Multiplicities in restricted root systems of split real rank one groups

Question

In Knapps book, Lie groups beyond an introduction, p.372-373, the restricted roots of $SU(n,1)$ and $SO_e(n,1)$ and their multiplicities are computed. Does anyone know a source where this is done for $Sp(n,1)$? Mainly I am interested just in the multiplicities. Thanks in advance.

Answer 1

The restricted root system is of type $BC_1$, i.e. after choosing a simple root $\lambda$ it consists of $\pm \lambda, \pm 2 \lambda$. This can be read off the Satake-Tits diagram easily -- not that there are many other options for rank 1 root systems anyway.

As for multiplicities, different from an erroneous claim I made in a comment, according to Onishchik/Vinberg, table 9, p. 312/313 (penultimate row), if your original group is of type $C_\ell, \ell \ge 3$, then the multiplicity of $\lambda$ is $4\ell-8$, and the multiplicity of $2\lambda$ is $3$.

That table contains all such multiplicities you could ask for; I doubt however that the reference contains a calculation for those.

However, it's easy to get the following: the total dimension of the root spaces in the complexified version ($2\ell^2$) minus the ones that are in the anisotropic kernel of the group (which is something of type $A_1 \times C_{\ell-2}$, giving $2+2(\ell-2)^2$) tells us that we need a total dimension of root spaces of $8 \ell -10$, hence the multiplicities of $\lambda$ and $2\lambda$ have to add up to $4\ell-5$. That's consistent with the above result, and now it would suffice to prove that the multiplicity of $2\lambda$ is $3$. But for this one might have to write down some matrix representation after all.