On the commutativity of the algebra of invariant differential operators on certain nilpotent homogeneous spaces (Q2723472)

Let \(G\) be a simply connected nilpotent Lie group with Lie algebra \(g,H\) be a closed subgroup with Lie algebra \(h\). For \(\beta\in h^*\) with \(\beta ([h,h])=0\), there exists a unitary character \(\chi_\beta\) of \(H\). Let \(\tau=\text{Ind}^G_H \chi_\beta\) be the unitary representation of \(G\) induced from \(\chi_\beta\), \(C^\infty(G,H, \beta)\) be the space of smooth functions on \(G\) covariant like \(\beta\) along right \(H\)-cosets, and \({\mathcal D} (G,H, \beta)\) be the algebra of differential operators leaving \(C^\infty (G,H,\beta)\) invariant and commuting with \(\tau\). Corwin and Greenleaf proved that if \(\tau\) is of finite multiplicity, then the algebra \({\mathcal D}(G,H,\beta)\) is commutative. The authors consider the inverse of the above statement and prove that in many cases, for example:NEWLINENEWLINENEWLINE(1) if \((g,h)\) is a reductive pair, i.e., there exists a vector subspace \(m\) of \(g\) such that \(g=h+m\) and \([h,m] \subset m\);NEWLINENEWLINENEWLINE(2) if \(H\) is a connected closed normal subgroup of \(G\),NEWLINENEWLINENEWLINEthen for \(\beta\in h^*\) such that \(\beta([h, h])=0\), the commutativity of the algebra \((G,H,\beta)\) implies that \(\tau\) is of finite multiplicity.