Non-split groups of Lie type and iSchur algebras with unequal parameters

Abstract

We present a geometric framework for realizing quantum Schur algebras with unequal parameters, via constructible sheaves on flag varieties arising from non-split groups of Lie type. In the special cases of general unitary groups and minus-type orthogonal groups, this framework yields the iSchur algebras with unequal parameters, which are closely related to the quasi-split iquantum groups of type AIII. Such a geometric realization enables the construction of canonical bases, integral forms, and monoidal categorification for these unequal-parameter iSchur algebras and the corresponding iquantum groups. This material is not a formal generalization from the equal-parameter case to the unequal-parameter case. A nontrivial arithmetic structure is imposed on the geometric structure of split groups, giving rise to many new phenomena. For example, the support varieties of generators of an integral form may not be rationally smooth, the categorification is provided by the category of descent data of semisimple complexes fixed by the twisted Frobenius, and the positivity of the canonical basis is established using a refined covering.