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**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.

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.

Reviewer: Mark R.Sepanski (Waco)

### MSC:

17B20 | Simple, semisimple, reductive (super)algebras |

20H15 | Other geometric groups, including crystallographic groups |

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\textit{T. Chmutova} and \textit{V. Ostrik}, Exp. Math. 11, No. 1, 99--117 (2002; Zbl 1027.17006)

### References:

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