New Generalized Verma Modules and Multilinear Intertwining Differential Operators

The present paper contains two interrelated developments. First are proposed new generalized Verma modules. They are called k Verma modules, k ∈ IN , and coincide with the usual Verma modules for k = 1. As a vector space a k Verma module is isomorphic to the symmetric tensor product of k copies of the universal enveloping algebra U(G), where G is the subalgebra of lowering generators in the standard triangular decomposition of a simple Lie algebra G = G⊕H⊕G. The second development is the proposal of a procedure for the construction of multilinear intertwining differential operators for semisimple Lie groups G. This procedure uses k Verma modules and coincides for k = 1 with a procedure for the construction of linear intertwining differential operators. For all k central role is played by the singular vectors of the k Verma modules. Explicit formulae for series of such singular vectors are given. Using these are given explicitly many new examples of multilinear intertwining differential operators. In particular, for G = SL(2, IR) are given explicitly all bilinear intertwining differential operators. Using the latter, as an application are constructed (n/2)-differentials for all n ∈ 2IN , the ordinary Schwarzian being the case n = 4. As another application, in a Note Added we propose a new hierarchy of nonlinear equations, the lowest member being the KdV equation.