Invariants on projective space

Abstract

This article is concerned with the construction of invariant nonlinear differential operators for projective space, pn := P(R n+1) There is no loss in working at first on the projective n-sphere Sn, that is the space of rays in Rn+l. The consequences for pn will be explained at the end of the article. Working on Sn avoids notational difficulties arising from the nonorientablity of pn when n is even. We shall use the general notion of a homogeneous bundle as in [2]. Suppose E and F are homogeneous vector bundles on X, a homogeneous space for some Lie group G. The polynomials of degree at most d on a vector space V themselves form a vector space, namely EhdO 0h V* (where o is the symmetric tensor product). Therefore, the G-invariant differential operators E --+ F of order k and which are polynomial in the jets of E are precisely the G-invariant homomorphisms d C)h Jk E --* F