We define the i-restriction and i-induction functors on the category O of the cyclotomic rational double affine Hecke algebras. This yields a crystal on the set of isomorphism classes of simple modules, which is isomorphic to the crystal of a Fock space.

We show that the product in the quantum K-ring of a generalized flag manifold G/P involves only finitely many powers of the Novikov variables. In contrast to previous approaches to this finiteness question, we exploit the finite difference module structure of quantum K-theory. At the core of the proof is a bound on the asymptotic growth of the J-function, which in turn comes from an analysis of the singularities of the zastava spaces studied in geometric representation theory. An appendix by H. Iritani establishes the equivalence between finiteness and a quadratic growth condition on certain shift operators.

In this paper, we study the relation between the cocenter \overline {{ilde {\mathcal H}}} and the representations of an affine pro-\overline {{ilde {\mathcal H}}} Hecke algebra \overline {{ilde {\mathcal H}}} . As a consequence, we obtain a new criterion on supersingular representations: a (virtual) representation of \overline {{ilde {\mathcal H}}} is supersingular if and only if its character vanishes on the non-supersingular part of the cocenter \overline {{ilde {\mathcal H}}} .

We prove that every finite group$G$can be realized as the group of self-homotopy equivalences of infinitely many elliptic spaces$X$. To construct those spaces we introduce a new technique which leads, for example, to the existence of infinitely many inflexible manifolds. Further applications to representation theory will appear in a separate paper.

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