Some perverse equivalences of $SL(2,q)$ in its defining characteristic

In this article, we study the modular representations of the special linear group of degree two over a finite field in defining characteristic. In particular, we study the automorphisms of derived category of representations. We have been able to obtain a new type of autoequivalence. This autoequivalence has some uncommon features. It is more conveniently conceived and proved using the representation theory of its Brauer correspondence but at the same time it can be very neatly described, using a type of derived equivalence called perverse equivalence, in global settings.