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Using the realization idea of simultaneous shape
invariance with respect to two different parameters of the
associated Legendre functions, the Hilbert space of spherical
harmonics $Y_{n\,m}(\theta,\varphi)$ corresponding to the motion of a
free particle on a sphere is splitted into a direct sum of
infinite dimensional Hilbert subspaces. It is shown that these
Hilbert subspace constitute irreducible representations for the
Lie algebra $u(1,1)$. Then by applying the lowering operator of
the Lie algebra $u(1,1)$, Barut-Girardello coherent states are
constructed for the Hilbert subspaces consisting of $Y_{m\,m}
(\theta,\varphi)$ and $Y_{m+1\,m} (\theta,\varphi)$.
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