  To a vector space V equipped with a non-quasiclassical involutary solution of
the quantum Yang-Baxter equation and a partition $\lambda$, we associate a
vector space $\Vl$ and compute its dimension. The functor $V\mapsto \Vl$ is an
analogue of the well-known Schur functor. The category generated by the objects
$\Vl$ is called the Schur-Weyl category. We suggest a way to construct some
related twisted varieties looking like orbits of semisimple elements in
sl(n)^*. We consider in detail a particular case of such "twisted orbits",
namely the twisted non-quasiclassical hyperboloid and we define the twisted
Casimir operator on it. In this case, we obtain a formula looking like the Weyl
formula, and describing the asymptotic behavior of the function
$N(\la)=\{\sharp \la_i\leq\la\}$, where $\la_i$ are the eigenvalues of this
operator.
