Higher mod $p$ Satake transforms: explicit formulas and images for ${\mathrm{G}\mathrm{L}}_2$

Heyer’s left adjoint to derived parabolic induction gives rise to higher mod $p$ Satake transforms $${\mathcal{S}}_V^i:{\mathcal{H}}_G(V)⟶{\mathcal{H}}_M(H^i(K_U,V)),$$ whose extreme degrees recover Herzig’s invariant and coinvariant transforms. For $G=\mathbf{G}(F)$, where $\mathbf{G}$ is a connected reductive group defined over a finite extension $F\mathrm{/}{\mathbb{Q}}_p$, we describe these maps explicitly using Mackey decompositions and corestriction in group cohomology. For $G={\mathrm{G}\mathrm{L}}_n(F)$, we evaluate them on the standard Hecke generators, showing that their support lies in a single Weyl orbit and that their values on the individual support cosets are concrete corestriction operators. For $G={\mathrm{G}\mathrm{L}}_2(F)$, we determine the image in every cohomological degree for each non-Steinberg Serre weight. The endpoint images are the classical polynomial Satake images, while every interior image is of the form $$k[x,y^{\pm 1}]\mathrm{/}(x^r)$$ for some positive integer $r$. Thus, every interior image is a finite nilpotent thickening of the Laurent polynomial algebra $k[y^{\pm 1}]$. We give explicit bounds for its thickness, prove that they are sharp, and show that in the unramified case every interior image is either reduced or a square-zero thickening.