Local coefficients and Gelfand-Graev representations for non-split covers on SL(2)

A purely local approach has been developed by Krishnamurthy and Kutzko to compute Langlands-Shahidi local coefficient for $\mathrm{S}\mathrm{L}(2)$ via types and covers à la Bushnell-Kutzko. In this paper, we extend their method to the non-split case and complete their project. We also study the algebraic structure of Gelfand-Graev representations, which generalizes the results of Chan-Savin and Mishra-Pattanayak to $\mathrm{S}\mathrm{L}(2)$ over non-archimidean local fields without any restriction on the characteristic.