  Dirac submanifolds are a natural generalization in the Poisson category for
symplectic submanifolds of a symplectic manifold. In a certain sense they
correspond to symplectic subgroupoids of the symplectic groupoid of the given
Poisson manifold.
  In particular, Dirac submanifolds arise as the stable locus of a Poisson
involution. In this paper, we provide a general study for these submanifolds
including both local and global aspects.
  In the second part of the paper, we study Poisson involutions and the induced
Poisson structures on their stable locuses. We discuss the Poisson involutions
on a special class of Poisson groups, and more generally Poisson groupoids,
called symmetric Poisson groups (and symmetric Poisson groupoids). Many
well-known examples, including the standard Poisson group structures on
semi-simple Lie groups, Bruhat Poisson structures on compact semi-simple Lie
groups, and Poisson groupoids connecting with dynamical $r$-matrices of
semi-simple Lie algebras are symmetric, so they admit a Poisson involution.
  For symmetric Poisson groups, the relation between the stable locus Poisson
structure and Poisson symmetric spaces is discussed. As a consequence, we show
that the Dubrovin-Ugaglia-Boalch-Bondal Poisson structure on the space of
Stokes matrices $U_{+}$ appearing in Dubrovin's theory of Frobenius manifolds
is indeed a Poisson symmetric space for the Poisson group $B_{+}*B_{-}$.
