Invariant differential operators and eigenspace representations on an affine symmetric space (Q5960966)

The author studies a certain preferred polynomial algebra \(D\) of \(G\)-invariant differential operators on an affine symmetric space \(G/H\) of split rank \(r\). This preferred polynomial algebra \(D\) is generated by \(r\) elements. The author shows that the space of \(K\)-finite joint eigenfunctions of \(D\) on \(G/H\) constitutes an admissible \((\mathfrak g, K)\)-module. Generically, these representations are shown to be isomorphic to a direct sum of explicit irreducible principal series for \(G/H\). In the nongeneric case, the representation is shown to have the same composition series as the corresponding direct sum of principal series for \(G/H\). In addition, the author gives a Poisson transform for \(\tau\)-spherical eigenfunctions on \(G/H\) based on Eisenstein integrals.NEWLINENEWLINE Editorial remark: The paper has been withdrawn [ibid. 186, No. 1, 319 (2017; Zbl 1368.22009)].