Canonical Bases in Quantum Cluster Algebras and the Positivity of Macdonald Polynomials via Double Affine Hecke Algebras

This monograph establishes a rigorous isomorphism between the spherical subalgebra of the Double Affine Hecke Algebra (DAHA) of type GLn and a specific quantization of the cluster algebra associated with the moduli space of G-local systems on a punctured torus. We demonstrate that Lusztig's dual canonical basis for the quantum cluster algebra maps injectively to a generalized basis of Macdonald polynomials P(q,t). By exploiting the positivity of structure constants in the cluster canonical basis—a consequence of the monoidal categorification of cluster algebras—we provide a non-combinatorial, representation-theoretic proof of the strong Macdonald positivity conjecture. Furthermore, we analyze the t limit, recovering the connection between q-Whittaker functions and quantum Q-systems, thereby unifying the discrete integrable systems of Fomin-Zelevinsky with the spectral theory of Cherednik operators.