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G type finite
Y name stable
Q Arthur parameter ψ paramodular
P Arthur parameter ingredients global base point πψ
B contribution to L2disc(G(F)G(A),χ) cuspidality condition for πΠψ
F parity condition spin L-function L(s,πψ,ρ4)
tempered standard L-function L(s,πψ,χ,ρ5)

Arthur packets for GSp(4)

Decomposition of the space L2disc(G(F)G(A),χ) for G=GSp(4), where F is a number field, A its ring of adeles, and χ is the central character.
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type name Arthur parameter ingredients contribution to L2disc parity condition tempered finite stable para global base point πΠψ cuspidal spin L-function standard L-function
(G) general type, GL(4) type μν(1) μ is a χ-self-dual, symplectic, unitary, cuspidal, automorphic representation of GL(4,A). ψ(G)πΠψπ none globally generic always L(s,μ)  (primitive) L(s,μ,Λ2)L(s,χ)
(Y) Yoshida type (μ1ν(1))(μ2ν(1)) μ1 and μ2 are distinct, unitary, cuspidal, automorphic representations of GL(2,A) with the same central character χ. ψ(Y){πΠψ:,π=1}π (1)n=1, where n=#{v|πv is non-generic}. globally generic if πL2 L(s,μ1)L(s,μ2) L(s,μ1×μ2)L(s,χ)
(Q) Klingen packets, Soudry type μν(2) μ is a χ-self-dual, unitary, cuspidal, automorphic representation of GL(2,A) with central character ωμχ. (μ is then of orthogonal type.) ψ(Q)πΠψπ none Langlands quotient of ||ω1μχ||1/2μ if ππψ L(s+12,μ)L(s12,μ) L(s+1,ωμ)L(s1,ωμ)L(s,μ×μ)L(s,χ)
(P) Siegel packets, Saito-Kurokawa type (μν(1))(σν(2)) μ is a unitary, cuspidal, automorphic representation of GL(2,A) with central character ωμ=χ, and σ is a Hecke character with σ2=χ. ψ(P){πΠψ:,π=ε(1/2,σ1μ)}π (1)n=ε(1/2,σ1μ), where n=#{v|πv is not the base point in Πψv}. Langlands quotient of ||1/2σ1μσ||1/2 if (πL2 and ππψ) or if (π=πψ and ε(1/2,σ1μ)=1 and L(1/2,σ1μ)=0) L(s,μ)L(s+12,σ)L(s12,σ) L(s+12,σμ)L(s12,σμ)L(s,χ)
(B) Borel packets, Howe - Piatetski-Shapiro type (σ1ν(2))(σ2ν(2)) σ1 and σ2 are distinct unitary Hecke characters with σ21=σ22=χ. ψ(B){πΠψ:,π=1}π (1)n=1, where n=#{v|πv is not the base point in Πψv}. Langlands quotient of ||σ1σ12×σ1σ12||1/2σ2 if πL2 and ππψ L(s+12,σ1)L(s12,σ1)L(s+12,σ2)L(s12,σ2) L(s+1,σ1σ2)L(s,σ1σ2)2L(s1,σ1σ2)L(s,χ)
(F) finite-dimensional σν(4) σ is a unitary Hecke character with σ2=χ. ψ(F)σ none Langlands quotient of ||2×||σ||3/2 never L(s+32,σ)L(s+12,σ)L(s12,σ)L(s32,σ) L(s+2,χ)L(s+1,χ)L(s,χ)L(s1,χ)L(s2,χ)