In this paper the authors give an explicit description of the eigenforms and spectrum of the Laplacian acting on $p$-forms on the round sphere $S^n$, apparently citing an unpublished computation of Calabi.
Although the AMS reviewer says that there is a "minor typographical errors" in such computation.
Can you help me to find the errors or suggest me a correct reference for the laplacian on p-forms on the round sphere, possibly more explicit than Ikeda and Taniguchi's 1978 paper?
Thanks
David
I like the following description of the spectrum of $S^{2m-1}$ via representation theory of $SO(2m)$. Assume $n=2m-1$. Let $\mathcal E_0=\{0\}$ and $$\mathcal E_p=\{\lambda_{k,p}:=k^2+k(2m-2)+(p-1)(2m-1-p): k\in \mathbb N\},$$ for $1\leq p\leq m$. One can check that $\mathcal E_p\cap\mathcal E_{p+1}$ is empty for every $0\leq p\leq m-1$.
The eigenvalues of the Hodge-Laplace operator $\Delta_p$ on $p$ forms on $S^{2m-1}$ are $\mathcal E_p\cup\mathcal E_{p+1}$ for $0\leq p\leq m-1$. The multiplicity is given by $$ \textrm{mult}(\lambda) = \begin{cases} \dim \pi_{k,p} & \text{ if } \lambda=\lambda_{k,p},\\ \dim \pi_{k,p+1} & \text{ if } \lambda=\lambda_{k,p+1}. \end{cases} $$
Here, $\pi_{k,p}$ denotes the irreducible representation of $SO(2m)$ having highest weight $(k\varepsilon_1+\dots+\varepsilon_p)$ when $1\leq p\leq m-1$, and $\pi_{k,m}$ denotes the sum of the irreducible representations with highest weight $(k\varepsilon_1+\dots+\varepsilon_m)$ and $(k\varepsilon_1+\dots+\varepsilon_{m-1}-\varepsilon_m)$ respectively.
You can compute $\dim \pi_{k,p}$ by using Weyl's dimension formula.
