\(M\)-spherical \(K\)-modules of a rank one semisimple Lie group (Q1424386)

Let \(G_\circ=K_\circ A_\circ N_\circ\) be the Iwasawa decomposition of a connected non-compact real semisimple Lie group with finite centre and let \(M_\circ\) be the centralizer of \(A_\circ\) in \(M_\circ\). Let \({\mathfrak g}={\mathfrak k}\oplus {\mathfrak p}\) be the complexification of the corresponding Cartan decomposition of \(\text{ Lie\,}G_\circ\). It was proved by B.~Kostant that, for any \(M\)-spherical \(K\)-module \(V\), there exists a unique \(d\) (the Kostant degree of \(V\)) such that \(V\) can be realized as a submodule of the space of all \(\mathfrak k\)-harmonic polynomials of degree \(d\) on \(\mathfrak p\). In the paper under review, the authors give an algorithm for obtaining a highest weight vector from any \(M\)-invariant vector in an irreducible \(M\)-spherical \(K\)-module. This algorithm allows to compute a sharp bound for the Kostant degree, \(d(v)\), of any \(M\)-invariant vector \(v\) in a locally finite \(M\)-spherical \(K\)-module \(V\). This method computes \(d(v)\) effectively if the real rank of \(G_\circ\) is equal to one.