rt.representation theory - Finite dimensional spherical representation of $SO(n,1)(mathbb{R})$

Bonsoir Ludo! I am puzzled by the fact that your title asks something more restrictive than the OP, since the latter does not contain the word "spherical". Let me answer the latter first. Any finite-dimensional representation of $SO(n,1)(mathbb{R})$ extends to a representation of the complexification, which is $SO_{n+1}(mathbb{C})$. By Weyl's unitary trick, those are in 1-1 correspondence with unitary finite-dimensional representation of the maximal compact subgroup of the complexification, here $SO_{n+1}$. The finite-dimensional, unitary, irreducible representations of such a group are parametrized by their highest weight, and can be described via Verma modules, see Chapter IV in Knapp's "Representation theory of semi-simple groups" (Princeton UP, 1986).

Now, if you need only spherical irrep's, this amounts to consider irreducible $SO_{n+1}$-representations having non-zero $SO_n$-invariant vectors; or, equivalently (by an easy case of Frobenius reciprocity), irreducible $SO_{n+1}$-sub-representations of $L^2(S^n)$ (where $S^n=SO_{n+1}/SO_n$ is the $n$-sphere). These correspond to homogeneous harmonic polynomials in $n+1$ variables.

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