A representation of the chamber set Hecke algebra of an affine building
Helgason's conjecture that any joint-eigenfunction of all invariant differential operators on a symmetric space can be given by the Poisson integral can be re-formulated for affine buildings in terms of their Hecke algebras. In a series of articles, the authors have dealt with affine buildings of rank 3 and showed, case by case, that Helgason's conjecture fails with respect to one type of special vertices or all types of special vertices if and only if a certain matrix \(\mathbb{P}\) is singular. In the paper under review, the authors introduce a finite-dimensional representation of a particular subalgebra of the Hecke algebra of an affine building. They then use this to show that the non-singularity of the matrix \(\mathbb{P}\) is equivalent to a condition for the bijectivity of the Poisson integral found by \textit{S. Kato} [Math. Ann. 257, 1--7 (1981; Zbl 0452.43014)] in the case of buildings from \(p\)-adic reductive groups.
scientific article; zbMATH DE number 6378227
scientific article