Geodesic
The Explicit Fourier Decomposition of L2SO(n)/SO(n - m))
Canadian journal of mathematics, Tome 27 (1975) no. 2, pp. 294-310

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The decomposition of L2SO(n)/SO(n - m)) into a direct sum of irreducible representations of SO(n) is given abstractly by the branching theorem and the Frobenius reciprocity theorem [1]. The goal of this paper is to obtain this decomposition explicitly, generalizing the theory of spherical harmonics (m = 1). The case m = 2 has been studied in Levine [6], and the case 2m ≦ n in Gelbart [3]. Our results shed more light on these cases as well as revealing new phenomena which only occur when 2m > n. 
Strichartz, Robert S. The Explicit Fourier Decomposition of L2SO(n)/SO(n - m)). Canadian journal of mathematics, Tome 27 (1975) no. 2, pp. 294-310. doi: 10.4153/CJM-1975-036-x
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