Generic Hecke algebras in the infinite

Abstract

Aiming for a revival of the theory of crystallographic complex reflection groups, we compute (minimal) Coxeter-like reflection presentations for the infinite families of those non-genuine groups which satisfy Steinberg's fixed point theorem. These new presentations behave \`{a} la Coxeter, encoding many of the group's properties at a glance, and their signature feature -- named the $x$-relation -- is fully understood in terms of configuration spaces. Crucially, the presentations further achieve the braid theorem, allowing to deform into the generic Hecke algebra. In particular, we revisit the affine GDAHA family in deformation terms of the most general class of Steinberg crystallographic complex reflection groups.