The principal series 2-representation for ${\mathrm{G}\mathrm{L}}_n\times {\mathrm{G}\mathrm{L}}_n$ over a 2-dimensional local field

Let $F={\mathbb{F}}_q((u))((t))$ and $H={\mathrm{G}\mathrm{L}}_n(F{)}_L\times {\mathrm{G}\mathrm{L}}_n(F{)}_R$. An ordered tuple $\alpha =({\alpha }_1,\dots ,{\alpha }_n)$ defines a cross-$K_2$ multiplier on the doubled Borel. Twisted equivariantization then yields an exact uniformly smooth categorical principal series $2$-representation of $H$. For unitary parameters, the $\mathbb{R}((X))$-measure gives a positive Hom-valued formal Hermitian inner product on the spherical part. These constructions provide two-dimensional categorical analogues of selected unramified structures.