狄拉克指数的平移原理
Translation principle for Dirac index
摘要：
设$G$是$GL(n,\bR)$的闭连通转置稳定子群的有限覆盖，其复化李代数为$\frg$ .设$K$是$G$的极大紧子群，并假定$G$和$K$具有相等的秩.证明了虚$(\frg,K)$ -模的Dirac指标的平移原理.作为一个副产品，每个连贯的家庭，这样的模块，我们附加一个多项式的对偶的紧凑Cartan子代数的$\frg$。这个"指数多项式“生成包含在相干连续表示中的Weyl群的不可约表示。我们证明了指数多项式是King特征多项式的紧Cartan子群上的精确模拟。在\cite{K1}中定义了最大可裂Cartan子群的特征多项式，证明了特征多项式在一个标量倍数下等于Goldie秩多项式.在Gelfand-Kirillov维数至多为$G/K$维数一半的表示情形下，我们还推测了我们的指数多项式与相应凝聚族的伴随圈中不可约分量的重数之间的显式关系.
英文摘要：
Let $G$ be a finite cover of a closed connected transpose-stable subgroup of $GL(n,\bR)$ with complexified Lie algebra $\frg$. Let $K$ be a maximal compact subgroup of $G$, and assume that $G$ and $K$ have equal rank. We prove a translation principle for the Dirac index of virtual $(\frg,K)$-modules. As a byproduct, to each coherent family of such modules, we attach a polynomial on the dual of the compact Cartan subalgebra of $\frg$. This ``index polynomial'' generates an irreducible representation of the Weyl group contained in the coherent continuation representation. We show that the index polynomial is the exact analogue on the compact Cartan subgroup of King's character polynomial. The character polynomial was defined in \cite{K1} on the maximally split Cartan subgroup, and it was shown to be equal to the Goldie rank polynomial up to a scalar multiple. In the case of representations of Gelfand-Kirillov dimension at most half the dimension of $G/K$, we also conjecture an explicit relationship between our index polynomial and the multiplicities of the irreducible components occuring in the associated cycle of the corresponding coherent family.
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