G-Valued Galois Deformation Rings when $\ell \neq p$

Abstract: For a smooth group scheme $G$ over an extension of $\mathbf{Z}p$ such that the generic fiber of $G$ is reductive, we study the generic fiber of the Galois deformation ring for a $G$-valued mod $p$ representation of the absolute Galois group of a finite extension of $\mathbf{Q}\ell$ with $\ell \neq p$. In particular, we show it admits a regular dense open locus, and that it is equidimensional of dimension $\dim G$.