Pure and Applied Mathematics Quarterly

Volume 17 (2021)

Number 4

Special Issue: In Memory of Prof. Bertram Kostant

Guest Editors: Shrawan Kumar, Lizhen Ji, and Kefeng Liu

From conjugacy classes in the Weyl group to semisimple conjugacy classes

Pages: 1159 – 1189

DOI:  https://dx.doi.org/10.4310/PAMQ.2021.v17.n4.a1


Jeffrey Adams (Department of Mathematics, University of Maryland, College Park, Md., U.S.A.)

Xuhua He (Department of Mathematics, University of Maryland, College Park, Md.,U.S.A.; and Institute of Mathematical Sciences and Department of Mathematics, Chinese University of Hong Kong)

Sian Nie (Academy of Mathematics and Systems Science, University of Chinese Academy of Sciences, Beijing, China)


Suppose $G$ is a connected complex semisimple group and $W$ is its Weyl group. The lifting of an element of $W$ to $G$ is semisimple. This induces a well-defined map from the set of elliptic conjugacy classes of $W$ to the set of semisimple conjugacy classes of $G$. In this paper, we give a uniform algorithm to compute this map. We also consider the twisted case.


algebraic groups, Weyl groups, elliptic conjugacy classes, semisimple conjugacy classes

2010 Mathematics Subject Classification

Primary 20G07. Secondary 20E45, 20F55.

To Bert Kostant with admiration.

X. H. was partially supported by NSF DMS-1801352.

S. N. is supported in part by QYZDB-SSW-SYS007 and NSFC grant (Nos. 11501547, 11621061 and 11688101).

Received 11 March 2019

Accepted 26 March 2020

Published 22 December 2021