A subset $A$ of a Riemannian symmetric space is called an antipodal set
if the geodesic symmetry $s_x$ fixes all points of $A$ for each $x \in A$.
This notion was first introduced by Chen and Nagano.  In this talk, using
the $k$-symmetric structure, first we describe an antipodal set of a complex
flag manifold.  Tanaka and Tasaki proved that the intersection of two real
forms $L_1$ and $L_2$ in a Hermitian symmetric space of compact type is an
antipodal set of $L_1$ and $L_2$.  We can observe the same phenomenon for
the intersection of certain real forms in a complex flag manifold.
  As an application, we calculate the Lagrangian Floer homology of a pair
of real forms in a monotone Hermitian symmetric space.  Then we obtain
a generalization of the Arnold-Givental inequality.
  This talk is based on a joint work with Hiroshi Iriyeh and Hiroyuki Tasaki.