A generalization of combinatorial identities for stable discrete series constants

Pub Date : 2019-12-01 DOI:10.4171/jca/62
R. Ehrenborg, S. Morel, Margaret A. Readdy
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引用次数: 3

Abstract

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 and Herb 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 discrete series character identities proved in a previous paper of the second author. We also introduce a signed convolution of valuations on polyhedral cones in Euclidean space and show that the resulting function is a valuation. This gives a theoretical framework for the valuation appearing in Appendix A of the Goresky--Kottwitz--MacPherson article. In Appendix B we extend the notion of $2$-structures (due to Herb) to pseudo-root systems.
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稳定离散序列常数组合恒等式的推广
本文讨论了Harish Chandra关于实还原群的稳定离散级数的特征公式中出现的常数,尽管它不需要任何关于实还原基或离散级数的知识。在Harish Chandra的工作中,我们所掌握的关于这些常数的唯一信息是,它们是由电感性质唯一确定的。后来Goresky——Kottwitz——MacPherson和Herb给出了这些常数的不同公式。在本文中,我们将这些公式推广到任意有限Coxeter群的情况(在这种情况下,离散级数不再有意义),并给出了两个公式一致的直接证明。我们实际上证明了一个稍微更一般的恒等式,它也暗示了在第二作者先前的论文中证明的离散序列特征恒等式背后的组合恒等式。我们还引入了欧几里得空间中多面体锥上估值的有符号卷积,并证明了所得函数是估值。这为Goresky——Kottwitz——MacPherson文章附录a中的估价提供了一个理论框架。在附录B中,我们将$2$-结构的概念(由于Herb)扩展到伪根系统。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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