Special values for conformally invariant systems associated to maximal parabolics of quasi-Heisenberg type

Abstract

In this paper we construct conformally invariant systems of first order and second order differential operators associated to a homogeneous line bundle L s → G 0 /Q 0 with Q 0 a maximal parabolic subgroup of quasi-Heisenberg type. This generalizes the results by Barchini, Kable, and Zierau. To do so we use techniques different from ones used by them.

Systems of Differential Operators and Generalized Verma Modules

“…In [35], under the assumption that q is not of type D n (n − 2), we found the special values s 2 for Ω 2 systems for the Type 1a and Type 2 constituents. The technique that we used allowed us to handle each case uniformly.…”

“…A project for conformally invariant systems started with the work of [1] and [2]. Other progress on the project are reported in [18,19,20,21,22] and [33,34,35]. To see a recent development of the theory of conformally invariant systems the introduction of the work [22] of Kable is very helpful.…”

“…The present work is also part of the project. The aim of this paper is to close the cases that were left open in [35] and [34]. To describe our work of this paper more precisely, we now briefly review the works in the papers.…”

“…In [35], the parabolic subalgebra q = l ⊕ n was taken to be a maximal parabolic subalgebra of quasi-Heisenberg type, that is, a maximal parabolic subalgebra with nilpotent radical n with conditions that [n, [n, n]] = 0 and dim([n, n]) > 1. In this setting the Ω k systems for k ≥ 5 are zero.…”

“…To describe the open cases, let us now start explaining briefly the classification for the irreducible constituents W that contribute to Ω 2 systems. In [35], we first observed that if irreducible constituents W contribute to Ω 2 systems then their highest weights are of the form µ+ , where µ is the highest weight for g (1) and is some weight for g (1). We called such irreducible constituents special and classified as Type 1a, Type 1b, Type 2, and Type 3 with respect to certain technical conditions for the highest weight µ + .…”