B型有理Cherednik代数的酉表示与晶体组合学
Unitary representations of type B rational Cherednik algebras and crystal combinatorics
摘要：
我们比较晶体组合学的水平2 Fock空间的分类酉不可约表示的B型有理数Cherednik代数的研究如何酉行为下的抛物限制。首先，我们证明了这样一个代数的任何有限维酉不可约表示都被一个由一个分量中的矩形划分和另一个分量中的空划分组成的双划分所标记。这是一个新的证明结果，可以推导出定理的Montarani和Etingof-Stoica。其次，我们证明了删除盒子的晶体算子保留了么正性的组合条件，并且对晶体进行分类的抛物限制函子将不可约的么正酉表示发送到酉表示。第三，我们找到了酉表示的支持。
英文摘要：
We compare crystal combinatorics of the level 2 Fock space with the classification of unitary irreducible representations of type B rational Cherednik algebras to study how unitarity behaves under parabolic restriction. First, we show that any finite-dimensional unitary irreducible representation of such an algebra is labeled by a bipartition consisting of a rectangular partition in one component and the empty partition in the other component. This is a new proof of a result that can be deduced from theorems of Montarani and Etingof-Stoica. Second, we show that the crystal operators that remove boxes preserve the combinatorial conditions for unitarity, and that the parabolic restriction functors categorifying the crystals send irreducible unitary representations to unitary representations. Third, we find the supports of the unitary representations.
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