A note on the continuation of Borel-de Siebenthal discrete series out of the canonical chamber (Q1204439)

Let \(\mathfrak{g}_ 0\) be the Lie algebra of a real reductive group \(G\) with maximal compact subgroup \(K\) such that \(\text{rank} G=\text{rank} K\). Let \({\mathfrak k}_ 0\) be the Lie algebra of \(K\). Fix a compact Cartan subalgebra \({\mathfrak t}_ 0\) of \({\mathfrak g}_ 0\). Let \(\mathfrak g\), \(\mathfrak k\) and \(\mathfrak t\) be the complexifications of \(\mathfrak{g}_ 0\), \({\mathfrak k}_ 0\) and \({\mathfrak t}_ 0\) respectively, let \(\Delta\) be the set of \(\mathfrak t\)-roots of \(\mathfrak g\) and \(\Delta({\mathfrak k})\) be the set of \(\mathfrak t\)- roots of \(\mathfrak k\). Fix a positive system \(\Delta^ +\subset\Delta\) with simple roots \(\alpha_ 1,\alpha_ 2,\dots,\alpha_ n\) and assume that \(\alpha_ i\) are all compact except \(\alpha_ \ell\). Assume also that the coefficient of \(\alpha_ \ell\) in the expansion of the maximal root is two. Let \(\mathfrak q\) be the maximal parabolic defined by \(\alpha_ \ell\). There exists an element \(\zeta\) in \({\mathfrak t}^*\) such that \((\alpha_ i,\zeta)=\delta_{i\ell}\). Let \(\xi=((\alpha_ \ell,\alpha_ \ell)/2)\zeta\) and \(\lambda_ 0\) be the unique point on the line \(z\xi\), \(z\in\mathbb{R}\), such that the sum of \(\lambda_ 0\) and half the sum of positive roots lies on the wall of the \(\mathfrak q\)- antidominant chamber. Let \(\lambda(z)=\lambda_ 0+z \xi\), \(z\in\mathbb{R}\). By \textit{T. J. Enright} and \textit{J. A. Wolf} [Mém. Soc. Math. Fr., Nouv. Sér. 15, 139-156 (1984; Zbl 0582.22013)], the discrete series coming from the Borel-de Siebenthal chamber are realized as Zuckerman functors of generalized Verma modules. In this paper, it is proved that there exist certain values \(c_ 0\), \(c_ 1\), such that the Zuckerman functor of highest weight module corresponding to \(\lambda(z)\), \(z \in \mathbb{R}\) is a nonzero unitary representation if \(\lambda(z)\) is \(\Delta({\mathfrak k})\)- integral and \(z < c_ 0\), and is zero if \(c_ 0 < z < c_ 1\). Moreover, the values \(c_ 0\), \(c_ 1\) are computed for types \(B_ n\), \(C_ n\), \(D_ n\) and \(F_ 4\) in this paper.