What is the exact value of the first eigenvalue of the laplacian for complex projective space viewed as $SU(n+1)/S(U(1)\times U(n))$?

The spectrum of the Laplacian of $\mathbb C P^n$ with the FubiniStudy metric is
So, the first nonzero 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 nonzero 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 $k(k+n1)$, and its multiplicity is $\binom{n+k}{k}\binom{n+k1}{k1}$, see e.g. [Berger, Gauduchon,Mazet, "Le spectre d'une variété riemannienne", Lecture Notes in Mathematics, Vol. 194 SpringerVerlag]. 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, $2k(n+2k1)$, $4k(n+k)$ 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. 


See SPECTRA AND EIGENFORMS OF THE LAPLACIAN ON $S^n$ AND $P^(C)$ (osaka j math 1977, you can skip to page 529 or if really lazy look at Theorem 5.2 


Not an answer but advise. Laplace probably(?) comes(should I explain this "comes"???) from quadratic Casimir in U(g) up to scalar factor which depends on volume. Casimir can be written as \sum_i e^ie_i, where e^i and e_i are dual basises in g, with respect to CartanKilling form. So the question is what is the minimal eigen of quadratic Casimirs in finitedim representations which enter decomposition of L^2(CP^n). I guess(?) standard vector representation of su(n) in C^n enters this decomposition. I guess(?) minimal eigen of quadratic Casimir in ALL irreps corresponds to this C^n. If all guesses are correct you need just to calculate e^ie_i value in C^n and also care about the scalar which is related to volume normalization. 

