The spectral decomposition conjecture for Hecke operators over quadratic extensions

Let $F$ be the base field, let $E/F$ be an extension as in Section 1.1, and let $L$ be a division $F$-algebra of rank $n$. Let $\mathcal S^L_E$ be the associated space of sections, and let $\mathcal D$ denote the indexing set of Hecke correspondences. For each $\underline D\in\mathcal D$, let $T_{\underline D}$ be the corresponding Hecke operator on $\mathcal S^L_E$.

Spectral decomposition conjecture. There exist a set $I^E$, a function $d^E:I^E\to\mathbb N$, and distinct functions $\lambda_i^E:\mathcal D\to\mathbb C$ for $i\in I^E$, such that, for every division algebra $L$ of rank $n$, there is a direct-sum decomposition

$$\mathcal S^L_E=\bigoplus_{i\in I^E}\mathcal S^L_E(i)$$

with

$$\dim\mathcal S^L_E(i)=d^E(i)$$

and

$$T_{\underline D}|_{\mathcal S^L_E(i)}=\lambda_i^E(\underline D)\operatorname{Id}_{\mathcal S^L_E(i)}$$

for every $\underline D\in\mathcal D$.

This is presented as an extension of the quadratic-extension algebra-isomorphism conjecture and predicts a uniform simultaneous eigenspace decomposition for the Hecke operators across all division algebras $L$. The source gives no resolution and suggests the conjecture as a goal of the article.