Symmetric Crystals and LLTA Type Conjectures for the Affine Hecke Algebras of Type B

In the previous paper “Symmetric Crystals and Affine Hecke Algebras of Type B”, we formulated a conjecture on the relations between certain classes of irreducible representations of affine Hecke algebras of type B and symmetric crystals for ${\text{gl}}_{\mathrm{\infty }}$. In the first half of this paper (sections 2 and 3), we give a survey of the LLTA type theorem of the affine Hecke algebra of type $A$. In the latter half (sections 4, 5 and 6), we review the construction of the symmetric crystals and the LLTA type conjectures for the affine Hecke algebra of type $B$.