Given semisimple linear connected Lie group $G$ with maximal compact subgroup $K$, It is a known result that the $L^2$-eigenspaces of the Laplacian on a symmetric space $G/K$ can be identified with the direct summand of $K$-invariants of discrete series representations, i.e. taking $X:=G/K$ \begin{equation} L^2(X, \Lambda^pT^*X)_{disc}\cong \bigoplus_{\pi\in \hat{G}_d}V_\tilde{\pi}\otimes [V_\pi\otimes \Lambda^p\mathfrak{p}^*]^K \end{equation} where right hand side corresponding to the discrete part of Laplacian, i.e., the $L^2$-eigenspaces, and $\hat{G}_d$ is the equivalence class of all discrete series representation.

To see the right hand side is a $G$-subspace of the left hand side is easy, since Casimir operators acting on $V_\pi^\infty$ by scalar. On the other hand,

Question: I cannot see why the total $L^2$-eigenspace can be written as a direct sum of discrete series representation.

I find a more general answer to this question, namely for any pseudifferential elliptic operator on equivariant setting this holds true. This was proven by Connes and Moscovici. Nonetheless, I was wondering if there is some way to get it easier (in fact almost straightforward) from Plancherel formula, as hinted by Martin Olbrich right in his paper before equation (9).


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