A two-sided q-analogue of the Coxeter complex

We define a complex of bimodules over the Iwahori-Hecke algebra associated to a finite Coxeter group, calculate its cohomology and show that it induces a derived equivalence over its module category extending the Morita equivalence given by a certain algebra automorphism. We show further that when tensored with the index representation this complex becomes isomorphic to the one-sided q-analogue of the Coxeter complex previously defined by V. Deodhar [On some geometric aspects of Bruhat orderings. II. The parabolic analogue of Kazhdan-Lusztig polynomials, J. Algebra 111 (2) (1987) 483-506] and A. Mathas [A q-analogue of the Coxeter complex, J. Algebra 164 (3) (1994) 831-848].