We define and study the index map for families of $G$-transversally elliptic operators and introduce the multiplicity for a given irreducible representation as a virtual bundle over the base of the fibration. We then prove the usual axiomatic properties for the index map extending the Atiyah–Singer results [1]. Finally, we compute the Kasparov intersection product of our index class against the K-homology class of an elliptic operator on the base. Our approach is based on the functorial properties of the intersection product, and relies on some constructions due to Connes–Skandalis and to Hilsum–Skandalis.

中文翻译：

---

$ G $的横向椭圆族的索引。一世

我们定义和研究了$ G $-横向椭圆算子族的索引图，并引入了给定不可约表示形式的多重性，作为在纤维化基础上的虚拟束。然后，我们证明了扩展Atiyah-Singer结果的索引图的通常公理属性[1]。最后，我们根据索引运算符的椭圆类算子的K-homology类，计算索引类的Kasparov交集。我们的方法基于相交积的函数性质，并依赖于因Connes–Skandalis和Hilsum–Skandalis的某些构造。