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 the Langlands–Shahidi local coefficients for SL(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 SL(2) over non-archimedean local fields without any restriction on the characteristic.