Harmonic analysis on compact polar homogeneous spaces
Motivated by extensions of harmonic analysis on semisimple symmetric spaces, the author introduces a new class of homogeneous spaces \(G/H\), called polar spaces. Here \(G\) is a compact semisimple Lie group, \(H\) is a closed subgroup, and the adjoint action of \(H\) on the tangent space to \(G/H\) at the origin is assumed to be a polar representation in the sense of Dadok; in particular, abelian Cartan subspaces are available in the Lie algebra of \(G\). Symmetric spaces are polar spaces, and polar spaces are spherical spaces. From Brion's classification of spherical pairs it follows that, at the Lie algebra level, the pairs \((B_3, {\mathfrak g}_2)\) and \(({\mathfrak g}_2, A_2)\) are the only simple non-symmetric polar pairs up to automorphisms. Similar definitions are given for noncompact \(G\), with the pairs \(({\mathfrak g}_{2(2)}, su(2,1))\), \(({\mathfrak g}_{2(2)}, sl(3,\mathbb{R}))\) and \((so(3,4)\), \({\mathfrak g}_{2(2)})\) as the only simple non-symmetric polar pairs. The main result of the paper states that \(G/H\) has a discrete series of representations if and only if it admits a compact Cartan subspace. The proof discusses the above three cases only, the case of symmetric spaces being settled by the work of Flensted-Jensen, Oshima and Matsuki.
scientific article; zbMATH DE number 1026460