2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/166220In 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 $V_θ(λ)$. In this paper, we prove the existence of crystal bases and global bases of $V_θ(0)$ 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.33 pagesRepresentation TheoryQuantum Algebratext