TY - JOUR

T1 - Local coefficients and gamma factors for principal series of covering groups

AU - Gao, Fan

AU - Shahidi, Freydoon

AU - Szpruch Alpert, Daniel

PY - 2019

Y1 - 2019

N2 - 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 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.

AB - 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 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.

UR - https://www.ams.org/cgi-bin/mstrack/accepted_papers/memo

U2 - https://arxiv.org/abs/1902.02686

DO - https://arxiv.org/abs/1902.02686

M3 - Article

SN - 0065-9266

JO - Memoirs of the American Mathematical Society

JF - Memoirs of the American Mathematical Society

ER -