We consider an involution on the affine Weyl group of type $A$ induced from the nontrivial automorphism on the (finite) Dynkin diagram. We prove that the number of left cells fixed by this involution in each two-sided cell is given by a certain Green polynomial of type $A$ evaluated at $-1$.

中文翻译：

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关于仿射Schützenberger对合

我们考虑了由（有限）Dynkin图上的非平凡自同构引起的$ A $仿射Weyl群的对合。我们证明了由该对合在每个两侧像元中固定的左像元的数量由类型为$ A $的某个Green多项式给出，其值为$ -1 $。