6
$\begingroup$

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of the eigenforms and spectrum of the Laplacian acting on $p$-forms on the round sphere $S^n$, apparently citing an unpublished computation of Calabi.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

$\endgroup$
  • $\begingroup$ Ref: Berger, Gauduchon, Mazet, Le Spectre d'un variete Riemannienne $\endgroup$ – Donu Arapura Apr 16 '15 at 23:24
  • 1
    $\begingroup$ I think it's perfect for the laplacian on functions but they don't say much about the one on forms $\endgroup$ – David P Apr 17 '15 at 7:35
  • $\begingroup$ 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 $\endgroup$ – Donu Arapura Apr 17 '15 at 8:20
7
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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 $\lambda_{k,p} = (k+p)(k+2m-p)$. $\endgroup$ – David P Apr 20 '15 at 10:03
  • $\begingroup$ I corrected the highest weights of $\pi_{k,p}$. I hope now this coincides with Ikeda-Taniguchi's work. $\endgroup$ – emiliocba Apr 21 '15 at 0:17

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.