絡作用素と p進簡約群の既約表現の構成 (印刷用)

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p

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Langlands

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F Archimedes G F P G F- , P = MU P Levi (_M Levi ) G :=_{G(F), P := P(F), M := M(F), U := U(F)...} IG P(σ) M σ G . (Langlands) . . . .. . . . G π , M σ ,π IG P(σ) (Langlands ) . ( ) 2 / 27

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rP Jacquet IG P , G π M σ HomM(rP(π), σ) ! HomG(π, IG_P(σ)). Langlands , . ⇒ ( ) . ( ) 3 / 27

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Irr(G) G Irru(G) G Π0(G) G Π2(G) G Πtemp(G) G

Π0(G)⊂Π2(G)⊂ Πtemp(G)⊂ Irru(G)⊂ Irr(G)

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. Plancherel µ(σ) IG P(σ) , . Harish-Chandra R- EndG(IG_P(σ)) . R- IG_P(σ) . ( ) 5 / 27

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X(_{M) M} F -aM := Hom(X(M), R),a∗_M = X(M) ⊗ R HM : M→ aM . exp_{'χ, H}M(m)( = |χ(m)|F (χ ∈ X(M)) M1 _{:= Ker H} M λ∈ a∗ M,C := a ∗ M⊗ C e λ_{∈ ˆA} M:= Hom(M/M1, C×) eλ : M/M1 _{+ m ,→ exp'λ, H} M(m)( ∈ C× . ( ) 6 / 27

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M (σ, V) , IG P(V) I G P(σ) (σλ, Vλ) := (σ_{⊗ e}λ_{, V) (λ}_{∈ a}∗ M,C) , IG P(Vλ) I G P(σλ) P, P- _{∈ P(M) M} _F -, P- :=_P-(F), P- = M-U -. ( ) . . . .. . . . JP-_|P(σλ)φ(g) := ! U∩U-_\U-φ(u -_g)du-_(g _{∈ G, φ ∈ I}G P(Vλ)) A_M _M , Σ_M G A_M , Σ_M ⊃ Σ_P P K G ( ) 7 / 27

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. (Waldspurger03, Konno03 ) . . . .. . . . (1)φ_{∈ I}G P(Vλ) v∨ ∈ V∨ , α∨(Re(λ)) >> 0, _{∀α ∈ Σ}_P_{\ Σ}_P -'JP-_|P(σ_λ)φ( g), v∨( := ! U∩U-_\U-'φ(u -_{g), v}∨_(du -,JP-_|P(σλ)φ( g)∈ (V∨)∨=V. (2) ˆ AM + λ ,→ JP-|P(σλ)φ|K ˆ AM indK_K_∩P(V)- . (3) IG P(Vλ) + φ ,→ JP-|P(σλ) ∈ I G P-(Vλ)φ HomG(IG_P(σλ), IG_P-(σλ)) . ( ) 8 / 27

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(Waldspurger03, Konno03, Arthur89 ) . . . .. . . . (1) λ_{∈ ˆA}M φ∈ IG_P(Vλ) , JP-|P(σλ)φ . (2) ( )φ_{∈ I}G P(V), φ ∨_{∈ I}G P-(V∨) 'JP-_|P(σ)φ, φ∨( = 'φ, J_P_|P-(σ∨)φ∨(. (3)_{P, P}- _{∈ P(M)} Weyl a+ P, a + P- ⊂ aM ,d(_{P, P}-) a+ P, a + P- . P1, P2, P3 ∈ P(M) d(P3,P1) = d(P3,P2) + d(P2,P1) , . JP3|P1(σλ) = JP3|P2(σλ)◦ JP2|P1(σλ) ( ) 9 / 27

Plancherel

(Waldpurger

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σ M(F) ˆ AM := Hom(M/M1_{, C}×_). . . (Waldspurger03, Sauvageot97) . . . .. . . . ˆ AM Zariski ΞP(σ) , eλ ∈ ΞP(σ) IG P(σλ) . P P opposite . Schur eλ_{∈ Ξ}P(σ) j(σλ) := J_P|P(σλ)JP| P(σλ)∈ EndG(IG_P(σλ))! C ˆ AM . ( ) 10 / 27

Plancherel

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Σred P ΣP α∈ Σred P Mα ⊃ M Levi Mα , Σ_P∩Mα = {α} . γ(G/M) γ(G/M) := ! UδP (mP( u))d u . δP P mP( u) u∈ U ( P = M U) G P M . . (Plancherel ) . . . .. . . . σ M .λ∈ a∗ M,C . µ(σλ) := j(σλ)−1 " α∈Σred P γ(Mα/M)2 ( ) 11 / 27

Plancherel

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Placnerel Waldspurger . Plancherel

ˆ A1 M . G Plancherel . Plancherel Harish-Chandra . Harish-Chandra Eisenstein c , Plancherel . µ(σλ) M P . µ(σλ) Weyl W . µ(w_σ_{λ) = µ(σλ),} _{∀w ∈ W.} ( ) 12 / 27

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R

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R -. (Harish-Chandra, Waldspurger) . . . .. . . . π∈ Πtemp(G) ,G Levi M σ∈ Π2(M) (_{M, σ)} G P ∈ P(M) π IG_P(σ) . ⇒ Πtemp(G) IG_P(σ) (σ∈ Π2(M)) . ( ) 13 / 27

R

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. ( , Arthur89) . . . .. . . . Levi _M σ _{∈ Π}2(M) AMˆ {rP-_|P(σ_λ)}_{P, P}-_∈P(M) ⊂ C( Â_M) . (1) RP-_|P(σ_λ) := r_P-_|P(σ_λ)−1J_P-_|P(σ_λ) λ∈ Â_M , . RP3|P1(σλ) = RP3|P2(σλ)◦ RP2|P1(σλ) (P1, P2, P3 ∈ P(M)) (2) ( )φ_{∈ I}G P(V), φ∨∈ I G P-(V∨) . 'RP-_|P(σλ), φ∨( = 'φ, RP|P-(σ∨ λ)φ ∨₍ (3)RP-_|P(σλ) Re(λ)∈ a+ P(Weyl ) . (4)RP-_|P(σλ) λ∈ Â1 M :={χ ∈ ÂM | χ } . ( ) 14 / 27

R

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(σ, V) M(F) W(M) := NG(M)/M, W(M)σ:={w ∈ W(M) | wσ! σ} w_{∈ W(M)}σ M+_w :='M, w( σ+ w ind M+ w M σ (σ M + w ) δ+ P,w P δP M + w A(σ+ w) : IG_w−1_P(V)+ φ ,→ δ + P,w(w)σ + w(w)φ(w−1∗) ∈ IGP(V) . w_{∈ W(M)}σ . RP|P(w, σ) : IG_P(σ)→ IG_P(σ) . RP|P(w, σ) : IG_P(V) R_{w−1 P}_|P(σ) −→ IG w−1_P(V) A(σ+ w) −→ IG P(V) ( ) 15 / 27

R

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. . . . .. . . . W(M)0 σ:={w ∈ W(M)σ | RP|P(w, σ) } . W(M)0 σ W(M)σ W(M)σ) W(M)0σ. . (R- ) . . . .. . . . Rσ:= W(M)σ/W(M)0σ . (Harish-Chandra, ) . . . .. . . . EndG(IG_P(σ)) {RP|P(w, σ)| w ∈ W(M)σ} . . (Silberger78) . . . .. . . . {RP|P(w, σ)| w ∈ Rσ} EndG(IG_P(σ)) . dim EndG(IG_P(σ)) = *Rσ. ( ) 16 / 27

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R

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Rσ+ w1, w2 , ζσ∈ C2(Rσ, C×) RP|P(w1w2, σ) = ζσ(w1, w2)RP|P(w1, σ)RP|P(w2, σ) . [ζσ] _{∈ H}2_(Rσ_{, C}×₎ ([Chao-Li]). [ζσ] Zσ⊂ H2(Rσ, C×) , 1_{→ Z}σ → ˜Rσ → Rσ→ 1 . H2_(Rσ_{, C}×₎ _{→ H}2_{( ˜Rσ}_{, C}×₎ _ζ_σ λσ ∈ C1( ˜Rσ, C×) ζσ(w1, w2) = λσ( ˜w1)λσ( ˜w2) λσ( ˜w1w2˜ ) ( ˜Rσ+ ˜wi ,→ wi ∈ Rσ, i = 1, 2). ( ) 17 / 27

R

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˜Rσ _{+ ˜r ,→ ˜R(˜r, σ) := λ}σ(˜r)−1R(r, σ)∈ AutG(IG_P(σ)). χσ := λσ|Zσ Zσ . Π( ˜Rσ, χσ) :=_{{ρ ∈ Irr( ˜R}σ) _{| ρ|}Zσ =χσ} . (Arthur93) . . . .. . . . (1) ˜Rσ_{× G} . IG_P(σ) _! # ρ∈Π( ˜Rσ,χσ) ρ∨_{⊗ π}ρ (2)JH(IG_P(σ)) IG_P(σ) . Π( ˜Rσ, χσ) ! JH(IG_P(σ)) ( ) 18 / 27

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P0 G ,P0 ⊃ M0 G Levi , Σ0 := ΣM0 Σ0 ⊃ ∆P0 ∆_P:=_{α0|aM | α0 ∈ ∆P0 \ ∆P0∩M} a∗,+ P :={λ ∈ a ∗ M | α ∨_{(λ) > 0}_{∀α ∈ ∆} P} W M0 G Weyl . (Langlands ) . . . .. . . . (1)(σ, V) _{∈ Π}temp(M), P ∈ P(M), λ ∈ a∗,+_P , . JG_P(σλ) := Im[J_P|P(σλ) : IG_P(σ)→ IG_P(σ)](Langlands ) (2) π _{∈ Irr(G)} ,π _{! J}G P(σλ) (P, σ, λ) W- , . ( ) 19 / 27

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. ( , Bernstein-Zelevinski77 ) . . . .. . . . π_{∈ Irr(G)} , Levi _Mc ρ ∈ Π0(Mc) G . [Mc, ρ] π . (i)_∃Pc ∈ P(Mc), π -→ IG_P c(ρ). (ii)∀Pc ∈ P(Mc), IG_P c(ρ) π . . (Langlands-Casselman , Konno03 ) . . . .. . . . π _{∈ Irr(G)} ,[Mc, ρ] . (i)π _{⇔ Re(Exp(r}Pc(π)))⊂ +_a∗ Pc, ∀Pc ∈ P(Mc). (ii)π ⇔ Re(Exp(rPc(π)))⊂+a∗_P c, ∀Pc ∈ P(Mc). ( ) 20 / 27

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. (Harish-Chandra, Silberger79 ) . . . .. . . . P = MU , IG P(σ) (σ∈ Π2(M)) W(M)σ! {1}, µ(σ) ! 0. ⇒ . (Harish-Chandra, Silberger79 ) . . . .. . . . P = MU ,σ _{∈ Π}2(M) . IG P(σλ0) (λ0 ∈ a∗_M\ a∗_G) µ(σλ) λ = λ0 . ⇒ (Langlands ) ( ) 21 / 27

G = U(2), U(3)

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E/F F F U(2), U(3) . U(2)(F) := $ g_{∈ GL}2(E) %% %% %% t g & 0 1 −1 0 ' g = & 0 1 −1 0 '( U(3)(F) :=         g ∈ GL3(E) %% %% %% %% % t_g    0 0 1 0 _{−1 0} 1 0 0    g =    0 0 1 0 _{−1 0} 1 0 0             B = MU(Borel ) M =        {diag(a, a−1₎ _{| a ∈ E}×_} ₍_{G = U(2))} {diag(a, t, a−1₎ _{| a ∈ E}×_{, t} _{∈ E}1_{} (G = U(3))}. E1 ₌_U(1)(F)_. ( ) 22 / 27

G = U(2), U(3)

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σ_{∈ Irr}u(M) = Π2(M) χ∈ Irru(E×), η∈ Irru(E×/F×)

      

σ(diag(a, a−1_{)) = χ(a)} ₍_{G = U(2))}

σ(diag(a, t, a−1)) = χ(a)ηu(t) (G = U(3))

. ηu : E1 + z/ z ,→ η(z) ∈ C1. λ_{∈ C}         

σλ(diag(a, a−1)) = χ(a)_|a| λ

2

E (G = U(2))

σλ(diag(a, t, a−1) = χ(a)_|a| λ

2

Eηu(t) (G = U(3))

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Irru(E×, δ) :={χ ∈ Irru(E×)| χ|F× = δ} (δ = 1_F×, ω_E/F)

*Rσ≤ *W(M) = 2 , ⇒Harish-Chandra Langlands ( ) 23 / 27

G = U(2), U(3)

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. (Plancherel , Keys-Shahidi88 ) . . . .. . . . (1)_{G = U(2)} µ(σλ) = e1(λ) LF(1− λ, (χ|F×)−1)LF(1 + λ, χ|F×) LF(λ, χ|F×)L_F(−λ, (χ|_F×)−1) .

(2)_{G = U(3)} ,ωE/F E/F

µ(σλ) =e2(λ) LE(1− λ, χ−1η)LF(1− 2λ, ωE/F(χ|F×)−1) LE(λ, χη−1)LF(2λ, ωE/Fχ|F×) × LE(1− λ, χη −1_)L_F_{(1 + 2λ, ω}_E/Fχ_|_F_×₎ LE(−λ, χ−1η)LF(−2λ, ωE/F(χ|F×)−1) . e1(λ), e2(λ) µ(σλ) . ( ) 24 / 27

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G = U(2), U(3)

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. (IG B(σλ) ,G = U(2) ) . . . .. . . . χ_{∈ Irr}u(E×), λ∈ R . (1)χ_|NE/F(E×) ! 1F× ,I G B(σλ) . (2)χ = µ∈ Irru(E×, ωE/F) . (i)IG B(σ0) ! π(µ) +_{⊕ π(µ)}−_{, π(µ)}± _{∈ Πtemp}_{(G) (λ = 0)}_. (ii) IG B(σλ) (λ! 0) . (3)χ = η _{∈ Irr}u(E×, 1F) . (i)λ =±1 0 _{→ St}G_(η)_{→ I}G B(σ1) → ηu(det)→ 0 0 _{→ η}u(det) → IG_B(σ−1) → StG(η)→ 0 . StG_{(η) = η} u(det)⊗ StG(1E×) ∈ Π₂(G) StG(1_E×) Steinberg . ηu(det) = JG_P(σ1) . (ii)λ_{! ±1} IG B(σλ) . ( ) 25 / 27

G = U(2), U(3)

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. (IG B(σλ) ,G = U(3) I) . . . .. . . . χ∈ Irru(E×), η∈ Π(E×, 1F×), λ∈ R . (1)χ_|NE/F(E×)! 1F× , I G B(σλ) . (2)χ = η1 ∈ Irru(E×, 1F×), η₁ ! η , (i)IG_B(σ0) (λ = 0) ! π(η1, η)+⊕ π(η1, η)−, π(η1, η)± ∈ Πtemp(G). (ii) IG B(σλ) (λ! 0) . (3)χ = η . (i)λ =_±2 0 _{→ St}G(η)_{→ I}G_B(σ2) → ηu(det)→ 0 0 _{→ η}u(det) → IG_B(σ−2) → StG(η)→ 0 . StG(η) = ηu(det)⊗ StG(1E×) StG(1_E×) Steinberg , η(det) = JG P(σ2) . (ii) IG B(σλ) (λ! ±2) . ( ) 26 / 27

G = U(2), U(3)

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. (IG B(σλ) ,G = U(3) II) . . . .. . . . (4)χ = µ_{∈ Irr}u(E×, ωE/F) . (i)λ_{∈ {±1}} . 0_{→ π}2_{(µ, η)} _{→ I}G B(σ1) → π nt_{(µ, η)}_{→ 0} 0_{→ π}nt(µ, η)_{→ I}G_B(σ₋₁) _{→ π}2(µ, η) _{→ 0} . π2_{(µ, η)}_{∈ Π} 2(G), πnt(µ, η)∈ Irru(G)\ Πtemp(G) . (ii)λ_{! ±1} , IG B(σλ) . ( ) 27 / 27