Canonical bases arising from quantum symmetric pairs of Kac-Moody type

Canonical bases arising from $\imath$quantum covering groups

“…More recent developments have made it apparent that quantum symmetric pairs play an important role in representation theory at large. In a series of papers, H. Bao and W. Wang proposed a program of canonical bases for quantum symmetric pairs [BW18a,BW18b,BW18c]. They performed their program for the Type AIII/IV symmetric pairs (sl 2N , s(gl N ×gl N )) and (sl 2N +1 , s(gl N ×gl N +1 )) and applied it to tensor products of their U ı -modules, establishing a Kazhdan-Lusztig theory and irreducible character formula for the category O of the orthosymplectic Lie superalgebra osp(2n+1 | 2m).…”

“…For the negative half U − of the quantum group in rank one U = U q (sl 2 ), the Lusztig divided powers are monomials in a single variable F , and they form the canonical basis for U − . The canonical basis for U ı in rank one is formed by the ı-divided powers, introduced in [BW18b,BW18c] and further explored in [BeW18]. Instead of being monomials, they are polynomials in a single variable B.…”

“…The ı-divided powers and their expansion formulas in [BeW18] formed a cornerstone of the construction of the Serre presentation for quasi-split ı-quantum groups established in H. Chen, M. Lu and W. Wang in [CLW18]. In [BW18b,BW18c], ı-divided powers for i ∈ I with τ i = i were defined using the same formulas, and then shown to generate as an algebra the integral form A Uı of the modified quantum group. In [C19], the ı-divided powers above are shown to have a generalization to U ı π , the ı π -divided powers B (m) i, 1 and B (m) i, 0 which are given in the formulas (3.7) and (3.8) below for i ∈ I with τ i = i.…”

“…In the case of the diagonal quantum symmetric pair, the quasi-K-matrix arises naturally from Lusztig's quasi R-matrix. The quasi-K-matrix is applied in [BW18b,BW18c] to transform involutive based U-modules (U-modules with distinguished bases compatible with the bar-involution ψ on U), into involutive based U ı -modules, compatible with the bar-involution ψ ı on U ı .…”

“…In section 4, the quasi-K-matrix Υ for U ı π is constructed and in section 5 the integrality of its action is established, by which we mean that Υ preserves the integral A-forms on integrable highest weight U π -modules and their tensor products. We conclude by constructing the ı-canonical basis for based U ı π -modules in a section 6 followed by canonical basis for the modified form Uı π in section 7 generalizing [BW18b,BW18c].…”