Parameters q_i as indeterminates or general ring elements in the masure-based Hecke algebra

Ascertain whether, in the Iwahori–Hecke algebra attached to a group acting strongly transitively on a thick masure (with parameters q_i depending on simple reflections), the parameters q_i can be treated as independent indeterminates or, more generally, taken as elements of an arbitrary commutative ring.

Background

Beyond split Kac–Moody groups, Bardy–Panse–Gaussent–Rousseau define Hecke algebras for groups acting on masures, introducing possibly distinct parameters q_i. The present paper adapts certain support arguments to this setting.

However, the foundational choice of parameters remains unclear in full generality: whether one may consider the q_i as formal indeterminates or assign them arbitrarily in a coefficient ring is not settled, and relates to a conjecture cited by the authors.