Matrix valued commuting differential operators with \(A_2\) symmetry (Q1625128)

Let \(G = \mathrm{SL}(3,F) (F=\mathbb R, \mathbb C, \mathbb H)\) and \(K\) the usual maximal compact subgroup of \(G=KAN\). \(\tau\) is the standard representation of \(K\), \(E_\tau\) be the associated homogeneous vector bundle over \(G/K\), and \(\mathbb D (E_\tau)\) be the algebra of the left \(G\)-invariant differential operators on \( E_\tau\). \textit{A. Deitmar} [J. Reine Angew. Math. 412, 97--107 (1990; Zbl 0712.43006)] proved that \(\mathbb D(E_\tau)\) is commutative if and only if \( \tau_{| M}\) is multiplicity-free, where \(M\) is the centralizer of \(A\) in \(K\). Even when \(\mathbb D(E_\tau)\) is commutative, it seems to be hard to understand its structure and representations except the case of one-dimensional \(K\)-types. Under the author's hypothesis \(\mathbb D(E_\tau)\) is commutative because the Weyl group \(S_3\) acts transitively on constituents of \(\tau_{| M}\). In this case, \(\mathbb D(E_\tau )\simeq (S(\mathfrak a_\mathbb C)^{\{I, S_{e_1-e_2} \}}\). The author gives radial parts of two explicit generators. The author generalizes these to matrix-valued commuting differential operators by allowing the root multiplicity to be a continuous parameter and also to the case of an elliptic potential function. For the entire collection see [Zbl 1392.42001].