(G) |
general type, GL(4) type |
μ⊠ν(1) |
μ is a χ-self-dual, symplectic, unitary, cuspidal, automorphic representation of GL(4,A). |
⨁ψ∈(G)⨁π∈Πψπ |
none |
∙ |
∙ |
∙ |
∙ |
globally generic |
always |
L(s,μ) (primitive) |
L(s,μ,Λ2)L(s,χ) |
(Y) |
Yoshida type |
(μ1⊠ν(1))⊞(μ2⊠ν(1)) |
μ1 and μ2 are distinct, unitary, cuspidal, automorphic representations of GL(2,A) with the same central character χ. |
⨁ψ∈(Y)⨁{π∈Πψ:⟨⋅,π⟩=1}π |
(−1)n=1, where n=#{v|πv is non-generic}. |
∙ |
∙ |
|
|
globally generic |
if π∈L2 |
L(s,μ1)L(s,μ2) |
L(s,μ1×μ2)L(s,χ) |
(Q) |
Klingen packets, Soudry type |
μ⊠ν(2) |
μ is a χ-self-dual, unitary, cuspidal, automorphic representation of GL(2,A) with central character ωμ≠χ. (μ is then of orthogonal type.) |
⨁ψ∈(Q)⨁π∈Πψπ |
none |
|
|
∙ |
|
Langlands quotient of |⋅|ω−1μχ⋊|⋅|−1/2μ |
if π≠πψ |
L(s+12,μ)L(s−12,μ) |
L(s+1,ωμ)L(s−1,ωμ)L(s,μ×μ)L(s,χ) |
(P) |
Siegel packets, Saito-Kurokawa type |
(μ⊠ν(1))⊞(σ⊠ν(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)⨁{π∈Πψ:⟨⋅,π⟩=ε(1/2,σ−1μ)}π |
(−1)n=ε(1/2,σ−1μ), where n=#{v|πv is not the base point in Πψv}. |
|
∙ |
|
∙ |
Langlands quotient of |⋅|1/2σ−1μ⋊σ|⋅|−1/2 |
if (π∈L2 and π≠πψ) or if (π=πψ and ε(1/2,σ−1μ)=1 and L(1/2,σ−1μ)=0) |
L(s,μ)L(s+12,σ)L(s−12,σ) |
L(s+12,σμ)L(s−12,σμ)L(s,χ) |
(B) |
Borel packets, Howe - Piatetski-Shapiro type |
(σ1⊠ν(2))⊞(σ2⊠ν(2)) |
σ1 and σ2 are distinct unitary Hecke characters with σ21=σ22=χ. |
⨁ψ∈(B)⨁{π∈Πψ:⟨⋅,π⟩=1}π |
(−1)n=1, where n=#{v|πv is not the base point in Πψv}. |
|
|
|
|
Langlands quotient of |⋅|σ1σ−12×σ1σ−12⋊|⋅|−1/2σ2 |
if π∈L2 and π≠πψ |
L(s+12,σ1)L(s−12,σ1)L(s+12,σ2)L(s−12,σ2) |
L(s+1,σ1σ2)L(s,σ1σ2)2L(s−1,σ1σ2)L(s,χ) |
(F) |
finite-dimensional |
σ⊠ν(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(s−12,σ)L(s−32,σ) |
L(s+2,χ)L(s+1,χ)L(s,χ)L(s−1,χ)L(s−2,χ) |