We consider an n-fold Brylinski–Deligne cover of a reductive group over a p-adic field. Since the space of Whittaker functionals of an irreducible genuine representation of such a cover is not one-dimensional, one can consider a local coefficients matrix arising from an intertwining operator, which is the natural analogue of the local coefficients in the linear case. In this paper, we concentrate on genuine principal series representations and establish some fundamental properties of such a local coefficients matrix, including the investigation of its arithmetic invariants. As a consequence, we prove a form of the Casselman–Shalika formula which could be viewed as a natural analogue for linear algebraic groups. We also investigate in some depth the behaviour of the local coefficients matrix with respect to the restriction of genuine principal series from covers of GL2 to SL2. In particular, some further relations are unveiled between local coefficients matrices and gamma factors or metaplectic-gamma factors.
Bibliographical notePublisher Copyright:
c 2023 American Mathematical Society.
- Covering groups
- Plancherel measure
- Whittaker functionals
- gamma factor
- local coefficients matrix
- scattering matrix