Invariant differential operators

11.01.2018, 16:15  –  Haus 9, Raum 1.10
Forschungsseminar Differentialgeometrie

Vít Tuček (Prag)

It is well known that Laplace and Dirac operators on $\mathbb{R}^{p, q}$ admit strictly larger Lie algebra of symmetries than just the orthogonal ones, namely they are invariant with respect to infinitesimal generators of conformal symmetries. It turns out that operators invariant with respect to this Lie algebra can be completely classified and that in some cases they have natural "extensions" to any manifold with a conformal structure. I will review how the question of existence of invariant operators between sections of homogeneous bundles can be reduced to a purely algebraic problem. This algebraic reformulation yields complete classification in some cases and in some other cases it can be reduced to solving systems of PDEs on polynomials. Existence of natural extensions of these operators to any manifold with appropriate geometric structure is much more difficult problem. General construction of these extensions for overdetermined operators (such as operators whose kernel are conformal Killing fields) was given by Cap-Slovak-Soucek. I will present a simplified construction by Calderbank-Diemer which I generalized in my PhD thesis to construct extensions of the conformally invariant modifications of the Laplace-Beltrami and Dirac operators. The crucial point of this construction is convergence of certain Neumann series, which at the moment is formal i.e. at the level of formal power series.