GL 2 1 Lie SL(2, R) GL(2, A) Gelbart [Ge] 1 3 [Ge] Jacquet-Langlands [JL] Bump [Bu] Borel([Bo]) ([Ko]) ([Mo]) [Mo] 2 2.1 H = {z C Im(z) > 0} Γ SL(2, Z) Γ N N Γ (N) = {γ SL(2, Z) γ = 1 2 mod N} g SL(2, R) H f, ( ) f k g(z) = j(g, z) k a b f(g(z)), g =, j(g, z) = cz + d, g(z) = az + b c d cz + d j(g 1 g 2, z) = j(g 1, g 2 (z))j(g 2, z) (f, g) f k g G H 1
2.1 H f 3, SL(2, Z) Γ k (1) f H (2) γ Γ f k γ = f (3) f Γ \H cusp γ SL(2, Z) f k γ Fourier f k γ = a γ (n)e 2πinz/N n=0 (3) γ SL(2, Z) a γ (0) = 0 f cusp M k (Γ ), cusp S k (Γ ) 2.1 f M k (Γ ) cusp M y k/2 f(z) < M Γ Fourier Hecke f S k (Γ ) a(n) = O(n k/2 ) Eisenstein theta Fourier Γ Γ 0 (N) = {( a c ) } b SL(2, Z) d c 0 mod N ψ N Dirichlet ( ) ψ Γ 0 (N) ψ (( a c )) b = ψ(d) d N M k (Γ 0 (N), ψ) = {f M k (Γ (N)) f k γ = ψ(γ)f, γ Γ 0 (N)} Fourier Dirichlet L 2
2.2 (L 1) f S k (Γ 0 (N), ψ) s C Fourier f = n=1 a(n)e2πinz L(f, s) = a(n)n s n=1 f L Re(s) L Mellin 0 f(iy)y s 1 dy = = 0 a(n)e 2πny y s 1 dy n=1 a(n)(2πn) 1 s e 2πny (2πny) s 1 dy (2.1) n=1 = (2π) s Γ (s)l(s, f) 2.2 (1) L(s, f) s (2) Λ(s, f) = N s/2 (2π) s Γ (s)l(s, f) Λ(s, f) = i k Λ(k s, f) f(z) = ( Nz) k f( (Nz) 1 ) (3) Λ(s, f) C > 0 V (C) = {s C Re(s) < C} 0 2.2 Hecke Fourier Hecke Hecke ( ) T p 1 0 Γ 0 (N) Γ 0 (N) Γ 0 (N) Fourier 0 p 2.3 f(z) = n=1 a(n)e2πinz S k (Γ 0 (N), ψ), T p f(z) i) (p, N) = 1 T p f(z) = n=1 ( ( )) n a(np) + ψ(p)p k 1 a e 2πinz, p ( p n a ( ) n = 0) p 3
ii) p N T p f(z) = a(np)e 2πinz. n=1 T p f S k (Γ 0 (N), ψ) Atkin-Lehner [AL] ii) U p - T p - (p, N) = 1 p T p Petersson S k (Γ 0 (N), ψ) T p (p N) Atkin-Lehner new form Sk new (Γ 0 (N), ψ) p N p T p 2.4 (L 2) f T p T p λ p L(s, f) = a(1) p (1 λ p p s + ψ(p)p k 1 2s ) 1 a(1) f Fourier e 2πiz Re(s) 0 2.3 f S k (Γ 0 (N), ψ) T p - 2.2 2.4 2.2 L 2.4 Euler f L 2.2 f 2.4 Euler 2.4 L SL 2 Dirichlet Fourier Siegel 2.2 Dirichlet Koecher-Maass 2.2 2.4 2.4 ( 1 ) f, g S k (Γ 0 (N), ψ) T p f g 3 SL 2 (R) 3.1 Γ = Γ 0 (N) f M k (Γ ) G = SL 2 (R) K = SO(2) G f H G/K H, g g(i) f G 4
3.1 f S k (Γ ) G φ f φ f (g) = j(g, i) k f(g(i)) φ f C (G) 3.1 φ f (1) φ f Γ - γ Γ φ f (γg) = φ f (g) ( ) cos θ sin θ (2) K r θ = φ f (gr θ ) = e ikθ φ f (g) sin θ cos θ (3) φ f C M φ f (g) < C g M ( g = Tr( t gg) ) ( ) (4) sl(2, C) X = 1 1 i X φ f = 0 (X φ f 2 i 1 ) f cusp (5) δ SL(2, Z) Γ δ = δ 1 Γ δ δ SL(2, Z) g G φ f (δng) dn = 0. N Γ δ \N {( ) } 1 a N = G a R (4) 2.1 (1) f X Cauchy-Riemann (1) f, (2) K i, (3) 2.1 (3) f cusp ) (1), (2) (3) g G z = x + yi H ( x 1/2, z > 1) ( ) y 1/2 xy 1/2 g = δg z r θ, δ SL(2, Z), g z = 0 y 1/2 φ f (g) = y k/2 e ikθ f k δ(z) y < C g M f k δ Fourier n 0 (4) (5) N Γ δ = {( 1 hm ) m Z } 5
h z = g(i) H ( ) ) 1 1 hn φ f (δng) dn = φ f (δ g dn N Γ δ \N 0 ( ( ) ) ( ( ) k 1 1 hn 1 hn = f δ g(i) j δ g, i) dn = 0 1 0 f(δ(z + hn))j(δ, z + hn) k j(g, i) k dn 1 = j(g, i) k (f k δ)(z + hn) dn 0 f k δ f cusp (1) φ f C (Γ \G) G g G ψ C (Γ \G) gψ(h) = ψ(hg) gψ C (Γ \G) (2) φ f K Cφ f 1 K (4) G 3.2 G Lie G Hilbert G G- (g, K)- Lie G K K- (Peter-Weyl ) G- (π, H) K- H = ˆ m(δ)h δ, δ K m(δ) < K- (admissible) K = SO(2) 1 m Z K r θ e imθ SO(2) Z 6
G Lie g 0 G g 0 G Hilbert (π, H) g 0 = Lie(G) H v H X g 0 d dt π(exp(tx))v π(exp(tx))v v = lim (3.1) t=0 t 0 t v H H g 0 H Lie g := g 0 R C G (π, H) G- g- K- (K G ) H K- H K = {v H dim(π(k)v) < } H K H = ˆ δ m(δ)h δ H K = δ m(δ)h δ H K H H K H dense g K H K (g, K)- H G- H K (g, K)- (g, K)- [Ma] f M k (Γ 0 (N), ψ) X φ f = 0 X ( ) g (3.1) z = x + yi H y 1/2 xy 1/2 g z = 0 y 1/2 g z (i) = z g G g = g z r θ φ f K r θ X r 1 θ X = 1 2 ( 1 0 0 1 ) + i 2 ( 0 1 0 0 = e 2iθ X g = g z ) ( ) + i 0 0 2 1 0 A, B, C A φ f (g) = 1 (( ) ( )) d y 1/2 xy 1/2 e t 0 2 dt φ f t=0 0 y 1/2 0 e t = 1 d ( 2 dt e kt y k/2 f(x + iye 2t ) ) t=0 = k 2 yk/2 f(z) + y k/2+1 y f(z) 7
B φ f (g) = i (( ) ( )) d y 1/2 xy 1/2 1 t 2 dt φ f t=0 0 y 1/2 = i d 2 dt y k/2 f(x ty + iy) t=0 = iyk/2+1 2 x f(z) C φ f (g) = i (( ) ( )) d y 1/2 xy 1/2 1 0 2 dt φ f t=0 0 y 1/2 t 1 = i ( d 2 dt ( ti + 1) k y k/2 f x ty ) t=0 t 2 + 1 + i y t 2 + 1 = k 2 yk/2 f(z) iyk/2+1 2 x f(z) f ( X φ f (g) = iy k/2+1 x + i ) f(z) = 0 y 3.3 f S k (Γ 0 (N)) 2.1 φ f L 2 (Γ \G) G Hilbert L 2 (Γ \G) K- L 2 (Γ \G) K (g, K)-φ f (g, K)-? (g, K)- ( ) H = i, X + = 1 1 0 2 ( ) ( 1 i, X = 1 i 1 2 1 i i 1 ) g = sl(2, C) SL 2 -triple [X +, X ] = H [H, X + ] = 2X + [H, X ] = 2X exp(iθh) = r θ K (g, K)- V v m r θ v = e imθ v Hv = mv X + v, X v m + 2, m 2 k 1 (g, K)- : 8
V = Cv k+2m v n r θ v n = e inθ v n Hv n = nv n m=0 X + v n Cv n+2 X v n Cv n 2 X v k = 0 D + k k 2 (holomorphic discrete series) k = 1 (limit of discrete series) L 2 (G) K (L 2 (G) K- ) v k lowest weight vector D + k [Kn, Chapter II, 5-6] X + X V = (anti-holomorphic discrete series) D k D + k m=1 Cv k 2m sl(2, C) lowest weight k k 0 f S k (Γ ) φ f L 2 (Γ \G) 3.1 φ f D + k vector lowest weight 3.2 S k (Γ ) Hom (g,k) (D k, L 2 (Γ \G)) f S k (Γ ) D k lowest weight vector φ f (g, K)- L 2 (Γ \G) G 3.1 X φ f = 0 U(g) g U(g) = 1 4 (H2 + 2X + X + 2X X + ) ( U(g) ) Casimir U(g) Z(g) Z(g) = C[ ] f M k (Γ ) ( k ( )) k 2 2 1 φ f = 0 9
Z(g) = C[ ] Schur X G 3.2 G f (1) f Γ - γ Γ, g G f(γg) = f(g). (2) f K- g G dim f(gk) k K C < (3) f Z(g)- 0 p(x) p( )f = 0. (4) f C, M f(g) < C g M. A(Γ, K) f cuspidal f(ng) = 0 a. a. g, N f cusp cusp A 0 (Γ, K) (3) f U(g) (2) (3) f smooth (cf. [Bo, Theorem 2.13]). ψ L 2 (Γ \G) 3.1 (5) N ψ(ng) dn = 0 φ cuspidal cuspidal L 2 cusp(γ \G) G L 2 (Γ \G) K L 2 disc (Γ \G) K L 2 disc (Γ \G) K = V L 2 (Γ \G) K V : L 2 (Γ \G) L 2 disc (Γ \G) K L 2 (Γ \G) K V 10
3.3 (1) L 2 0 (Γ \G) (g, K)- L 2 0(Γ \G) K = m(π)v π, m(π) < (2) (g, K)- π: L 2 disc (Γ \G) K = L 2 0(Γ \G) K H res H res Eisenstein residue Γ = SL(2, Z) C [Bo, Theorem 16.6] 4 4.1 SL(2, R) Hecke L- ( ) 1 0 Hecke 0 p SL(2) GL(2) G = GL(2), A = A Q Q A f = p< Q p, G Q = GL(2, Q) G = GL(2, R), G + = GL + (2, R) := {γ GL(2, R) det g > 0} G Af G A G - 1 K 0 G Af det : K f A ( ) G A = G Q G + K 0 (4.1) 4.1 Γ = K 0 SL(2, Q) Γ \H G Q R + \G A/K 0 Γ \SL(2, R) G A G Q 11
Example 4.2 N {( ) } a b K 0 (N) = GL(2, c d Ẑ) c 0 mod N = GL 2 (Z p ) K p p N p N ( K p = {( a c ) } b GL(2, Z p ) d c 0 mod p ) Γ = K 0 (N) SL(2, Q) = Γ 0 (N) ψ modn Dirichlet f M k (Γ 0 (N), ψ) ψ A ω ψ ω ψ : A A /Q R + = Z p p N Z p (Z/NZ) ψ C 1 ω ψ K 0 (N) ( ) a b ω ψ : ω ψ (d) C 1 c d 4.3 f M k (Γ 0 (N), ψ) G A Φ f (4.1) G A g = γg k 0 Φ f (g) = f(g (i))j(g, i) k ω ψ (k 0 ) 1 well-defined (i) γ G Q Φ(γg) = Φ(g) ( G Q - ) (ii) k 0 K 0 (N) Φ f (gk 0 ) = Φ(g)ω ψ (k 0 ) 1 (iii) Φ f (gr θ ) = e ikθ Φ f (g) (iv) Φ f G + smooth Φ f = k ( ) k 2 2 1 Φ f 12
(v) z Z A A Φ f (zg) = ω ψ (z) 1 Φ(g) (vi) Φ f c > 0 G A Ω C N g Ω a > c a A (( ) ) a 0 Φ f g C a N (vii) f S k (Γ 0 (N), ψ) (( ) ) 1 x Φ f g dx = 0 Q/A a.a. g (ii), (iii) K = K K 0 (1)- (vi) Z(g)- ω Hecke ω : A /Q R + C1 L2 (G Q \G A, ω) G A 1. γ G Q Φ(γg) = Φ(g) 2. z Z A Φ(zg) = ω(z)φ(g) 3. Φ(g) 2 dg < Z A G Q \G A f S k (Γ 0 (N), ψ) Φ f L 2 (G Q \G A, ω 1 ψ ) 4.2 G A Hecke L 2 (G Q \G A, ω) G A G A G = GL(2, R) G p = GL(2, Q p ) G 4.3 (iii), (iv) 2 G p G p (g, K)- G p (π, V ) K p = GL(2, Z p ) V = σ K p V (σ), dim V (σ) < 13
V ( ) K- f ( f G p Hecke ) π(f)v = f(g)π(g)v dg G p (π, V ) L 2 (G Q \G A, ω) π(f)φ f(g) = f(g 1 ) Φ G p Hecke ( 4.4 ) Φ L 2 (G Q \G A ) ( ω ) T (p)φ Φ H p = p 0 K p K p T (p)φ = Φ(gh) dh (4.2) H p f S k (Γ 0 (N)) p k/2 1 T (p)φf = Φ Tpf ) f S k (Γ 0 (N)) (4.2) g G A (4.1) g = γg k (γ G Q, g G, k K 0 (N)) Φ f G Q γ = 1 2 h k p- 1 2 p N (p N ) H p K p ( ) ( ) p 1 p b 1 0 H p = K p K p 0 p b=0 ω = ω 1 ψ (4.2) ( )) ( ( )) p 1 p b 1 0 T (p)φ f (g) = Φ f (g + Φ f g 0 p b=0 K p Haar vol(k p ) = 1 ( ) Φ f f p b G Q G K 0 (N) p- 1 2 ( ) G A g G A p- 1 2 p b γ = G Q ( ) ( ) ( ) p b p b p 1 bp 1 g = g k p = γ g k 0k 0 14
( ) 1 k 0 p- 1 p b 2 K 0 (N) ( )) ( ) p b z + b Φ f (g = p k/2 1 f p ( g (i) = z ) ( )) 1 0 Φ f (g = p k/2 f(pz) 0 p f S k (Γ 0 (N), ψ) L 2 (G Q \G A, ω 1 ψ ) G A R H p h (( )) a b h = ω ψ (d) c d R(h)Φ f Hecke 1 2.4 L 2 0 (G Q\G A, ω) L 2 (G Q \G A, ω) cuspidal G A R 0 ω 4.5 ( 1 ) G A (R 0 ω, L 2 0 (G Q\G A, ω)) 1 Φ Fourier Whittaker Whittaker [Ko] Whittaker [AL] A.O.L. Atkin and J. Lehner Hecke operators on Γ 0 (m), Math. Ann. 185 (1970), p134-160. [Bo] A. Borel, Automorphic forms on SL 2 (R), Cambridge Tracts in Mathematics, 130. Cambridge University Press, Cambridge, (1997). [Bu] D. Bump, Automorphic forms and representations, Cambridge University Press, (1997). 15
[Ge] [JL] S.S.Gelbart. Automorphic forms on adele groups, Annals of Math.Studies. Princeton University Press and University of Tokyo Press, Princeton, N.J., 83, 1975. H. Jacquet, and R.P. Langlands, Automorphic forms on GL(2)., Lecture notes in Mathematics 114, (1970) Springer Verlag. [Kn] A.W. Knapp, Representation theory of semisimple groups, An overview based on examples. Princeton Mathematical Series, 36. Princeton University Press, Princeton, NJ, 1986. [Ko], GL2 L, 16 L (2009), p37-136. [Ma], Weil Howe duality [Mo], Hecke, 18 (2010), p1-20. 16