Symmetric Genuine Spherical Whittaker Functions on

Abstract

Abstract. Let F be a p-adic field of odd residual characteristic. Let GSp 2n (F) and Sp 2n (F) be the metaplectic double covers of the general symplectic group and the symplectic group attached to the 2n dimensional symplectic space over F. Let σ be a genuine, possibly reducible, unramified principal series representation of GSp 2n (F). In these notes we give an explicit formulas for a spanning set for the space of Spherical Whittaker functions attached to σ. For odd n, and generically for even n, this spanning set is a basis. The significant property of this set is that each of its elements is unchanged under the action of the Weyl group of Sp 2n (F). If n is odd then each element in the set has an equivariant property that generalizes the uniqueness result of Gelbart, Howe and Piatetski-Shapiro proven in [?]. Using this symmetric set, we construct a family of reducible genuine unramified principal series representations which have more then one generic constituent. This family contains all the reducible genuine unramified principal series representations induced from a unitary data and exists only for n even.

R-GROUP AND WHITTAKER SPACE OF SOME GENUINE REPRESENTATIONS

“…In this subsection, we apply Theorem 5.6 to the double cover of GSp 2r and show that it recovers [48,Corollary 6.6]. Meanwhile, we also show that the analogue of Conjecture 5.7 fails for such covers.…”

“…Oy (w) is represented by the matrix γ(w,χ) • S R (w,i(χ)) Oy , where γ(w,χ) is the γ-factor associated to w and S R (w,i(χ)) Oy is a so-called scattering matrix. As an application of this theorem, we show in Section 5.5 that a result of Szpruch [48] on the double cover of GSp 2r can be recovered from it (see Theorem 5.10). Here Theorem 1.3 also implies that Conjecture 1.1 is equivalent to the following (compare Conjecture 5. for every w ∈ R χ , where the left-hand side denotes the trace of the matrix S R (w,i(χ)) Oy .…”

“…The number Q(e 0 ) ∈ Z/2Z determines whether the similitude factor F × corresponding to the cocharacter e 0 splits into GSp 2r or not. To recover the classical double cover of GSp 2r (see [48]), we take Q(e 0 ) to be an even number in this subsection.…”

R-group and Whittaker space of some genuine representations

“…Here σ Wh Oy (Û) is represented by the matrix γ(w, χ) • S R (w, i(χ)), where γ(w, χ) is the gamma-factor associated to Û and S R (w, i(χ)) a so-called scattering matrix. As an application of the above theorem, we show in §5.5 that a result of D. Szpruch [Szp15] on double cover of GSp 2r could be recovered from it, see Theorem 5.10. Here Theorem 1.3 also implies that Conjecture 1.1 is equivalent to the following (cf.…”

“…The number Q(e 0 ) ∈ Z/2Z determines whether the similitude factor F × corresponding to the cocharacter e 0 splits into GSp 2r or not. To recover the classical double cover of GSp 2r (see [Szp15]), we take Q(e 0 ) to be an even number in this subsection.…”