On the Local Converse Theorem for Depth $\frac{1}{N}$ Supercuspidal Representations of $\text{GL}(2N,F)$

In this paper, we use type theory to construct a family of depth $\frac{1}{N}$ minimax supercuspidal representations of $\text{GL}(2N,F)$ which we call middle supercuspidal representations. These supercuspidals may be viewed as a natural generalization of simple supercuspidal representations, i.e. those supercuspidals of minimal positive depth. Via explicit computations of twisted gamma factors, we show that middle supercuspidal representations may be uniquely determined through twisting by quasi-characters of $F^{\times }$ and simple supercuspidal representations of $\text{GL}(N,F)$.