A geometric approach to a description of the spectrum of the Hecke algebra on the subspace generated by unramified Eisenstein series

Let $E$ be a global field, $\mathbb{A}$ it ring of adeles and $\mathcal{O} \subset \mathbb{A}$ the subring of integers. Let $G$ be a split reductive group, $\Lambda$ be the lattice of coweights of the maximal torus $T \subset G$.
Let $Z = \Lambda \otimes R$ if $\mathrm{char}(F) = 0$ and $Z = \Lambda$ if $\mathrm{char}(F) 6= 0$. Denote by $L$ be the algebra of compactly supported continuous complex valued functions on $Z$ and by $H \subset L$ the subalgebra of $W$-invariant real-valued functions. We denote by $\mathrm{Eis} : L \rightarrow L^2(G(\mathcal{O})\backslash G(\mathbb{A})/G(E))$ the morphism defined by pseudo-Eisenstein series and by $V \subset L^2(G(\mathcal{O})\backslash G(\mathbb{A})/G(E))$ the closure of it's image. The natural action of $H$ on $L$ defines an action of $H$ on $V$ by commuting selfadjoint operators.
I will present our work with A.Okuonkov on a geometric approach to a description of the spectum of $H \subset V$ as a C*-algebra.