Langlands 1 1. Langlands p GL n Langlands [HT] The local Langlands conjecture is one of those hydra-like conjectures which seems to grow as it gets pr

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1 Langlands 1 1. Langlands p GL n Langlands [HT] The local Langlands conjecture is one of those hydra-like conjectures which seems to grow as it gets proved. ([HT], p.1) hydra [KP] Langlands Langlands Langlands G G = GL n Langlands 1 tetsushi@math.kyoto-u.ac.jp 1

2 Langlands Langlands 2 Langlands 2.1 G GL n U 2, U 3 [ ], [ ] G Langlands 3 (inner form) Langlands-Vogan Kottwitz, 3.1 Langlands 4 GL n (F q ) Deligne-Lusztig 5 Rapoport-Zink 6 Rapoport-Zink Kottwitz Rapoport-Zink Lubin-Tate Drinfeld L [RZ], [Ra], [Far1], [SW] Langlands 2. p Langlands Langlands Galois Weil Weil-Deligne Galois Langlands p F q ((t)) p R C Langlands Vogan [Vog] p F p Q p F p p F Galois Γ F := Gal(F /F ) F Galois G F Langlands G(F ) G F F Galois Γ F F F q Frob q Γ Fq := Gal(F q /F q ) 2

3 Frobenius Frob q Frob q (x) = x 1/q (x F q ) Γ Fq Γ F Γ Fq Frob i q Γ Fq U i Γ F I F := U 0 F W F := i Z U i F Weil U i Γ F U i W F W F I F = U 0 W F W F Artin Art F : F = W ab F U 1 W F W F [W F, W F ] G F G F - G(F ) G(F ) G(F ) p l mod p mod l Langlands C (admissible) [ ] Art F 1 1 1:1 GL 1 (F ) π ϕ: W F C / = : ϕ ϕ WF ab C GL 1 (F ) = F Art F WF ab ϕ C π π := ϕ Art F GL 1 (F ) = F GL 1 (F ) 1 G G(F ) W F Langlands : G(F ) π W F (Galois ) ϕ Langlands L G Ĝ G C C- [ ] [Bor], [Cog] Ĝ W F G L L G := Ĝ W F L G 1 Ĝ L G W F 1 ϕ: W F SL 2 (C) L G 4 ϕ G L Langlands ϕ(sl 2 (C)) Ĝ. ϕ SL 2 (C) ϕ SL2 (C) : SL 2 (C) Ĝ C (id,1) ϕ W F W F SL 2 (C) L G pr 2 W F pr i i Frobenius σ W F (pr 1 ϕ)(σ) Ĝ 3

4 L ϕ, ϕ Ĝ- gϕg 1 = ϕ g Ĝ G = GL n Ĝ = GL n(c), L G = GL n (C) W F GL n L W F SL 2 (C) SL 2 (C) W F 1 1 G = GL 1 Ĝ = C ϕ SL2 (C) : SL 2 (C) C C GL 1 L ϕ: W F C 1 1 Langlands 2.1 ( Langlands ). LLC G : G(F ) π G L ϕ / = /Ĝ- G(F ) L Ĝ- LLC G (π) π L Langlands ϕ Π G ϕ := LLC 1 G (ϕ) L G LLC G G LLC G [Bor] LLC G LLC G LLC G 2.1 LLC G (1) GL 1 LLC GL1 Langlands (2) GL n Zelevinsky π L ε n Zelevinsky [ ] GL n Langlands [ ] LLC GLn [He1] [He3] n LLC GLn [HT] [He2] [LRS] GL n Langlands LLC GLn [Laf] Scholze LLC GLn GL n Langlands [Scho] [Scho] LLC GLn [HT], [He2] LLC GLn (3) G GL n G(F ) GL n (F ) LLC G G Sp 2n SO n U n LLC G [MT], [Mœ] [MT], [Mœ] LLC G 4

5 G twisted endoscopy LLC G U 3 [Ro] [ ] Sp 2n, SO n [A3] U n [Mo] SO 2n [A4] G(F ) π LLC G G = SL n SL n GL n GL n G LLC SLn [LL], [HS] G = Sp 4 GSp 4 G(F ) GL 2 (F ) GL 4 (F ) LLC Sp4 LLC GSp4 [GT1], [GT2] (4) LLC G [DR] 0 LLC G Deligne-Lusztig Bruhat-Tits GL 2 0 [ ] LLC G 3. Langlands-Vogan Langlands G G (inner form) Langlands Vogan Langlands-Vogan [Vog] 2.1 Langlands G p F G F (quasi-split) 2.1 LLC G : G(F ) L Π G = ϕ G L ϕ /Ĝ- J G (inner form) G = GL n J = GL m (D) m n D F dim F D = n 2 /m 2 Ĝ = Ĵ, L G = L J G L J L J 2.1 LLC J : J(F ) L Π J ϕ G L ϕ /Ĝ- J J G = J LLC J 2 LLC 1 G LLC J : J(F ) L Π J ϕ G(F ) L Π G ϕ Jacquet-Langlands endoscopy endoscopic transfer LLC G, LLC J 5

6 endoscopic transfer [ ] L ϕ Π G ϕ, ΠJ ϕ L (endoscope) endoscopy Arthur ϕ S Sϕ Arthur ( := π ) 0 CentĜ(Im ϕ)/z(ĝ)w F L Arthur [A1] L ϕ: W F SL 2 (C) L G Im ϕ L G Ĝ L G CentĜ(Im ϕ) := g Ĝ x Im ϕ, gx = xg CentĜ(Im ϕ) Ĝ G L L G = Ĝ W F W F Ĝ W F Ĝ Z(Ĝ) Z(Ĝ)W F := g Z(Ĝ) w WF, wgw 1 = g L G Z(Ĝ)W F Ĝ H := Cent Ĝ (Im ϕ)/z(ĝ)w F C H H 0 Arthue S Sϕ Arthur π 0 (H) ) Arthur Π G ϕ SArthur = π 0 (H) = H/H 0 H H ϕ 1 1 : Sϕ Arthur 1:1 (?) Π G ϕ G = GL n Schur S ϕ L 1 [HT], [He2], [Scho] U 3 Arthur [ ] G Whittaker 1 [Vog], [GGP] Vogan J L G (pure inner form) Arthur S S 1 1 S Vogan ϕ / = S Vogan ϕ := π 0 ( CentĜ(Im ϕ) ) / = 1:1 (?) J G Vogan [Vog], [GGP] Kottwitz Kottwitz Hasse-Weil [Ko2] Kottwitz [Ra], [Kal] Langlands-Vogan Kottwitz G Z(G) Ĝ Ĝder := [Ĝ, Ĝ] Kottwitz 1 1 Z(Ĝder) W F C 1:1 H 1( Γ F, G ad (F ) ) = J J G 6 Π J ϕ / =

7 Kottwitz [Ko1, Proposition 6.4] G ad := G/Z(G) G χ: Z(Ĝ)W F C χ Z(Ĝder) W F Z(Ĝ)W F χ: Z(Ĝ)W F C H 1( Γ F, G ad (F ) ) = J J G χ G J χ Z(Ĝder) W F / = Z(Ĝ)W F χ J χ G = GL n 3.1 ( Langlands-Vogan Kottwitz ). G Z(G) CentĜ(Im ϕ)/z(ĝ)w F ϕ (elliptic) χ: Z(Ĝ)W F C LLC ϕ,χ : ρ : CentĜ(Im ϕ) ρ = χ 1:1 Π Jχ Z(Ĝ)W F ϕ χ ρ Sϕ Arthur 3.1 Arthur 3.1 Langlands 2.1 Arthur 3.1 CentĜ(Im ϕ) Sϕ Arthur, S Vogan ϕ 5, 6 Rapoport-Zink Langlands [HS] SL n L G Ĝder [A2] 3.3. ρ π Π Jχ ϕ Langlands-Vogan J χ (1) ϕ (2) L Π J χ ϕ (3) L Π J χ ϕ ϕ (1) L SL 2 (C) ϕ SL2 (C) (2) L Π J χ ϕ ϕ SL2 (C) ρ π Mœglin, Tadić G ρ π Langlands [MT], [Mœ] 7 / =

8 4. GL n (F q ) Langlands (toy model) GL n (F q ) q F q q GL n (F q ) GL 2 (F q ) SL 2 (F q ) 100 Frobenius, H. Jordan, Schur [Jo], [Schu] Green 60 GL n (F q ) [Gr1] Green p [ ] n = n n r (r 2) n M n1,...,n r := GL n1 GL nr GL n M n1,...,n r P n1,...,n r GL n P n1,...,n r GL n i = 1,..., r (ρ i, V i ) GL ni (F q ) P n1,...,n r (F q ) M n1,...,n r (F q ) = GL n1 (F q ) GL nr (F q ) ρ 1 ρ r GL(V1 V r ) ρ Ind GL n(f q ) P n1,...,nr (F q) ρ (ρ 1,..., ρ r ) GL n (F q ) GL n (F q ) θ : F q n C θ, θ q,..., θ qn 1 θ θ, θ θ = θ qi i 1 θ θ x F q n xqn = x θ qn = θ θ Green GL n (F q ) 4.1 (Green ([Gr1])). 1:1 GL n (F q ) θ : F qn C / = GL n (F q ) θ GL n (F q ) π θ Langlands 2.1 Harish-Chandra Lie θ L 4.1 GL n (F q ) Langlands θ π θ [ ] GL 2 (F q ) Green 4.1 Brauer π θ Deligne-Lusztig GL n (F q ) [DL] Deligne-Lusztig U n Tate-Thompson [Ta, pp. 102] SL 2 Drinfeld GL n F q G 8 /

9 G(F q ) GL n (F q ) X 1 DL := v = F n q det ( ) q 1 X qj 1 i = 1. X n det(x qj 1 i ) (i, j) X qj 1 i n DL F q (n 1) GL n Deligne-Lusztig DL F q - DL q = 2 n = 2 DL q 1 det(x qj 1 i ) = det ( X1 X q 1 X 2 X q 2 ) q 1 = ( X 1 X q 2 Xq 1 X 2) q 1 = 1 Drinfeld SL 2 Deligne-Lusztig XY q X q Y = 1 (q 1) GL n (F q ) F qn DL (g, α) GL n (F q ) F qn v DL (g, α) v = g αx 1. αx n v GL n (F q ) F q n (g, α) v DL l H i c( DL, Ql ) GLn (F q ) F q n Q l π θ Deligne-Lusztig 4.2 (Deligne-Lusztig ([DL])). l q Q l = C GL n (F q ) Hom F q n ( ) Hc i ( ) π θ i = n 1 DL, Ql, θ = 0 i n 1 F qn 1 θ θ DL L θ π θ DL GL n (F q ) 4.2 DL Green π θ 4.3 (Green ([Gr1])). 4.1 θ π θ g GL n (F q ) g GL n (F q ) P g (T ) = det(t g) F q [T ] P g (T ) = Q(T ) e Q(T ) F q [T ] Q(T ) α F q d m = dim Fq d Ker ( g α id ) deg Q(T ) 1 Tr π θ (g) = ( 1) n 1 i=0 9 θ qi (α) m 1 j=1 (1 q j deg Q(T ) )

10 P g (T ) = Q(T ) e Q(T ) F q [T ] Tr π θ (g) = 0. GL 2 (F q ) 4.1, 4.3 [ ] 4.3 g Jordan α Jordan m Tr π θ (g) L θ m 1 (1 S j deg Q(T ) ) j=1 g Jordan Green S = q F q Deligne-Lusztig [Lu1], [Lu2] Green 4.1, 4.3 Deligne- Lusztig [DL] 4.1, 4.3 [DL] Green [Ma] [Lu1] Green [DL] GL n (F q ) [Gr1] 44 [Gr2] Lusztig [DL] G G(F q ) Deligne-Lusztig [ ] Green Deligne-Lusztig 4 Langlands Green Deligne-Lusztig 2 [DR] 0 Langlands-Vogan Deligne-Lusztig Bruhat-Tits [Kal] Langlands Rapoport-Zink 5 Rapoport-Zink Deligne- Lusztig [Yo1], [Hara], [Vol], [VW] Rapoport-Zink [Yo1], [ImT] Deligne-Lusztig Langlands Deligne-Lusztig Rapoport-Zink Deligne-Lusztig p Rapoport-Zink Deligne-Lusztig Rapoport-Zink Lubin-Tate Drinfeld 1970 [De], [Dr3], [Dr1], [Dr2] Deligne-Lusztig Drinfeld SL 2 Drinfeld Tate-Thompson [Ta, pp. 102] Tate G(F q ) Langlands [DL] Langlands 10

11 5. Rapoport-Zink Rapoport-Zink [RZ] [Ra], [Far1], [SW] (1) Rapoport-Zink Rapoport-Zink M (2) Rapoport-Zink Q p G J E/Q p (3) Rapoport-Zink M Hc( i ) M, Q l G(Qp ) J(Q p ) W E Langlands-Vogan Jacquet-Langlands Kottwitz G G Rapoport-Zink G Rapoport-Zink Deligne-Lusztig [RZ] Rapoport-Zink p p Rapoport-Zink p p p [DOR] p Langlands Rapoport-Zink Rapoport-Zink 9 ( F, B,, OB, V,,, b, µ, L ) F p B F : B B (xy) = y x ( x, y B), 2 = id Q p - O B B O F - V B, : V V Q p b B, v, w V bv, w = v, b w b G ( Qur ) p basic isocrystal Q ur p Q p p µ: G m G Q p Hodge isocrystal Hodge [RZ] F p O B p X 0 L V O B - lattice chain Rapoport-Zink parahoric [RZ] Rapoport-Zink EL PEL 9 PEL EL, 7 ( E, G, J, χb, r µ, X 0, λ ) 11

12 E µ Q p E G Q p PEL G(Q p ) = (x, a) GL B (V ) Q p v, w V, gv, gw = a v, w EL G(Q p ) = GL B (V ) Rapoport-Zink G J G b basic Rapoport-Zink J G Levi χ b : Z(Ĝ)W F C r µ : Ĝ GL(V µ) G C Ĝ r µ µ: G m G Hodge V µ C- X 0 F p p B p X 0, X 0 f Hom ( X 0, X 0) Zp Q p (quasi-isogeny) g f = id g Hom ( X 0, X 0) Zp Q p Q p - B ( End X ) Zp Q p X B PEL λ 0 : X 0 X0 X 0 : X 0 p X0 λ 0 : X 0 X0 λ 0 : ( X0 ) = X0 X0 λ 0 = λ 0 λ 0 p X 0 (quasi-polarization) Rapoport-Zink M Langlands p 2 ( 1) EL ( 2) PEL ( 3) PEL F Q p Q p Q p B Q p Q p Q p 2 O B Z p Z p B V Q r p Q 2r p B r+s, ( ) V V L p i Z r p i Z p i Q 2r p i Z p i O r+s B i Z E Q p Q p B (r s ) Q p (r = s ) G GL r GSp 2r GU r,s Rapoport-Zink b, µ, L Rapoport-Zink ( 1) Lubin-Tate [HT] GL n Langlands Rapoport- Zink ( 2) Siegel 12

13 [KO], [LO], [Hara] ( 3) [Vol], [VW], [Zh] p X 0 X 0 ( 1): X 0 F p r 1 p r = 1 X 0 = Gm [p ] r = 2 X 0 = Ess [p ] E ss F p ( 2): X 0 = (E ss [p ]) r λ 0 : X 0 X 0 E ss ( 3): X 0 = (E ss [p ]) (r+s) λ 0 : X 0 X0 ( 2) B ι: B ( ) End X0 Zp Q p x O B ι(x) Lie LieX 0 diag ( ϕ(x),..., ϕ(x), ϕ(x) p,..., ϕ(x) p ) r s ϕ: O B F p 2 mod p diag ϕ(x) r ϕ(x) p s G J B p X 0 X 0 X 0 B λ 0 Q p J(Q p ) := q-isog OB, Q p λ 0 ( X0 ) J(Q p ) p isocrystal J(Q p ) G Q p - Rapoport-Zink M p Spf O E ur M O E ur E ur M p M 0 0 Rapoport-Zink p p n n Rapoport-Zink M n M n M 0 p Rapoport-Zink M := lim M n n M n n 0 Rapoport-Zink M Êur p p p Rapoport-Zink p Berkovich adic (pre-)perfectoid (pre-)perfectoid [SW] M C p - M (C p ) C p E p M (C p ) p M M (C p ) 5 ( ) M (C p ) = X, ι, λ, ρ, η X, ι, λ, ρ, η X O Cp p O Cp C p ι: B ( End X ) Zp Q p Q p - Lie Kottwitz Hodge µ 13 /

14 PEL λ: X X ( ) ( ) ρ: X 0 OCp Fp /po Cp X OC OCp /po p Cp B λ0 λ Q p η : V = Vp X X : X Tate V p X := ( lim X[p s ](O s Cp ) ) Zp Q p EL B PEL λ V p X V p X V p X V p ( Gm [p ] ) =: Q p (1) 1 Q p Q p (1) = Q p V p X V p X Q p η V, V p X λ V p X Q p η η Q p (1) = Q p 5 (X, ι, λ, ρ, η), (X, ι, λ, ρ, η ) f : X X ι = f ι, λ = f λ f 1, ρ = ( f (mod p) ) ρ, η = f η G(Q p ) J(Q p ) W E Rapooprt-Zink l l p Hc( i ) M, Q l := lim H i ( ) n c Mn ÊurC p, Q l G(Q p ) J(Q p ) M C p - G(Q p ) V J(Q p ) X 0 (g, j) G(Q p ) J(Q p ) (g, j) ( X, ι, λ, ρ, η ) = ( X, ι, λ, ρ j 1, η g 1 ) W E I E W E I E = ΓÊur := Gal ( Êur /Êur) ΓÊur C p I E H i c( M ÊurC p, Q l ) Galois Γ E Weil descent Weil W E [RZ, p.100, 3.48], [Far1, p.71, 4.4] 6. Langlands Kottwitz Rapoport-Zink H i c( M, Q l ) G(Qp ) J(Q p ) W E Langlands Kottwitz l p Q l = C 5 Rapoport-Zink (F, B,, O B, V,,, b, µ, L) (E, G, J, χ b, r µ, X 0, λ) Rapoport-Zink M G G Z(G) G J Langlands 2.1 Langlands-Vogan Kottwitz 3.1 ϕ: W F SL 2 (C) L G G L 14

15 ϕ 3.1 SL 2 (C) ϕ SL2 (C) Ĝ L L G = Ĝ W F E W E Ĝ W E = Ĝ W E r µ : Ĝ GL(V µ) W E W E ϕ WE L G pr 1 Ĝ r µ GL(V µ ) Im(pr 1 ϕ WE ) CentĜ(Im ϕ) W E CentĜ(Im ϕ) V µ : W E CentĜ(Im ϕ) GL(V µ ) CentĜ(Im ϕ) ρ ( ) W (ρ) := Hom CentĜ(Im ϕ) ρ, Vµ V µ W E CentĜ(Im ϕ) W (ρ) W E τ Π J ϕ L ϕ J L ρ τ 3.1 τ CentĜ(Im ϕ) ρ τ Z( Ĝ) W F = χ b χ b Rapoport-Zink χ b G J 6.1 (Kottwitz ([Ra], Conjecture 5.1)). G(Q p ) W E ( ) Hom J(Qp) Hc i ( ) π W (ρ τ ρ π ) i = dim M M, Q l, τ = π G(Q p )-smooth 0 i dim M G(Q p )-smooth G(Q p ) π G L π Π G ϕ 3.1 π Cent (Im ϕ) Ĝ ρ τ Z( = 1 Ĝ) W G F 6.1 J(Q p ) τ Π J ϕ M G(Q p ) π Π G ϕ W E L ϕ r µ M Langlands 4.2 Deligne-Lusztig G Drinfeld Rapoport-Zink [Ra, Conjecture 5.1] i Deligne- Lusztig Lubin-Tate Boyer [Boy] GSp 4 GU 1,2 [ItM1], [ItM2], [It] L π Hdim M c

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