New Generalized Verma Modules and Multilinear Intertwining Differential Operators

TL;DR

This paper introduces k-Verma modules as a generalization of Verma modules and develops a method to construct multilinear intertwining differential operators for semisimple Lie groups, with explicit formulas and applications.

Contribution

The paper proposes k-Verma modules and a new procedure for constructing multilinear intertwining differential operators, extending existing linear methods.

Findings

Explicit formulas for singular vectors of k-Verma modules.

Construction of all bilinear intertwining differential operators for SL(2,R).

Application to (n/2)-differentials, including the Schwarzian.

Abstract

The present paper contains two interrelated developments. First, are proposed new generalized Verma modules. They are called k-Verma modules, k\in N, 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^+ \oplus h \oplus 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,R) are given explicitly all bilinear intertwining differential operators. Using the latter, as an application are constructed (n/2)-differentials for all n\in 2N, 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.