Local factors of nongeneric supercuspidal representations of $${ GSp}_4$$ G S p 4

Zeta integrals for GSp(4) via Bessel models

“…as the gcd of all Z(s, B, Φ, µ), as B runs through B(π, Λ, β) and Φ runs through S 0 (V ). The same observation was made in Proposition 18 i) of [6].…”

“…For B ∈ B(π, Λ, β) and s ∈ C, we define the simplified zeta integralsζ(s, B, µ) = F × B([ x 1 ])µ(x)|x| s−3/2 d × x. (4.2.1)The same integrals appear in Proposition 18 of[6]. Using the general form(4.1.18) of the functions B([ x 1 ]), which holds both in the split and the non-split case, it is easy to see that ζ(s, B, µ) converges to an element C(q −s ) for real part of s large enough.…”

Newforms of Saito-Kurokawa lifts

“…Now the estimation (5.5) also holds by the proof of Lemma 45. If τ is not St, and F is odd residual, (5.18) was proved by Danishman [4]. We give a proof (5.18) for the case where τ is St in this subsection (c.f.…”

“…For this case, in sect. 4 we introduce nonsplit paramodular groups, and study a Hecke theory for Bessel vectors. But, since it seems to be difficult to compute a Hecke operator for Bessel vectors fixed by the above K 2m in general, we introduce other arithmetic subgroups, and consider a refinement (4.21) of the sequence (1.2).…”

Invariant Vectors for Weak Endoscopic and Saito-Kurokawa Lifts to GSp(4)

“…3.1] and this implies (4) by the local functional equation. Next, for non-generic and cuspidal π v ∼ = θ − (σ v ) at a place v with odd residue characteristic, (4) is implied by[6, Thm. 4.4 and Cor.…”