Local Coefficients and Gamma Factors for Principal Series of Covering Groups

Abstract

We consider an n n -fold Brylinski–Deligne cover of a reductive group over a p 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 G L 2 GL_2 to S L 2 SL_2 . In particular, some further relations are unveiled between local coefficients matrices and gamma factors or metaplectic-gamma factors.

R-GROUP AND WHITTAKER SPACE OF SOME GENUINE REPRESENTATIONS

“…• In Section 6.2 we consider the Whittaker space Wh ψ π un χ from the perspective of unramified Whittaker functions. Using an analogue of the Casselman-Shalika formula proved in [19], we show in Theorem 6.5 a result on dimWh ψ π un χ , which is compatible with Theorem 6.1.…”

“…In this paper, it is the latter that is used and plays a crucial role in determining dim Wh ψ (π σ ). We hope that this phenomenon also helps justify our viewpoint in [19] that both the local coefficient matrix and the scattering matrix are important objects and should be studied together.…”

“…Fix a Whittaker datum B = T U,ψ for G, where U is the unipotent radical of the Borel subgroup B and ψ : U → C × is a nondegenerate character. It is well known that genuine representations of covering groups could have high-dimensional ψ-Whittaker space (i.e., the space of ψ-Whittaker functionals; see the introductions of [18,19] for brief literature reviews). In particular, dim Wh ψ (I(χ)) increases as the degree of covering increases.…”

Hecke $L$-functions and Fourier coefficients of covering Eisenstein series

“…We call the matrix [τ (w, χ, γ, γ ′ )] a scattering matrix (see [GSS,§3.6]). It satisfies some immediate properties:…”

“…The investigation in [KP84] relies on the so-called scattering matrix arising from a map between two Whittaker spaces (i.e., the space of Whittaker functionals) induced from intertwining operators, while the method of [KP86] is via trace formula. With the same focus on such scattering matrices, the Casselman-Shalika formula in the covering setting was generalized in [Pat87,CO13,McN16,Suz97,GSS]. In a somewhat different direction, various forms of the Casselman-Shalika formula were also proved in connection with the theory of crystal basis, Demazure operators and representations of quantum groups, see [BBF11a,BN10,McN11,LLS14,LLL19,KL11].…”