We study
$K$
-orbits in
$G/P$
where
$G$
is a complex connected reductive group,
$P\,\subseteq \,G$
is a parabolic subgroup, and
$K\,\subseteq \,G$
is the fixed point subgroup of an involutive automorphism
$\theta$
. Generalizing work of Springer, we parametrize the (finite) orbit set
$K\,\backslash \,G/P$
and we determine the isotropy groups. As a consequence, we describe the closed (resp. affine) orbits in terms of
$\theta$
-stable (resp.
$\theta$
-split) parabolic subgroups. We also describe the decomposition of any
$(K,\,P)$
-double coset in
$G$
into
$(K,\,B)$
-double cosets, where
$B\,\subseteq \,P$
is a Borel subgroup. Finally, for certain
$K$
-orbit closures
$X\,\subseteq \,G/B$
, and for any homogeneous line bundle
$\mathcal{L}$
on
$G/B$
having nonzero global sections, we show that the restriction map
$\text{re}{{\text{s}}_{X}}\,:\,{{H}^{0}}\,\left( G\,/\,B,\,\mathcal{L} \right)\,\to \,{{H}^{0}}\,\left( X,\,\mathcal{L} \right)$
is surjective and that
${{H}^{i}}\,\left( X,\mathcal{L} \right)\,=\,0$
for
$i\,\ge \,1$
. Moreover, we describe the
$K$
-module
${{H}^{0}}\left( X,L \right)$
. This gives information on the restriction to
$K$
of the simple
$G$
-module
${{H}^{0}}\,\left( G\,/\,B,\mathcal{L} \right)$
. Our construction is a geometric analogue of Vogan and Sepanski’s approach to extremal
$K$
-types.