Standard homological properties for quantum \(GL_ n\)
The author studies the standard homological properties for quantum \(GL_n\). The quantum version of \(GL_n\) with which he works throughout was introduced by \textit{R. Dipper} [Proc. Lond. Math. Soc., III. Ser. 63, No. 1, 165-211 (1991; Zbl 0734.20018)]. He establishes the standard homological properties for the induction functor in the quantum setting, including the representation theory of various subgroups of quantum \(GL_n\), Kempf vanishing and further representation theoretic properties. The results given in this paper are valid for all \(q\neq 0\) and (except for the Grothendieck vanishing) are obtained without invoking the corresponding results in the classical case. In particular, his proof of Kempf's vanishing theorem does not rely on properties of the cohomology of projective varieties, instead, it is based on the Koszul resolution defined by a maximal parabolic subgroup. Another feature of the proofs is that no use is made of Hecke algebras.
scientific article; zbMATH DE number 883758