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Projections of orbital measures for classical Lie groups. (English. Russian original) Zbl 1360.22025

Funct. Anal. Appl. 50, No. 3, 228-232 (2016); translation from Funkts. Anal. Prilozh. 50, No. 3, 76-81 (2016).
Let \(G_n\) be one of the compact groups \(\mathrm{SO}(2n+1)\), \(\mathrm{Sp}(2n)\) or \(\mathrm{SO}(2n)\). Let \(g_n\) be the corresponding Lie algebra. Each orbit \(X\) of an adjoint action of \(G_n\) onto \(g_n\) carries a unique invariant probability measure \(\mu_X\). The set of all orbits is denoted by \(X_n\).
Take a natural projection \(p_n^k:g_n\rightarrow g_k\), \(k<n\). The projection of \(\mu_X\) can be represented as a continuous combination of orbital measures in the following manner \[ p_n^k(\mu_X)(S)=\int_{Y\in X_k} \mu_Y(S) \nu_{X, k}(dY), \]
where \(\nu_{X, k}(dY)\) is a certain probability measure on \(X_n\).
The main result of the paper is an explicit formula for \(\nu_{X, k}(dY)\).
The case of \(U(n)\) was considered in [G. Olshanski, J. Lie Theory 23, No. 4, 1011–1022 (2013; Zbl 1281.22003)] and [J. Faraut, Adv. Pure Appl. Math. 6, No. 4, 261–283 (2015; Zbl 1326.15058)].

MSC:

22E60 Lie algebras of Lie groups
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References:

[1] Curry, H. B.; Schoenberg, I. J., No article title, J. Anal. Math., 17, 71-107 (1966) · Zbl 0146.08404 · doi:10.1007/BF02788653
[2] Faraut, J., No article title, Adv. Pure Appl. Math., 6, 261-283 (2015) · Zbl 1326.15058 · doi:10.1515/apam-2015-5012
[3] Harish-Chandra, No article title, Amer. J. Math., 79, 87-120 (1957) · Zbl 0072.01901 · doi:10.2307/2372387
[4] Olshanski, G., No article title, J. Lie Theory, 23, 1011-1022 (2013) · Zbl 1281.22003
[5] G. M. Phillips, Interpolation and approximation by polynomials, CMS Books in Math, vol. 14, Springer-Verlag, New York, 2003. · Zbl 1023.41002 · doi:10.1007/b97417
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