If $A$ is a quasi-hereditary algebra then its Ringel dual $A'=End_A(T)$ is the endomorphism algebra of a (minimal) full tilting module $T$ for $A$. The algebra $A'$ is again quasi-hereditary, although with respect to the opposite partial order on the weights/simple modules.

There is a left exact Ringel duality functor $F$ from $A$-mod to $A'$-mod which acts on modules by $F(M)=Hom_A(T,M)$. It is known that Ringel duality sends tilting modules to projectives, costandard modules to standards, injective modules to projectives, and that it is exact on the subcategory of $\nabla$-filtered $A$-modules. In short, Ringel duality plays nicely with anything that has a $\nabla$-filtration.

All these facts can be found, for example, in the appendix to Donkin's book "The $q$-Schur Algebra".

My question: what is known about what Ringel duality does to the simple $A$-modules?

There are some things that I can say about this. I am more curious about what is already known.



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