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Rankin-Cohen brackets for orthogonal Lie algebras and bilinear conformally invariant differential operators

Petr Somberg

Abstract

Based on the Lie theoretical methods of algebraic Fourier transformation, we classify in the case of generic values of inducing parameters the scalar singular vectors corresponding to the diagonal branching rules for scalar generalized Verma modules in the case of orthogonal Lie algebra and its conformal parabolic subalgebra with commutative nilradical, thereby realizing the diagonal branching rules in an explicit way. The complicated combinatorial structure of singular vectors is conveniently determined in terms of recursion relations for the generalized hypergeometric function ${}_3F_2$. As a geometrical application, we classify bilinear conformally equivariant differential operators acting on homogeneous line bundles on the flag manifold given by conformal sphere $S^n$.

Rankin-Cohen brackets for orthogonal Lie algebras and bilinear conformally invariant differential operators

Abstract

Based on the Lie theoretical methods of algebraic Fourier transformation, we classify in the case of generic values of inducing parameters the scalar singular vectors corresponding to the diagonal branching rules for scalar generalized Verma modules in the case of orthogonal Lie algebra and its conformal parabolic subalgebra with commutative nilradical, thereby realizing the diagonal branching rules in an explicit way. The complicated combinatorial structure of singular vectors is conveniently determined in terms of recursion relations for the generalized hypergeometric function . As a geometrical application, we classify bilinear conformally equivariant differential operators acting on homogeneous line bundles on the flag manifold given by conformal sphere .

Paper Structure

This paper contains 7 sections, 7 theorems, 95 equations.

Table of Contents

  1. Introduction and Motivation
  2. F-method and diagonal branching problem for generalized Verma modules
  3. The construction of singular vectors for diagonal branching rules applied to scalar generalized Verma modules for $so(n+1,1,\mathbb R)$
  4. Description of the representation
  5. Reduction to scalar differential equation in two variables
  6. Polynomial solutions of the differential equation in two variables produced by the F-method
  7. Application - the classification of bilinear conformally equivariant differential operators on line bundles

Key Result

Corollary 2.1

Let with standard maximal parabolic subalgebras ${\mathfrak p}\oplus{\mathfrak p},\, \mathrm{diag}({\mathfrak p})$ given by omitting the first simple root in the corresponding Dynkin diagram. Then the multiplicity $m(\nu,(\lambda,\mu))$ of ${\fam2 M}^{{\mathfrak g}}_{{\mathfrak p}}(\mathbb C_\nu)\equiv{

Theorems & Definitions (10)

  • Corollary 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Example 3.3
  • Example 3.4
  • Theorem 3.5
  • Lemma 3.6
  • Example 3.7
  • Theorem 3.8
  • Theorem 4.1