Timeline for First eigenvalue of the Laplacian on Berger spheres
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12 events
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| Jun 19, 2011 at 22:28 | vote | accept | Renato G. Bettiol | ||
| Jun 19, 2011 at 22:27 | answer | added | Renato G. Bettiol | timeline score: 5 | |
| Jun 17, 2011 at 21:57 | comment | added | Claudio Gorodski | I think I see your objection. The Laplacian in $G$ is in general ´$\sum_i E_i^2 -\nabla_{E_i}E_i$´ for (E_i) orthonormal basis. In the Killing metric in $G$ the linear terms disappear and it is represented by the Casimir element, whic acts on a certain representation by the formula you quoted. An arbitrary left-invariant metric in $G$ can be diagonalized wrt Killing metric and the eigenvectors specify 1-parameter groups which are still geodesics. In particular, for the metrics you have, I think you get no terms of order $1$ in the Laplacian. | |
| Jun 17, 2011 at 16:59 | comment | added | Claudio Gorodski | No, the Laplacian of the normal homogeneous metric corresponds to the Casimir element and has the eigenavalues you described, but for the other metrics in the 1-parameter family you need to modify the Casimir element accordingly and correct the numerical value of the eigenvalues. Construct the Casimir element using a Killing orthonormal basis adapted to the submersion $SU(n+1)\to SU(n+1)/SU(n)$ and renormalize in one direction. For the normal homogeneous metric, Helgason's "Groups and geometric analysis" is also a good reference. | |
| Jun 17, 2011 at 16:48 | history | edited | Renato G. Bettiol |
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| Jun 17, 2011 at 16:39 | comment | added | Renato G. Bettiol | If this is indeed the result you were mentioning, to obtain the Berger metrics on $SU(n+1)/SU(n)$ we consider a 1-parameter family of left-invariant metrics on $SU(n+1)$ (and the same should hold in the other homogeneous spaces $Sp(n+1)/Sp(n)$ and $Spin(9)/Spin(7)$). Even if we take, in the above formula for $c(\gamma)$, the inner product on $\mathfrak h^*$ (the Lie algebra of the maximal torus) to be the one induced by this 1-parameter family, why should $c(\gamma)$ be the eigenvalues of $\Delta_t$, even for the $\gamma$'s that have a non-zero vector fixed by $SU(n)$ (or $Sp(n),Spin(7)$)? | |
| Jun 17, 2011 at 16:32 | comment | added | Renato G. Bettiol | Hi Claudio, I believe you are refering to Lemma 5.6.4 on p. 124 of Wallach's book (the 1973 print), where it is shown that the Laplacian on a compact Lie group $G$ acts on the factors $\Gamma_\gamma E$ of the left regular representation decomposition as the scalar $c(\gamma)=\langle \Lambda_\gamma+\rho,\Lambda_\gamma+\rho\rangle-\langle\rho,\rho\rangle$, where $\gamma$ is the irreducible representation in question, $\Lambda_\gamma$ is the highest weight of $\gamma$ and $\rho$ is half the sum of the positive roots of $G$. However I still don't see how to apply this on any homogeneous space... | |
| Jun 13, 2011 at 13:02 | comment | added | Claudio Gorodski | In fact, the one-parameter family of metrics do not consist of normal homogeneous metrics, except for the basic one. It is still possible to adapt the method nontheless. | |
| Jun 13, 2011 at 1:34 | comment | added | Claudio Gorodski | Hi, Renato! It can be done probably this way. The Berger sphere $G/K=S^{2n+1}=SU(n+1)/SU(n)$ is a normal homogeneous space. Peter-Weyl decomposes $L^2(G)$ wrt left regular representation and this induces a decomposition of $L^2(G/K)$ into $G$-irreducible representations. The Laplacian acting on $L^2(G/K)$ is the same as the Casimir element of $\mathfrak g$ acting on those representations, and it acts as a scalar, for which there is an explicit and easy formula. The details are in Nolan Wallach's book on Harmonic Analysis (if I rememeber well), I think you can figure out. | |
| Jun 13, 2011 at 0:04 | history | edited | Renato G. Bettiol |
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| Jun 12, 2011 at 23:21 | history | edited | Renato G. Bettiol |
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| Jun 12, 2011 at 22:57 | history | asked | Renato G. Bettiol | CC BY-SA 3.0 |