NEW GENERALIZED VERMA MODULES AND MULTILINEAR INTERTWINING DIFFERENTIAL-OPERATORS

This paper contains two interrelated developments. First are proposed new generalized Verma modules. They are called k-Verma modules, k is a n element of N, and coincide with the usual Verma modules for k = 1. A s a vector space a k-Verma module is isomorphic to the symmetric tenso r product of k copies of the universal enveloping algebra U(G(-)), whe re G(-) is the subalgebra of towering generators in the standard trian gular decomposition of simple Lie algebra G = G(+) + H + G(-). The sec ond development is the proposal of a procedure for the construction of multilinear intertwining differential operators for semisimple Lie gr oups G. This procedure uses k-Verma modules and coincides for k = 1 wi th a procedure for the construction of linear intertwining differentia l operators. For all k a central role is played by the singular vector s of the k-Verma modules. Explicit formulae for series of such singula r vectors are given. Using these are given explicitly many new example s of multilinear intertwining differential operators. In particular, f or G = SL(2,R) are given explicitly all bilinear intertwining differen tial operators. Using the latter, as an application are constructed 1/ 2n-differentials for all n is an element of 2N, the ordinary Schwarzia n being the case n = 4. (C) 1998 Elsevier Science B.V.