all other measures on \(G\). In the paper under review the authors study a class of singular, central measures. The class they study are the orbital measures. The orbital measure \(\mu_g\), supported on the conjugacy class \(C(g)\) containing \(g \in G\), is defined by \[ \int_G f \,d\mu_g = \int_G f(tgt^{-1})\,dm_G(t) \] where \(f\) is a continuous function on \(G\). Orbital measures are continuous if and only if \(g \notin Z(G)\), where \(Z(G)\) is the center of \(G\). The main theorem in this paper is that \(\widehat{\mu}^k \in \ell^2\) for all continuous orbital measures \(\mu\) on \(G\) if and only if \(k > k_0\), where \(k_0\) depends on the type of \(G\). For example, if \(G\) is of type \(B_n\), then \(k_0 = 2n - {1 \over 2}\).
Let \(\Phi\) be the set of roots for \((G,T)\), where \(T\) is a maximal torus. Let \(g \in G\) and set \(\Phi(g) = \{ \alpha \in \Phi: \alpha(g) \in 2\pi\mathbb{Z} \}\) and \(\Phi^+ (g) = \Phi(g) \cap \Phi^+\). Now \(\Phi(g)\) is a subroot system of \(\Phi\) and \(\Phi^+ (g)\) is a complete set of positive roots of the subroot system. Also, \(\Phi(g) = \Phi\) if and only if \(g \in Z(G)\). The authors list the maximal subroot systems, which were determined in a previous paper by K. E. Hare, D. C. Wilson and W. L. Yee [J. Aust. Math. Soc., Ser. A 69, No. 1, 61–84 (2000; Zbl 0994.43005)], for each type of group considered in the paper. The authors then divide this list into two parts, the better maximal subroot systems and the worst maximal subroot systems. It is proven that there exists \(g \in G\) such that \(g \notin Z(G), \Phi^+(g)\) is one of the worst maximal subroot systems and \(\widehat{\mu}_g^{k_0} \notin \ell^2\). It is also proven that if \(g \notin Z(G)\) and if \(\Phi^+(g)\) is not one of the worst maximal subroot systems, then \(\widehat{\mu}_g^{k_0} \in \ell^2\).
If \(\mu_g\) is the orbital measure associated to \(g \in G\), then the Fourier transform of \(\mu_g\) at the representation \(\lambda\) is given by \(\widehat{\mu}_g(\lambda) = Tr \lambda(g) / \deg \lambda.\) In order to prove the results stated above the authors use delicate arguments to find estimates of \(| {\operatorname{Tr} \lambda(g) \over \deg \lambda}| \) for \(g \notin Z(G)\).
In the final section of the paper the authors apply their main theorem to show that if \(g \notin Z(G)\) then \(\mu_g \ast L^p \subseteq L^2\) for \(p > p_0\), where \(p_0\) depends on the type of the group. For example, if \(G\) is of type \(A_n\) then \(p_0 = 2 - 4/(2n+3)\).
This is a clear and well written paper.

MSC:

43A80 Analysis on other specific Lie groups
22E46 Semisimple Lie groups and their representations
43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)

Citations:

Zbl 0994.43005
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References:

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