The distinguished involutions of the weyl group of type E 6

• Published in: Communications in algebra 2000-01, Vol.28 (1), p.193-208
• Main Author: Chengdong, Chen
• Format: Article
• Language: English
• Subjects: Distinguished involution; Hecke algebra; Supported by the National Natural Science Foundation of China
• ISSN: 0092-7872; 1532-4125
• DOI: 10.1080/00927870008826836

In [1] Kazhdan and Lusztig defined the concepts of left. right. and two-sided cells for the Coxeter group (W,S) which play an important role in the representation theory of its associated Hecke algebra. The concept of left cell has its origin in the work of Robinson and Shensted, who described a 1-1 correspondence between the synlmetric group S n and the set of pairs of standard Yong tableallx of the sanle shape, of size n. In this picture, a left cell of S n . appears as the subset of S n corresponding to the set of pairs with fixed. In 1982, Barbasch and Vogan clescribe explicitly the left cells of the Weyl group of the type B n , D n by using prin1itive ideals in the envelopillg algebra of a con1plex sen1isin1ple Lie algebra. In [8] Tong (jIlangqing describes explicitly the left cells of the Weyl group of the type E 6 by using the method introduced by Shiin [6]. Lusztig shows that each 1eft cell of W contains a unique distinguished involution which play an imporatant role in the ring J given in G. Lllsztig [3]. In this paper vve find all distinguished involutions for the Weyl group of type E 6 . The paper is organized as follows. In Section 1 we recall some general results on the relation the a-function, and distinguished involutions for Weyl groups. In Section:2 we describe canonical expressions for elernents of the Weyl group of type E 6 . In Section 3 we describe explicitly a11 distinguished involutions for the Weyl group W of type E 6 .