A construction of irreducible representations of the algebra of invariant differential operators on a homogeneous vector bundle and its applications (Q1335281)

Given a connected semisimple Lie group \(G\) with finite center and its maximal compact subgroup \(K\), every irreducible unitary representation \(\tau\) of \(K\) induces a \(G\)-homogeneous vector bundle \(E_ \tau\) over the symmetric space \(G/K\). The algebra \(D_ \tau\) of \(G\)-equivariant differential operators on \(E_ \tau\) acts on every admissible representation \(\pi\) of \(G\) that contains the \(K\)-type \(\tau\), i.e. \(\pi(\tau) \neq 0\). For principal series representations \(\pi\) this can be realized as the ``Poisson transform'', an integral transform that maps the \(\tau\)-type of \(\pi\) into the sections of \(E_ \tau\). The paper under consideration gives criteria for the nontriviality of \(\pi(\tau)\) for irreducible \(\pi\) and for the nontriviality of the Poisson transform.