TITLE: The first eigenvalue of the laplacian for complex projective space
QUESTION [15 upvotes]: What is the exact value of the first eigenvalue of the laplacian for complex projective space viewed as $SU(n+1)/S(U(1)\times U(n))$?

REPLY [18 votes]: The spectrum of the Laplacian of $\mathbb C P^n$ with the Fubini-Study metric is

$$Spec(\Delta_{\mathbb C P^n})=\{4k(n+k):k\in\mathbb N\} \quad\quad(*)$$

So, the first non-zero eigenvalue of $\mathbb C P^n$ is $\lambda_1=4n+4$.
Note this matches with the fact that $\mathbb C P^1$, with the FS metric, is isometric to the $2$-sphere of radius $1/2$, whose first non-zero eigenvalue is $\lambda_1=8$.

Let me quote a brief justification of (*) that I had written here:
Using the classic spherical harmonics theory, one obtains the $k$-th eigenvalue of the $n$-dimensional round sphere $S^n$ to be $k(k+n-1)$, and its multiplicity is $\binom{n+k}{k}-\binom{n+k-2}{k-2}$. By looking at eigenfunctions of the Laplacian on $S^n$,$S^{2n+1}$ and $S^{4n+3}$ (note they are the unit spheres of $\mathbb R^{n+1}$, $\mathbb C^{n+1}$ and $\mathbb H^{n+1}$) that are respectively invariant under the natural actions of $\mathbb Z_2$, $S^1$ and $S^3$, one can obtain the eigenfunctions hence the $k$-th eigenvalue of the projective spaces $\mathbb R P^n$, $\mathbb C P^n$ and $\mathbb H P^n$, respectively. These are, respectively, $2k(n+2k-1)$, $4k(n+k)$ and $4k(k+2n+1)$.
If you can understand some French, you will find a thorough explanation of the above in the book by Berger, Gauduchon, Mazet, "Le spectre d'une variete Riemannienne", Lecture Notes in Math, Springer, vol 194; but beware of a small typo [pointed out to me by G. Wei] on the multiplicity of the $k$th eigenvalue of $S^n$ (cf. above).
More generally, it is possible to compute the Laplace spectrum of a compact homogeneous space using some representation theory, as explained e.g. in Nolan Wallach's "Minimal immersions of symmetric spaces into spheres" or in more detail on his book "Harmonic analysis on homogeneous spaces" (Pure and applied mathematics, 19).