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A generalization of combinatorial identities for stable discrete series constants

机译：稳定离散级数常数的组合恒等式的推广

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This article is concerned with the constants that appear in Harish-Chandra???s character formula for stable discrete series of real reductive groups, although it does not require any knowledge about real reductive groups or discrete series. In Harish-Chandra???s work the only information we have about these constants is that they are uniquely determined by an inductive property. Later, Goresky???Kottwitz???MacPherson (1997) and Herb (2000) gave different formulas for these constants. In this article, we generalize these formulas to the case of arbitrary finite Coxeter groups (in this setting, discrete series no longer make sense), and give a direct proof that the two formulas agree. We actually prove a slightly more general identity that also implies the combinatorial identity underlying the discrete series character identities of Morel (2011). We deduce this identity from a general abstract theorem giving a way to calculate the alternating sum of the values of a valuation on the chambers of a Coxeter arrangement. We also introduce a ring structure on the set of valuations on polyhedral cones in Euclidean space with values in a fixed ring. This gives a theoretical framework for the valuation appearing in Goresky??? Kottwitz???MacPherson???s 1997 paper. In an appendix, we extend Herb???s notion of 2-structures to pseudo-root systems.
机译：这篇文章关注的常量出现在Harish-Chandra ? ?为稳定的离散的一系列真实的还原组,虽然它不需要任何了解真实的还原组或离散系列。我们有这些常数的信息他们是唯一地由一个归纳财产。(1997)和草(2000)给不同的公式对于这些常数。概括这些公式的情况下任意有限Coxeter组(在此设置,离散系列不再有意义),并给一个直接证明这两个公式一致。实际上是一个更一般的身份这也意味着组合身份潜在的离散系列角色莫雷尔的身份(2011)。从一般抽象定理给出一个身份的交错和计算方法估值的钱伯斯的价值观Coxeter安排。在多面体结构的估值锥在欧几里得空间在一个固定的值戒指。估值出现在Goresky ? ?Kottwitz ? ?麦克弗森? ?附录,我们扩展草? ?两个结构pseudo-root系统。

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