Dimensional Reduction of Invariant Fields and Differential Operators. II. Reduction of Invariant Differential Operators

In the present paper, which is a sequel of [1], we consider the dimensional reduction of differential operators (DOs) that are invariant with respect to the action of a connected Lie group G. The action of G on vector bundles induces naturally actions of G on their sections and on the DOs between them. In [1] we constructed explicitly the reduced bundle ξ, such that the set of all its sections, C∞(ξG), is in a bijective correspondence with the set C∞(ξ) of all G-invariant sections of the original vector bundle ξ. The main goal of the present paper is, given a G-invariant DO D : C∞(ξ) → C∞(η) to construct the reduced DO D : C∞(ξG) → C∞(ηG). Our construction of D uses the geometrically natural language of jet bundles which best reveals the geometry of the DOs and reduces the manipulations with DOs to simple algebraic operations. Since ξ was constructed in [1] by restricting a certain bundle to a submanifold of its base, an essential ingredient of the dimensional reduction of a DO is the restriction of the DO to a submanifold of the base. To perform such a restriction, one has to find splittings of certain short exact sequences of jet bundles, which in practice can be achieved by choosing an appropriate auxiliary DO – a highly nontrivial procedure involving arbitrary choices. However, in the case of a G-invariant DO D, this splitting is provided automatically by the Ginvariance of D (this uses the Lie derivative of the action of G). Certain properties of the DOs – in particular, their formal integrability – turn out to be crucial in our construction. We discuss this in detail and give an explicit example showing what can go wrong if one uses an auxiliary DO that is not formally integrable. Finally, we discuss the structure of the set of all G-invariant DOs that lead to the same reduced DO. Mathematics Subject Classification (2010). Primary 53C80; Secondary 58Z05, 37J15, 58D19, 70S10.