论文标题
混合霍奇模块和真实组
Mixed Hodge modules and real groups
论文作者
论文摘要
令$ g $为一个复杂的还原组，$θ\ colon g \ to g $ a divention和$ k = g^θ$。在Arxiv：1206.5547中，W。Schmid和第二名作者提出了一个计划，使用$ k $ equivariant Twisted Twisted Twisted Comply Hodge模块在$ G $及其极性上使用$ K $ Equivariant Twisted Twisted混合霍奇模块研究相应的真实形式$ G_ \ MATHBB {R} $。在本文中，我们迈出了实施该计划的第一个重要步骤。我们的第一个主要结果给出了标准模块中出现在lusztig-vogan多项式方面的组成序列中出现的Hodge数字的明确组合公式。我们的第二个主要结果是Jantzen猜想的两极化版本，表明组成因子上的Jantzen形式是基础Hodge模块的极化。我们的第三个主要结果指出，对于常规的Beilinson-Bernstein数据，一种不可约的Harish-Chandra模块的最小$ k $ type位于相应的Hodge模块的霍奇过滤的最低部分。我们结果的直接后果是Arxiv的签名多样性公式的Hodge理论证明：1212.2192，这是这项工作的灵感。
Let $G$ be a complex reductive group, $θ\colon G \to G$ an involution, and $K = G^θ$. In arXiv:1206.5547, W. Schmid and the second named author proposed a program to study unitary representations of the corresponding real form $G_\mathbb{R}$ using $K$-equivariant twisted mixed Hodge modules on the flag variety of $G$ and their polarizations. In this paper, we make the first significant steps towards implementing this program. Our first main result gives an explicit combinatorial formula for the Hodge numbers appearing in the composition series of a standard module in terms of the Lusztig-Vogan polynomials. Our second main result is a polarized version of the Jantzen conjecture, stating that the Jantzen forms on the composition factors are polarizations of the underlying Hodge modules. Our third main result states that, for regular Beilinson-Bernstein data, the minimal $K$-type of an irreducible Harish-Chandra module lies in the lowest piece of the Hodge filtration of the corresponding Hodge module. An immediate consequence of our results is a Hodge-theoretic proof of the signature multiplicity formula of arXiv:1212.2192, which was the inspiration for this work.
