  The unitary dual of $GL(n, \mathbb{R})$ was classified by Vogan in the 1980s.
Focusing on the irreducible unitary representations of $GL(n, \mathbb{R})$ with
half-integral infinitesimal characters, we find that Speh representations and
the special unipotent representations are building blocks. By looking at the
$K$-types of them, and by using a Blattner-type formula, we obtain all the
irreducible unitary $(\mathfrak{g}, K)$-modules with non-zero Dirac cohomology
of $GL(n, \mathbb{R})$, as well as a formula for (one of) their spin-lowest
$K$-types. Moreover, analogous to the $GL(n,\mathbb{C})$ case given in [DW1],
we count the number of the FS-scattered representations of $GL(n, \mathbb{R})$.
