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

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?