The socle filtrations of principal series representations of \(\mathrm{SL}(3, \mathbb{R})\) and \(\mathrm{Sp}(2, \mathbb{R})\) (Q6611876)

Let \(G\) be \(\mathrm{SL}(3,\mathbb{R})\) or \(\mathrm{Sp}(2,\mathbb{R})\). This paper aims to completely describe the socle filtrations of the principal series of \(G\) induced from minimal parabolic subgroups with non-singular infinitesimal characters. The most important part of the computation is to seek the candidates for the irreducible submodules of principal series in question using shift operators of \(K\)-types. Then, the author uses parity condition to narrow down the candidates for irreducible factors lying in each floor and uses horizontal symmetry to identify which floor a fixed irreducible factor lies in.