ABSTRACT
To a vector space V equipped with a non‐quasiclassical involutary solution arose from the quantum Yang‐Baxter equation and a partition λ, we associate a vector space V λ and compute its dimension. The functor V ↦ V λ is an analogue of the well‐known Schur functor. The categoiy generated by the objects V λ 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(λ) = {# λ i ≤ λ}, where λ i are the eigenvalues of this operator.
