I'm looking for an explicit description of all the finite dimensional irreducible representation of the Lie group $SO(n,1)(\mathbb{R})$. Can you tell me, where I can find this description ? Thank you.
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1$\begingroup$ What does "pherical" mean? $\endgroup$– José Figueroa-O'FarrillCommented Aug 26, 2010 at 16:00
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$\begingroup$ I am not sure what you mean by "explicit", but I think that they are all given by tensors of the ($n+1$)-dimensional vector representation. The representation ring of the Lie algebra $\mathfrak{so}(n,1)$ is generated by the vector and spinor representations, but the spinorial representations are not representations of $SO(n,1)$. $\endgroup$– José Figueroa-O'FarrillCommented Aug 26, 2010 at 16:03
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3$\begingroup$ My guess is that "pherical" should have been "spherical", which means "contains a K-invariant vector", where $K$ is a maximal compact subgroup. In that case, the answer is given by the symmetric powers of the defining representation. $\endgroup$– Victor ProtsakCommented Aug 26, 2010 at 17:12
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2$\begingroup$ No, the adjoint representation is not spherical for $n\geq 3.$ You need to take the defining representation of $SO(n,1)$ on $\mathbb{R}^{n+1},$ which is self-adjoint and spherical (the last basis vector is $K$-invariant), and its symmetric powers (which can be realized in the homogeneous polynomials of degree $d$) are spherical and, moreover, they exhaust all finite-dimensional spherical representations of $SO(n,1).$ I have no clue what you mean by "the Lie Algebra Unknown control sequence". $\endgroup$– Victor ProtsakCommented Sep 3, 2010 at 5:14
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1$\begingroup$ "Unknown control sequence" seems to refer to a TeX error... $\endgroup$– Alain ValetteCommented Aug 7, 2011 at 20:44
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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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$\begingroup$ Thanks Alain for your explanation. I'm only interest with the spherical representation. I make cup/paste mistake when I was taping the OP. I will look at Knapp's book. I want to undestand "geometrically" and "dynamically" this representation. $\endgroup$ Commented Aug 17, 2011 at 15:02