Local Coefficients and Gamma Factors for Principal Series of Covering Groups

We consider an n n -fold Brylinski–Deligne cover of a reductive group over alttext="p"> p encoding="application/x-tex">p -adic field. Since the space Whittaker functionals irreducible genuine representation such is not one-dimensional, one can local coefficients matrix arising from intertwining operator, which natural analogue in linear case. In this paper, we concentrate on principal series representations and establish some fundamental properties matrix, including investigation its arithmetic invariants. As consequence, prove form Casselman–Shalika formula could be viewed as for algebraic groups. also investigate depth behaviour with respect to restriction covers alttext="upper G upper L 2"> G L 2 encoding="application/x-tex">GL_2 S S encoding="application/x-tex">SL_2 . particular, further relations are unveiled between matrices gamma factors or metaplectic-gamma factors.