  This paper studies the Dirac cohomology of standard modules in the setting of
graded Hecke algebras with geometric parameters. We prove that the Dirac
cohomology of a standard module vanishes if and only if the module is not
twisted-elliptic tempered. The proof makes use of two deep results. One is some
structural information from the generalized Springer correspondence obtained by
S. Kato and Lusztig. Another one is a computation of the Dirac cohomology of
tempered modules by Barbasch-Ciubotaru-Trapa and Ciubotaru.
  We apply our result to compute the Dirac cohomology of ladder representations
for type $A_n$. For each of such representations with non-zero Dirac
cohomology, we associate to a canonical Weyl group representation. We use the
Dirac cohomology to conclude that such representations appear with multiplicity
one.
