The embeddings of discrete series into principal series for an exceptional real simple Lie group of type \(G_ 2\)
For an exceptional real simple Lie group \(G\) of type \(G_2\) the authors determine the embeddings of the discrete series into the principal series of \(G\). Let \(G=KAN\) be an Iwasawa decomposition of \(G\) and \(P=MAN\) the minimal parabolic subgroup of \(G\). Let \(\text{Ind}^G_P (\sigma \otimes e^{\mu+ \rho_P} \otimes 1_N)\) denote the principal series of \(G\) induced from \(P\) and \(\pi_\Lambda\) the discrete series of \(G\) with Harish-Chandra's parameter \(\Lambda\). To determine the embeddings the authors use the following theorem obtained by the second author [Jap. J. Math., New Ser. 16, 31-95 (1990; Zbl 0759.22019)]: if the Blattner parameter \(\lambda\) is far from the walls \[ \Hom_{({\mathfrak g}, K)} \bigl(\pi^*_\Lambda, \text{Ind}^G_P (\xi\times 1_N) \bigr) \simeq\Hom_{({\mathfrak a},M)} (\widetilde \xi^*, \text{Ker} D_{\lambda,1_N}), \] where \(D_{\lambda,1_N}\) is a gradient type differential operator on \(G\), and then they apply Zuckerman's translation functors to remove the restriction that \(\lambda\) is far from the walls. Actually, they explicitly determine the \(MA\)-module structure of \(\text{Ker} D_{\lambda,1_N}\) and calculate the parameters for which the left hand side is nontrivial.
scientific article; zbMATH DE number 938085