“…Namely, if ξ is diagonal, and ξ is not a scalar multiple of any of the non-negative powers of diag(q 2 , 1), then β ξ is surjective onto O(D \ GL q ). Moreover, for such ξ, the kernel of β ξ is the right ideal of O(M q ) generated by the elements τ 1 − τ 1 (ξ), τ 2 − τ 2 (ξ), where τ 1 , τ 2 are the basic β-coinvariants from [4]; that is, τ 1 = q −2 x 11 + q −4 x 22 , a weighted trace, and τ 2 = x 11 x 22 − qx 12 x 21 , the quantum determinant. When ξ is a scalar multiple of some non-negative power of diag(q 2 , 1), then the kernel of β ξ is larger, and the image is finite dimensional.…”