Abstract
We build an irreducible unitary representation of SO( ∞ ) from the usual one of SO( n ) in the space of harmonic homogeneous polynomials of degree m of ℝn. We give a characterization of these new representations which extends in a natural way the finite dimensional characterization. In the particular case of SO( ∞ ), we thus get some results of Ol‘shanskii (cf. [12]). This leads to a new proof of McKean conjecture about irreducible representations of ( infin ) (cf. [10]).
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Martini, C. Harmoniques Sphériques et Représentations Irréductibles de O ( ∞ ). Potential Analysis 10, 55–90 (1999). https://doi.org/10.1023/A:1008632801162
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DOI: https://doi.org/10.1023/A:1008632801162