On a characteristic function of the tensor \(K\)-module of inner type noncompact real simple groups
Midorikawa considers a simply connected complex Lie group \(G_c\) together with some noncompact real version \(G\) of it. Let \(K\) be a maximal compact subgroup of \(G\) and \({k}\oplus{p}\) the related Cartan decomposition of its Lie algebra. Then the complexified version \({k}_c\oplus{p}_c\) decomposes the complex Lie algebra of \(G_c\). If \(V\) is a \(K\)-module, so is \(L={ p}_c\otimes V\). Let \(V\) be simple with heighest weight \(\mu\). Moreover, if \( b\) is a maximal abelian subalgebra of \( k\), one may consider the root system of \({ b}_c\) and obtain a decomposition \(L=\oplus L_\omega\) where \(\omega\) runs over all noncompact roots such that \(\mu+\omega\) is compact. Either \(L_\omega\) is a simple \(K\)-submodule or trivial, i.e.\ \(L_\omega=\{0\}\). The purpose of the paper is to determine when it is nontrivial by checking whether some rational function is nonzero. The tensor space \({ p}_c\otimes V\) has been used in the literature to classify the unitary representations of \(G\) when \(G\) is either \(SL(2,R)\), the de Sitter group, \(SO(2n,1)\) or \(SU(n,1)\).
scientific article; zbMATH DE number 2064099
scientific article