Representations of general linear groups in the Verlinde category (Q6594722)

The author constructs affine group schemes \(GL(X)\) where \(X\) is any object in the Verlinde category \(Ver_p\) over an algebraically closed field \(k\) in characteristic \(\text{char}(k) = p>0\), which is a symmetric fusion category obtained by semisimplifying \(\text{Rep}_k(\mathbb{Z}/p\mathbb{Z})\), and classifies their irreducible representations. He begins by showing that for a simple object \(X\) of categorical dimension \(i\), this representation category is semisimple and is equivalent to the connected component of the Verlinde category for \(SL_i\). Subsequently, the author uses this along with a Verma module construction to classify irreducible representations of \(GL(nL)\) for any simple object \(L\) and any natural number \(n\). Finally, parabolic induction allows to classify irreducible representations of \(GL(X)\) where \(X\) is any object in \(Ver_p\).