Abstract
We extend the notion of ascent-compatibility from symmetric groups to all Coxeter groups, thereby providing a type-independent framework for constructing families of modules of 𝘖-Hecke algebras. We apply this framework in type 𝘉 to give representation–theoretic interpretations of a number of noteworthy families of type-𝘉 quasisymmetric functions. Next, we construct modules of the type-𝘉 𝘖-Hecke algebra corresponding to type-𝘉 analogues of Schur functions and introduce a type-𝘉 analogue of Schur Q-functions; we prove that these shifted domino functions expand positively in the type-𝘉 peak functions. We define a type-𝘉 analogue of the 𝘖-Hecke–Clifford algebra, and we use this to provide representation–theoretic interpretations for both the type-𝘉 peak functions and the shifted domino functions. We consider the modules of this algebra induced from type-𝘉 𝘖-Hecke modules constructed via ascent-compatibility and prove a general formula, in terms of type-𝘉 peak functions, for the type-𝘉 quasisymmetric characteristics of the restrictions of these modules.