Given a decomposition of a vector space
V
=
V
1
⊕
V
2
V=V_1\oplus V_2
, the direct product
X
\mathfrak {X}
of the projective space
P
(
V
1
)
\mathbb {P}(V_1)
with a Grassmann variety
G
r
k
(
V
)
\mathrm {Gr}_k(V)
can be viewed as a double flag variety for the symmetric pair
(
G
,
K
)
=
(
G
L
(
V
)
,
G
L
(
V
1
)
×
G
L
(
V
2
)
)
(G,K)=(\mathrm {GL}(V),\mathrm {GL}(V_1)\times \mathrm {GL}(V_2))
. Relying on the conormal variety for the action of
K
K
on
X
\mathfrak {X}
, we show a geometric correspondence between the
K
K
-orbits of
X
\mathfrak {X}
and the
K
K
-orbits of some appropriate exotic nilpotent cone. We also give a combinatorial interpretation of this correspondence in some special cases. Our construction is inspired by a classical result of Steinberg (1976) and by the recent work of Henderson and Trapa (2012) for the symmetric pair
(
G
L
(
V
)
,
S
p
(
V
)
)
(\mathrm {GL}(V),\mathrm {Sp}(V))
.