Hirzebruch genera on homogeneous spaces

We aim to present how the notion of universal toric genus can be applied in the case of homogeneous spaces to obtain the important results on some famous Hirzebruch genera such as Krichever genus, signature, elliptic genus and any genus defined by an odd power series.
Universal toric genus can be defined for any stable complex manifold with an equivariant torus action. It as an element of the complex cobordism ring of the classifying space of the acting torus. When the torus action has only isolated fixed points the toric genus can be localized meaning that it can be expressed in terms of signs and weights at the fixed points for the given torus action. When composed with a Hirzebruch genus, the toric genus gives arise to an equivariant genus which carries important information, such as rigidity or even triviality, on the initial Hirzebruch genus.
We explain how this approach can be used in the case of compact homogeneous spaces of positive Euler characteristic equipped with the canonical action of the maximal torus and an invariant almost complex structure. We prove that Krichever genus is rigid for this action just using representation theory for Lie groups. We also prove using representation theory that any Hirzebruch genus given by an odd power series is trivial on a large class of homogeneous spaces what then holds for the signature, $\hat{A}$-genus and the elliptic genus as well.
The talk is based on the joint work with Victor M. Buchstaber (Toric genera of homogeneous spaces and their fibrations, International Mathematics Research Notices, 2012).