  Bloch and Okounkov's correlation function on the infinite wedge space has
connections to Gromov-Witten theory, Hilbert schemes, symmetric groups, and
certain character functions of $\hgl_\infty$-modules of level one. Recent works
have calculated these character functions for higher levels for $\hgl_\infty$
and its Lie subalgebras of classical type. Here we obtain these functions for
the subalgebra of type $D$ of half-integral levels and as a byproduct, obtain
$q$-dimension formulas for integral modules of type $D$ at half-integral level.
