  Brasselet, the second author and Yokura introduced Hodge-theoretic
Hirzebruch-type characteristic classes $IT_{1, \ast}$, and conjectured that
they are equal to the Goresky-MacPherson $L$-classes for pure-dimensional
compact complex algebraic varieties. In this paper, we show that the framework
of Gysin coherent characteristic classes of singular complex algebraic
varieties developed by the first and third author in previous work applies to
the characteristic classes $IT_{1, \ast}$. In doing so, we prove the ambient
version of the above conjecture for a certain class of subvarieties in a
Grassmannian, including all Schubert subvarieties. Since the homology of
Schubert subvarieties injects into the homology of the ambient Grassmannian,
this implies the conjecture for all Schubert varieties in a Grassmannian. We
also study other algebraic characteristic classes such as Chern classes and
Todd classes (or their variants for the intersection cohomology sheaves) within
the framework of Gysin coherent characteristic classes.
