The $q$-deformed random-to-random family in the Hecke algebra

We generalize Reiner--Saliola--Welker's well-known but mysterious family of *$k$-random-to-random shuffles* from Markov chains on symmetric groups to Markov chains on the Type-$A$ Iwahori--Hecke algebras. We prove that the family of operators pairwise commutes and has eigenvalues that are polynomials in $q$ with non-negative integer coefficients. Our work generalizes work of Reiner--Saliola--Welker and Lafrenière for the symmetric group, and simplifies all known proofs in this case.