Nikolaus Conference 2018

This talk is about unexplained coincidences of decomposition numbers between seemingly unrelated objects. The decomposition matrix of a unipotent block of a finite group of Lie type in cross characteristic has a square submatrix indexed by the unipotent characters. Many low-rank examples of these decomposition matrices were computed in recent years by Dudas and Malle. In many cases, the matrices obtained are identical on the principal series characters, which are indexed by the irreducible characters of the Weyl group, to decomposition matrices I computed for the rational Cherednik algebra at a corresponding parameter. I will explain structural parallels and differences between the two theories and summarize the numerical data, and I will provide examples that show that we cannot in general expect the decomposition matrix of the Cherednik algebra to appear as a submatrix of the decomposition matrix of the finite group.