Eigentensors for Lichnerowicz Laplacian on $\mathbb{CP}^n$

Question

Consider the Lichnerowicz Laplacian arising in the study of the stability of Einstein metrics:

$\Delta_L h_{ij} := \nabla^* \nabla h_{ij} + 2 R_{i p j q} h_{pq}$.

I am interested to know, on $\mathbb {CP}^n$, as explicitly as possible, the first eigentensors for this operator on the space of traceless, divergence-free symmetric two-tensors. My understanding is that the answer is in the 1980 paper of Koiso, ``Rigidity and stability of Einstein metrics...,'' although it is (to me) a fairly abstract exercise in representation theory. Is it possible to describe these eigentensors in a more explicit way? As a further question, do any of these eigentensors have a nontrivial invariance group?

Answer 1

The eigenvalues and the corresponding eigentensors of the Lichnerowicz Laplacian on the complex projective space are explicitly known. See here:

https://www.sciencedirect.com/science/article/pii/S0393044010000926