A common structure in PBW bases of the nilpotent subalgebra of U_q(g) and quantized algebra of functions

For a finite-dimensional simple Lie algebra g, let U^+_q(g) be the positive part of the quantized universal enveloping algebra, and A_q(g) be the quantized algebra of functions. We show that the transition matrix of the PBW bases of U^+_q(g) coincides with the intertwiner between the irreducible A_q(g)-modules labeled by two different reduced expressions of the longest element of the Weyl group of g. This generalizes the earlier result by Sergeev on A_2 related to the tetrahedron equation and endows a new representation theoretical interpretation with the recent solution to the 3D reflection equation for C_2. Our proof is based on a realization of U^+_q(g) in a quotient ring of A_q(g).