# Spectrum of the Laplacian on p-forms on the sphere

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of the eigenforms and spectrum of the Laplacian acting on $p$-forms on the round sphere $S^n$, apparently citing an unpublished computation of Calabi.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

• Ref: Berger, Gauduchon, Mazet, Le Spectre d'un variete Riemannienne – Donu Arapura Apr 16 '15 at 23:24
• I think it's perfect for the laplacian on functions but they don't say much about the one on forms – David P Apr 17 '15 at 7:35
• Good point. The other, more relevant, reference that I forgot to mention is Folland, Harmonic Analysis of the de Rham complex of the sphere, Crelles 1989 – Donu Arapura Apr 17 '15 at 8:20

## 1 Answer

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.

• Did you do this by seeing in Ikeda-Taniguchi which irreps of SO(2m) appear in $C^\infty(\Lambda^p (S^{2m-1}))$ and seeing which engenvalue the Casimir has on them? If so I think it should be $\lambda_{k,p} = (k+p)(k+2m-p)$. – David P Apr 20 '15 at 10:03
• I corrected the highest weights of $\pi_{k,p}$. I hope now this coincides with Ikeda-Taniguchi's work. – emiliocba Apr 21 '15 at 0:17