References about properties of the spectra of the Laplace-Beltrami on p-forms over homogeneous spaces

Question

I am reading the paper "Specra and Eigenforms of the Laplacian on $\mathbb{S}^n$ by Ikeda-Taniguchi and $\mathbb{P}^n(\mathbb{C})$" and i want to know where do i can get a proof or some ideas to prove some statements about the Laplacian defined on Homogeneos Spaces, which are given in the text. The statements are the following:

1. The Laplacian is an strongly-elliptic operator on $p$-forms
2. The eigenvalues of $\Delta$ on $p$-forms form a discrete set of non-negative real numbers.
3. The eigenspaces of the Laplacian are all fininte dimensional spaces and the algebraic sum of all the eigenspaces is a dense subset of $C^\infty(\Lambda^pM)$.

I think the properties 2 and 3 follows from the self-adjointness of $\Delta$ and 1. But since i have never studied differential operators on manifolds, i have no ideia of good references to understand these statements, much less to prove them.

Somebody knows how do i get some material to understand this?