We introduce and study the index morphism for leafwise $G$-transversally elliptic operators on smooth closed foliated manifolds. We prove the usual axioms of excision, multiplicativity and induction for closed subgroups. In the case of free actions, we relate our index class with the Connes-Skandalis index class of the corresponding leafwise elliptic operator on the quotient foliation. Finally we prove the compatibility of our index morphism with the Gysin morphism and reduce its computation to the case of tori actions. We also construct a topological candidate for an index theorem using the Kasparov Dirac element for euclidean $G$-representations.

我们介绍并研究叶子方向的索引形态 $G$-光滑的封闭叶形流形上的横向椭圆算子。我们证明了封闭子群通常具有的切除，乘法和归纳法则。在自由动作的情况下，我们将商类的索引类与相应的叶式椭圆算子的Connes-Skandalis索引类相关联。最后，我们证明了索引形态与Gysin形态的相容​​性，并将其计算简化为tori动作的情况。我们还使用欧几里得的Kasparov Dirac元素构造索引定理的拓扑候选$G$-表示。