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I am interested in paper " Sharp constants in several inequalities on the Heisenberg group " of Rupert L.Frank and Elliott H.Lieb " http://arxiv.org/pdf/1009.1410v2.pdf. In this paper ( page 17 ), the authors mention about the Funk-Hecke theorem on the complex sphere. In particularly,

$ L^2(\mathbb{S}^{2n+1}) = \bigoplus_{j,k \geq} \mathscr{H}_{jk}$ (1), the space $\mathscr{H}_{jk}$ is the space of restrictions to $\mathbb{S}^{2n+1}$ of harmonic polynomials $p(z,\bar{z})$ on $\mathbb{C}^{n+1}$ that are homogeneous of degree $j$ in $z$ and $k$ in $\bar{z}$. In proposition 5.2, the authors claim that the operator on $\mathbb{S}^{2n+1}$ with kernel $K(\zeta \cdot \bar{\eta})$ is diagonal with respect decomposition (1).

The authors just explain briefly that the operator is diagonal follows from Schur's lemma and the irreducibility of the space $\mathscr{H}_{jk}$. And I can't understand this explanation.

I just know Schur's lemma in representation theory.Is there another Schur's lemma ? What reference should I read to understand about Funk-Hecke theorem.

Improve this question
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  • $\begingroup$ That is the same Schur's lemma: every automorphism of an irreducible representation is multiplication by a constant. Hence any operator commuting with the group action is a constant on each irrep, so is diagonal. $\endgroup$
    – Ben McKay
    Mar 7, 2016 at 14:30
  • $\begingroup$ The relevant group is the group of unitary linear transformations of $\mathbb{C}^{n+1}$ acting on the sphere $S^{2n+1}$ and thereby acting on the functions on the sphere. Because the unitary group is compact, the square integrable functions on the sphere split into an infinite series of finite dimensional irreps. $\endgroup$
    – Ben McKay
    Mar 7, 2016 at 15:11
  • $\begingroup$ Thanks for your explanation. " any operator commuting with the group action is a constant on each irrep ", I am really sorry that I don't know this proposition. What reference of representation theory should I read ? $\endgroup$
    – Unrated
    Mar 8, 2016 at 17:29
  • $\begingroup$ It should be enough to read en.wikipedia.org/wiki/Schur's_lemma. I am assuming that each irrep is invariant under the operator in question. $\endgroup$
    – Ben McKay
    Mar 8, 2016 at 17:34

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