等秩环群的多重表示与Kostant的Dirac算子
Multiplets of representations and Kostant's Dirac operator for equal rank loop groups
摘要：
设g是半单李代数，h是g中的最大秩约化子代数.给定g的任何不可约表示，考虑它与g中h的正交补相关的旋量表示的张量积。格罗斯，Kostant，拉蒙德，和斯滕伯格最近证明了一个推广的Weyl字符公式分解的签署字符的这一产品表示方面的字符的一组不可约表示的h，所谓的多重态。Kostant然后构造了一个正式的h-等变狄拉克算子在这样的产品表示，其核心正是多重的h-表示对应于给定的表示g。我们重现这些结果中的Kac-Moody设置的扩展循环代数Lg和Lh。我们证明了Weyl-Kac特征公式的齐次推广，它产生了Lh的不可约正能量表示的多重态，与Lg的任何不可约正能量表示相关联。我们构造了一个Lh-等变算子，类似于Kostant的Dirac算子，关于Lg的表示与关联到Lh在Lg中的补的旋量表示的张量积。然后，我们证明，该算子的Lh-多重态对应于Lg的原始表示。
英文摘要：
Let g be a semi-simple Lie algebra and let h be a reductive subalgebra of maximal rank in g. Given any irreducible representation of g, consider its tensor product with the spin representation associated to the orthogonal complement of h in g. Gross, Kostant, Ramond, and Sternberg recently proved a generalization of the Weyl character formula which decomposes the signed character of this product representation in terms of the characters of a set of irreducible representations of h, called a multiplet. Kostant then constructed a formal h-equivariant Dirac operator on such product representations whose kernel is precisely the multiplet of h-representations corresponding to the given representation of g. We reproduce these results in the Kac-Moody setting for the extended loop algebras Lg and Lh. We prove a homogeneous generalization of the Weyl-Kac character formula, which now yields a multiplet of irreducible positive energy representations of Lh associated to any irreducible positive energy representation of Lg. We construct a Lh-equivariant operator, analogous to Kostant's Dirac operator, on the tensor product of a representation of Lg with the spin representation associated to the complement of Lh in Lg. We then prove that the kernel of this operator gives the Lh-multiplet corresponding to the original representation of Lg.
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