Josef Šilhan

The conformal Killing equation on forms—prolongations and applications☆

Abstract We construct a conformally invariant vector bundle connection such that its equation of parallel transport is a first order system that gives a prolongation of the conformal Killing equation on differential forms. Parallel sections of this connection are related bijectively to solutions of the conformal Killing equation. We construct other conformally invariant connections, also giving prolongations of the conformal Killing equation, that bijectively relate solutions of the conformal Killing equation on k-forms to a twisting of the conformal Killing equation on ( k − l ) -forms for various integers l. These tools are used to develop a helicity raising and lowering construction in the general setting and on conformally Einstein manifolds.

Higher symmetries of the conformal powers of the Laplacian on conformally flat manifolds

A.R.G. gratefully acknowledges support from the Royal Society of New Zealand via Marsden Grant Nos. 06-UOA-029 and 10-UOA-113. J.S. was supported by the Max-Planck-Institute fur Math- ¨ ematik in Bonn and by the Grant agency of the Czech republic under the Grant No. P201/12/G028.

Equivariant quantizations for AHS-structures

Abstract We construct an explicit scheme to associate to any potential symbol an operator acting between sections of natural bundles (associated to irreducible representations) for a so-called AHS-structure. Outside of a finite set of critical (or resonant) weights, this procedure gives rise to a quantization, which is intrinsic to this geometric structure. In particular, this provides projectively and conformally equivariant quantizations for arbitrary symbols on general (curved) projective and conformal structures.

On a new normalization for tractor covariant derivatives

A regular normal parabolic geometry of type G/P on a manifold M gives rise to sequences

Conformal Operators on Forms and Detour Complexes on Einstein Manifolds

D_i

Second Order Symmetries of the Conformal Laplacian

of invariant differential operators, known as the curved version of the BGG resolution. These sequences are constructed from the normal covariant derivative

Prolongation of symmetric Killing tensors and commuting symmetries of the Laplace operator

\nabla^\omega

Conformal Operators on Weighted Forms; Their Decomposition and Null Space on Einstein Manifolds

on the corresponding tractor bundle V, where

Commuting Linear Operators and Decompositions; Applications to Einstein Manifolds

\omega

Conformally invariant quantization – towards the complete classification

is the normal Cartan connection. The first operator