  In the recent papers with Masaki Kashiwara, the author introduced the notion
of symmetric crystals and presented the Lascoux-Leclerc-Thibon-Ariki type
conjectures for the affine Hecke algebras of type $B$. Namely, we conjectured
that certain composition multiplicities and branching rules for the affine
Hecke algebras of type $B$ are described by using the lower global basis of
symmetric crystals of $V_\theta(\lambda)$. In this paper, we prove the
existence of crystal bases and global bases of $V_\theta(0)$ for any symmetric
quantized Kac-Moody algebra by using a geometry of quivers (with a Dynkin
diagram involution). This is analogous to George Lusztig's geometric
construction of $U_v^-$ and its lower global basis.
