This talk is about a family of infinite dimensional affine schemes recently introduced by myself and Bernard Leclerc called schemes of (G,c)-bands. Here G denotes a simple simply laced and simply connected algebraic group, and c is a Coxeter element of its Weyl group. These schemes offer a geometric framework for understanding certain cluster algebras which play a central role in the representation theory of quantum affine algebras, their Borel subalgebras, and shifted quantum affine algebras. The goal of the talk is to introduce these schemes and explore their connections with cluster algebras and representation theory. Time permitting, I will mention some possible developments. -- This seminar is part of the activities of the Dipartimento di Eccellenza CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.