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Class operators for compact Lie groups. (English) Zbl 0895.22005

The paper is concerned with interpretation and evaluation of the integral \[ \widehat{K}_H(\rho_H)=\int_{G/H}e^{i\rho(x)}d\mu_{G/H}(x) \] called the class operator related to \(G/H\). Here \(G\) is a compact Lie group, \(G/H\) its coset space representing a conjugacy class of \(G\), \(x\mapsto \rho(x)\) is the parametrization of an \(\text{ Ad}\)-orbit in \(L(G)\), the Lie algebra of \(G\), by elements of \(G/H\), with \(H\) being the centralizer in \(G\) of a fixed element \(\rho_H\in L(G)\) belonging to the chosen orbit, \(e\) denotes the exponential mapping and \(d\mu_{G/H}(x)\) is the invariant measure on \(G/H\). In the special case when \(G=SU(2)\) and \(H=U(1)\), the space \(G/H\) can be naturally identified with the two-dimensional unit sphere \(S^2\) and the integral reduces to \[ {1 \over 4\pi}\int^{2\pi}_0\int^\pi_0 \exp[i\psi(J_x\sin\theta\cos\varphi+ J_y\sin\theta\sin\varphi+ J_z\cos\theta)]\sin\theta d\theta d\varphi \] with \(J_x, J_y, J_z\) denoting the infinitesimal generators of rotations around the \(x, y, z\) axes. Computation of this latter integral in the paper of Fan Hongyi and Ren Yon [J. Phys. A, Math. Gen. 21, 1971-1976 (1988; Zbl 0659.22014)] has been a starting point for a series of papers, simplifying and generalizing the original method, in particular by two of the present authors [N. B. Backhouse, J. Phys. A, Math. Gen. 21, L1113–L1115 (1988; Zbl 0666.22005) and J. Rembieliński, ibid. 22, 591-592 (1989; Zbl 0683.22016)], the reviewer and A. Orłowski [J. Phys. A, Math. Gen. 27, 167-175 (1994; Zbl 0805.22010)] and E. Montaldi and G. Zucchelli [Rend. Semin. Mat. Fis. Milano 64, 217-222 (1994; Zbl 0849.22022)].
The authors interpret the former integral as a formal power series in the enveloping algebra of \(L(G)\) and show that it can be computed in terms of characters of irreducible representations of \(G\). In the special case of \(SU(3)\) they also give an explicit formula for the value of this integral in terms of the Casimir operators for the group.
Another approach to the extension of the class operator has been proposed by the reviewer together with A. Orłowski [cf., e.g., J. Math. Phys. 38, 2720-2727 (1997; Zbl 0874.22022)].

MSC:

22E30 Analysis on real and complex Lie groups
22E70 Applications of Lie groups to the sciences; explicit representations
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
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