Cambridge Journal of Mathematics

Volume 8 (2020)

Number 1

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

Pages: 149 – 241



Rong Zhou (Department of Mathematics, Imperial College London, United Kingdom)

Yihang Zhu (Department of Mathematics, Columbia University, New York, N.Y., U.S.A.)


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 Mirković–Vilonen basis of a certain weight space of a representation of the Langlands dual group.


twisted orbital integrals, affine Deligne–Lusztig varieties

2010 Mathematics Subject Classification

Primary 11G18. Secondary 22E35.

R. Z. is partially supported by NSF grant DMS-1638352 through membership at the Institute for Advanced Study.

Y. Z. is supported by NSF grant DMS-1802292.

Received 27 December 2018

Published 25 February 2020