Let S be a closed orientable surface of genus at least 2 and let G be a semisimple real algebraic group of non-compact type. We consider a class of representations from the fundamental group of S to G called positively ratioed representations. These are Anosov representations with the additional condition that certain associated cross ratios satisfy a positivity property. Examples of such representations include Hitchin representations and maximal representations. Using geodesic currents, we show that the corresponding length functions for these positively ratioed representations are well-behaved. In particular, we prove a systolic inequality that holds for all such positively ratioed representations.

中文翻译：

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正比表示

设 S 是一个至少为 2 的封闭可定向曲面，设 G 是一个非紧型的半单实代数群。我们考虑一类从 S 到 G 的基本群的表示，称为正比表示。这些是具有附加条件的 Anosov 表示，即某些关联的交叉比满足正性属性。这种表示的示例包括 Hitchin 表示和最大表示。使用测地线电流，我们表明这些正比表示的相应长度函数表现良好。特别是，我们证明了一个收缩不等式，它适用于所有这些正比表示。