Invariant differential operators and spherical sections on a homogeneous vector bundle
Let \(G\) be a connected Lie group and \(K\) a closed subgroup of \(G\) such that \(G/K\) is reductive. Let \(E_ \tau\) denote the homogeneous vector bundle over \(G/K\) associated to a representation \((\tau, V)\) of \(K\), \(D(E_ \tau)\) the algebra of \(G\)-invariant differential operators on \(E_ \tau\), and \((\chi,H)\) a finite dimensional representation of \(D(E_ \tau)\). The author first investigates the structure of \(D(E_ \tau)\) and obtains the algebra isomorphism \(D(E_ \tau)\cong U({\mathfrak g}_ c)^ K/ U({\mathfrak g}_ c)^ K\cap U({\mathfrak g}_ c) {\mathcal I}^ T\) under a certain condition, where \({\mathcal I}=\ker(d\tau)\) and \(T\) is the anti-automorphism of \(U({\mathfrak g}_ c)\) defined by \(1^ T=1\), \(x^ T=- x\), \((xy)^ T= y^ T x^ T\) \((x,y\in{\mathfrak g}_ c)\). Next, as a generalization of zonal spherical functions, he defines spherical sections of \(E_ \tau\) over \(G/K\) as follows. Let \({\mathcal B}(E_ \tau)\) be the space of hypersections of \(E_ \tau\) over \(G/K\), which can be regarded as a \(G\)-module by \(\pi\). A section \(u\) in \({\mathcal B}(E_ \tau)\) is called an eigensection if \(u\) belongs to a finite sum of \(D(E_ \tau)\)-invariant subspaces which are isomorphic to a quotient \(D(E_ \tau)\)-module of \(H\). Then, an eigensection \(u\) is called a spherical section if \(u\) satisfies \[ d(\tau) \int_ K \overline{\text{tr}(\tau(k))} \pi(k) u dk=u. \] He obtains an upper bound of the dimension of the space of spherical sections and, in particular, he determines it explicitly when \(G/K\) is a Riemannian symmetric space.
algebra of \(G\)-invariant differential operators
scientific article; zbMATH DE number 97922