Contravariant forms on generalized Verma modules and \(b\)-functions
Let \(G\) be a complex Lie group, \({\mathfrak g}\) its Lie algebra, \({\mathfrak p}\) a parabolic subalgebra such that \(({\mathfrak g},{\mathfrak p})\) is of ``commutative parabolic type'', i.e. correponds to an Hermitian symmetric space (most results of the paper are given for the special case where \(({\mathfrak g},{\mathfrak p})\) is of ``regular type'', which corresponds to symmetric spaces of tube type). For \(\lambda\in\Hom({\mathfrak p},\mathbb{C})\), let \(M(\lambda)\) be the generalized \(U({\mathfrak g})\)-module with heighest weight \(\lambda\). Two bilinear forms on \(M(\lambda)\) are defined and their ratios on the irreducible components of the Schmid decomposition are computed in terms of \(b\)-functions. The unitarizability of the irreducible quotient of \(M(\lambda)\) is studied as an application.
scientific article; zbMATH DE number 1310266