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 -