Calculating canonical distinguished involutions in the affine Weyl groups. (English) Zbl 1027.17006

Let \(G\) be a semisimple algebraic group over \(\mathbb C\). It is known there is a bijection from the set \(\mathfrak U\) of unipotent classes in \(G\) and the set \(X_+\) of dominant weights. This map is carried out by first using work of Lusztig to establish a bijection between \(\mathfrak U\) and the set of two-sided cells in the associated affine Weyl group, \(W_a\). If \(\mathcal O\) is a unipotent orbit and \(c_{\mathcal O}\) is the corresponding two-sided cell, work of Lusztig and Xi assign to \(c_{\mathcal O}\) a canonical left cell \(C_{\mathcal O}\). Finally, \(C_{\mathcal O}\) contains a distinguished involution \(d_{\mathcal O}\in W_a\) that is the shortest element in \(W d_{\mathcal O} W\) where \(W\) is the finite Weyl group. Finally, the bijection \(\mathcal L :\mathfrak U \rightarrow X_+\) is established by using the bijection from \(W\setminus W_a/W\) to \(X_+\). Thus explicitly calculating \(\mathcal L\) comes down to calculating the distinguished involutions.
In previous work, the first author conjectured an algorithm for calculating \(\mathcal L\). Due to results of Bezrukavnikov, the conjecture is known to be correct. In this paper, the authors explicitly calculate \(\mathcal L\) for \(GL_n\) and for almost all groups of rank at most seven (excluding some gaps for groups of type \(B_6\), \(C_6\), \(B_7\), \(C_7\), and \(E_7\)) as well as partial results for \(D_8\) and \(E_8\). The results of the calculations are listed in tables at the end of the paper.


17B20 Simple, semisimple, reductive (super)algebras
20H15 Other geometric groups, including crystallographic groups
Full Text: DOI arXiv EuDML


[1] Bezrukavnikov R., ”On tensor categories attached to cells in affine Weyl groups.” · Zbl 1078.20044
[2] Bezrukavnikov R., ”Quasi-exceptional sets and equivariant coherent sheaves on the nilpotent cone.” · Zbl 1065.20055
[3] Collingwood D., Nilpotent orbits in semisimple Lie algebras (1993) · Zbl 0972.17008
[4] Hinich V., Israel J. Math. 73 pp 3– (1991)
[5] DOI: 10.2307/2373130 · Zbl 0124.26802
[6] Lusztig G., J. Algebra 109 (2) pp 223– (1987) · Zbl 0654.20047
[7] Lusztig G., Adv. in Math. 72 (2) pp 284– (1988) · Zbl 0664.20028
[8] Lusztig G., J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (2) pp 297– (1989)
[9] McGovern W., Invent. Math. 97 (1) pp 209– (1989) · Zbl 0648.22004
[10] Ostrik V., Representation Theory 4 pp 296– (2000) · Zbl 0986.20045
[11] Panyushev, Funct. Anal. Appl. 25 (3) pp 225– (1991) · Zbl 0749.14030
[12] Spaltenstein N., Classes unipotentes et sous-groupes de Borel. (1982) · Zbl 0486.20025
[13] Vogan D., Representation theory of Lie groups (Park City, UT, 1998) pp 179– (2000)
[14] Xi N., ”The based ring of two-sided cells of affine Weyl groups of type à n–1” · Zbl 1001.20041
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.