Twisted orbital integrals and irreducible components of affine Deligne-Lusztig varieties

We analyze the asymptotic behavior of certain twisted orbital integrals arising from the study of affine Deligne-Lusztig varieties. The main tools include the Base Change Fundamental Lemma and $q$-analogues of the Kostant partition functions. As an application we prove a conjecture of Miaofen Chen and Xinwen Zhu, relating the set of irreducible components of an affine Deligne-Lusztig variety modulo the action of the $\sigma$-centralizer group to the Mirkovic-Vilonen basis of a certain weight space of a representation of the Langlands dual group.