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.
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