## 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.

### MSC:

 17B20 Simple, semisimple, reductive (super)algebras 20H15 Other geometric groups, including crystallographic groups
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### References:

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