Tilting modules of affine quasi-hereditary algebras

TL;DR

This paper studies tilting modules in affine quasi-hereditary algebras, providing existence criteria, and demonstrates their application in category equivalences and embeddings related to affine Hecke algebras.

Contribution

It establishes an existence theorem for indecomposable tilting modules and applies it to show the Arakawa-Suzuki functor yields a fully faithful embedding.

Findings

Existence of indecomposable tilting modules under certain conditions.

Criterion for functors to induce category equivalences.

Embedding of deformed BGG category into affine Hecke algebra modules.

Abstract

We discuss tilting modules of affine quasi-hereditary algebras. We present an existence theorem of indecomposable tilting modules when the algebra has a large center and use it to deduce a criterion for an exact functor between two affine highest weight categories to give an equivalence. As an application, we prove that the Arakawa-Suzuki functor [Arakawa-Suzuki, J. of Alg. 209 (1998)] gives a fully faithful embedding of a block of the deformed BGG category of $\mathfrak{gl}_{m}$ into the module category of a suitable completion of degenerate affine Hecke algebra of $GL_{n}$.