Coinvariants for a coadjoint action of quantum matrices

Let $K$ be a (algebraically closed) field. A morphism $A\mapsto g^{-1}Ag$, where $A\in M(n)$ and $g\in GL(n)$, defines an action of a general linear group $GL(n)$ on an $n\times n$-matrix space $M(n)$, referred to as an adjoint action. In correspondence with the adjoint action is the coaction $\alpha\colon K[M(n)]\to K[M(n)]\otimes K[GL(n)]$ of a Hopf algebra $K[GL(n)]$ on a coordinate algebra $K[M(n)]$ of an $n\times n$-matrix space, dual to the conjugation morphism. Such is called an adjoint coaction.
We give coinvariants of an adjoint coaction for the case where $K$ is a field of arbitrary characteristic and one of the following conditions is satisfied: (1) $q$ is not a root of unity; (2) $\operatorname{char}K=0$ and $q=\pm1$; (3) $q$ is a primitive root of unity of odd degree. Also it is shown that under the conditions specified, the category of rational $GL_q\times GL_q$-modules is a highest weight category.