  In this paper, we extend the local Shimura correspondence building upon the
groundwork laid by Gordan Savin. As preparations, we review part of the type
theory of Bushnell and Kutzko which equally applies to covering groups. By
adapting the method of linear algebraic groups, we describe the structure of
Hecke algebras associated with genuine principal series components and
construct the types. In particular, we show each of them shares the same affine
Hecke algebra part as one corresponding Hecke algebra of the principal
endoscopy group. However, we give a counter example to show they are not
isomorphic in general.
