In 1980, Lusztig introduced the periodic Kazhdan-Lusztig polynomials, whichare conjectured to have important information about the characters ofirreducible modules of a reductive group over a field of positivecharacteristic, and also about those of an affine Kac-Moody algebra at thecritical level. The periodic Kazhdan-Lusztig polynomials can be computed byusing another family of polynomials, called the periodic $R$-polynomials. Inthis paper, we prove a (closed) combinatorial formula expressing periodic$R$-polynomials in terms of the "doubled" Bruhat graph associated to a finiteWeyl group and a finite root system.
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机译：1980年，Lusztig引入了周期Kazhdan-Lusztig多项式，该多项式被认为具有关于正特性域上的还原性组的不可约模块的特征的重要信息，以及有关临界级的仿射Kac-Moody代数的重要信息。周期Kazhdan-Lusztig多项式可以通过使用另一个称为周期$ R $-多项式的多项式族来计算。在本文中，我们证明了一个（封闭的）组合公式，该公式用与有限Weyl基团和有限根系统相关的“加倍” Bruhat图来表示周期$ R $-多项式。
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