Eigenspaces of operator algebras on the fundamental apartment of an affine building
In [Ric. Mat. 62, No. 2, 341--362 (2013; Zbl 1310.51010)], the authors considered the Hecke algebra of an affine building of rank 3 and obtained a characterization of the bijectivity of the Poisson transform in terms of the cyclicity of a finite set of vectors under a certain representation of the Hecke algebra introduced there. In the paper under review, the authors' aim is to extend their result to affine buildings of arbitrary rank \(\geq 3\). The key ingredient in achieving this goal is knowledge of the dimension of a certain eigenspace of the operator algebra \(\widetilde{\mathcal{H}}_c(\mathbb{A})\) which is the retraction of the Hecke algebra with respect to a chamber onto a fundamental apartment \(\mathbb{A}\). They proceed by first considering the Hecke algebra of \(\mathbb{A}\), then the retraction with respect to a fixed boundary point and finally \(\widetilde{\mathcal{H}}_c(\mathbb{A})\). It is shown that for each character \(\chi\) of \(\mathbb{A}\) the dimension of the eigenspace in question is finite and at least the order of the Weyl group \(\mathbf{W}\). Equality occurs if and only if the stabilizer \(\mathbf{W}_\chi\) of \(\chi\) is a reflection group.
scientific article; zbMATH DE number 6378272
scientific article