Project B3: Spherical harmonic analysis of affine buildings and Macdonald theory
The focus of Project B3 is spherical harmonic analysis on affine buildings. This theory exhibits striking analogies with the corresponding Archimedean theory for Riemannian symmetric spaces. We shall study equivariant Poisson transformations and their inverse boundary value maps as well as positive definite spherical functions. Here, group representations play a crucial role. We shall also consider generalisations in the context of Macdonald--Cherednik theory, where the role of the symmetric space is replaced by an affine Hecke algebra. Furthermore, we will consider open questions in harmonic analysis related to associated families of Macdonald polynomials, such as the existence of positivity-preserving dual convolution structures.