On Shahidi local coefficients matrix

Abstract

In this article we define and study the Shahidi local coefficients matrix associated with a genuine principal series representation $$\mathrm{I}(\sigma )$$ of an n-fold cover of p-adic $$\mathrm{{SL}_{2}(\mathrm{F})}$$ and an additive character $$\psi $$ . The conjugacy class of this matrix is an invariant of the inducing representation $$\sigma $$ and $$\psi $$ and its entries are linear combinations of Tate or Tate type $$\gamma $$ -factors. We relate these entries to functional equations associated with linear maps defined on the dual of the space of Schwartz functions. As an application we give new formulas for the Plancherel measures and use these to relate principal series representations of different coverings of $$\mathrm{{SL}_{2}(\mathrm{F})}$$ . While we do not assume that the residual characteristic of $$\mathrm{F}$$ is relatively prime to n we do assume that n is not divisible by 4.