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This is a follow-up to my previous question. Given a semisimple symmetric space $M\simeq G/H$, in particular, the real hyperbolic space $H_{p,q}\simeq SO(p,q)/SO(p,q-1)$, and a vector bundle $E$ over $M$ (for simplicity let's consider only the tensor bundles), the $L^{2}$-sections of $E$ constitute an unitary representation $\rho_{E}$ of $G$.

I'd like to know if the decomposition of $\rho_{E}$ into unitary irreps of $G$ has been studied or not, at least for the real hyperbolic space.

If not, can the spherical function method (see e.g. the explanations in my previous question) be generalized to the case of vector bundles?

For the second question, I found in Harmonic Analysis for Vector Fields on Hyperbolic Spaces that the authors have generalized this method to the tangent bundle case, but completeness is ensured by a Jacobi transform.

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    $\begingroup$ You only get a representation of $G$ on the sections, if the bundle is homogeneous, that is, induced by a finite-dimensional representation $\tau$ of $H$. For the representation of $G$ to be unitary, $\tau$ needs to be unitary, but there aren't that many finite dimensional unitary reps of $H$, unless $H$ is compact, i.e., $q=1$. $\endgroup$
    – Antonius
    Commented Jan 24 at 15:35

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