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Lie L ( Introduction L Rankin-Selberg, Hecke L (,,, Rankin, Selberg L (GL( GL( L, L. Rankin-Selberg, Fourier, (=Fourier (= Basic identity.,,.,, L.,,,,., ( Lie G (=G, G.., 5, Sp(, R,. L., GL(n, R Whittaker Bump, Stade,., Whittaker,. GL(n Whittaker. Whittaker A Q, π = vπ v GL(n, A, ϕ π. Q\A ψ, GL(n N =

N n = {(x ij x ii =,x ij =0(i>j}, ψ N(Q\N(A ψ(x =ψ(x + x 3 + + x n,, x =(x ij N(A. Whittaker W ϕ,ψ W ϕ,ψ (g = ψ(ngψ (ndn, g GL(n, A N(Q\N(A. W ϕ,ψ (ng =ψ(nw ϕ,ψ (g (n N(A. Shalika Fourier (( γ ϕ(g = W ϕ,ψ g γ N (Q\GL(,Q. W(π, ψ ={W ϕ,ψ ϕ π}, GL(n, A. W(π, ψ π Whittaker. GL(n, Q v π v Whittaker. Q v ψ v N(Q v, W(ψ v ={W : GL(n, Q v C smooth W (ng =ψ v (nw (g}, GL(n, Q v. π v W(ψ v W(π v,ψ v, π v Whittaker., ξ v π v, Ψ W(π v,ψ v, Ψ(ξ v =:W ξv ξ v Whittaker.., v =. G = SL(n, R. N = N n (R, A = {a = diag(a,...,a n a i > 0, a i =}, K = SO(n G = NAK. M = Z K (A = {diag(m,...,m n A m i {±}}, I {,,...,n}, M σ I σ I (m = i I m i (m = diag(m,...,m n M. σ I = σ I c (I c {,...,n} I h := (I [n/]. ν Lie(A C, ν i = ν(e ii E n /n, ν (ν,...,ν n C n ( i ν i =0. π I,ν = C -Ind G MAN(σ I exp(ν + ρ N. ρ., I =, π I,ν. Whittaker n!,., π I,ν K-type, n! Whittaker ( Whittaker, Whittaker ( Whittaker. R n, G R n K st, π I,ν K-type h st. R n i v i, nc h = {A = {a,...,a h } a <a < <a h n}, {ξ A A n C h } h st. Whittaker {W ξa A n C h }., G = NAK, Whittaker A (A-. A : diag(a,...,a n A, y i = a i /a i+, y =(y,...,y R+.

.3 Whittaker π,ν ξ 0 Whittaker W ξ0 W ξ0 (ngk = ψ (nw (g (n N,g G, k K. K. Whittaker Jacquet., a(wng ν+ρ ψ (ndn. N g = n(ga(gk(g (n(g N,a(g A, k(g K. w Weyl S n., Jacquet,. G = SL(, R. W ξ0 A (y = yk ν (πy, K-Bessel K ν (πy. (a y ν (x + y ν+/ exp(π x dx, (b (c 0 πi R exp{ πy(t + t }t ν dt t, i i ( s + ν Γ Γ ( s ν (πy s ds., (a Jacquet, (b K-Bessel, (c Mellin- Barnes. Mellin-Barnes., (b (c exp( axx s dx = a s Γ(s 0. (b, (c, n =3 Vinogradov-Takhtadzhyan [], Bump [], Stade [9], [0] SL(n, R SL(n, R. Jacquet,., Stade, Stade [4] SL(n, R SL(n, R. Theorem.. Wν n (y =y ρ W n ν (y W ν (y =K ν (πy, ( W ν n (y = W t (ν,...,ν y,...,y t R + exp { (πy i t i } t i t t n (πy i (n iν t nν ( i, W n ν (y SL(n, R Whittaker., ν =(ν + ν /(n,...,ν n + ν /(n., W n ν (y Mellin Vν n (s Vν n (s,...,s = R + 3 W ν n (y (πy i s i dy i y i dt i t i.

, V n ν (s = n (πi n ( si z i Γ z,...,z n V ν (z,...,z n + (n iν ( si z i Γ iν (n (n n dz i. (., z i σ i i σ i + i, σ i R., Whittaker W ξ0 gl(n, C, W ξ0 (ngk =ψ (nw ξ0 (g, W ξ0 (y. Hashizume [] Harish-Chandra K, y =0 (= Whittaker., SL(n, R. m =(m,,m N C n m(ν C C n 0 (ν =, n m i m i m i+ + { ν i ν } i+ m i Cm n (ν = Cm e n i (ν (. (, e i R i., ν C n., Mν n (y =y ν+ρ Cm(ν n (πy i m i m N, {M n wν(y w S n }., Weyl S n ν C n. Whittaker Whittaker Wν n (y = [ ( νi + ν j w Γ w S n i<j n ] Mν n (y (.3. ( (., Bump, Stade,, [4] Theorem. compatible. Theorem.. C n m(ν = {k,...,k n } C (k,...,k n (ν,...,ν {(m i k i! (ν i ν n + mi k i }., (a n =Γ(a + n/γ(a (Pochhammer, {k,...,k n } 0 k i m i. C k (ν =/k!(ν + k. 4

, (. n. k i, n =3, n =4 4 F 3 (. Theorem., Stade Jacquet SL(n, R SL(n, R, Whittaker : Theorem. Wν, t,., Theorem., Mwν n (y., W ν n..4 Whittaker Whittaker,. Whittaker {W ξa A n C h }. Casimir, Whittaker K-type (Dirac-Schmid., Theorem. Whittaker, Whittaker,. n =3,n =4, [5]. Theorem.3. I = {i,...,i h } n C h, I 0 = {i,...,i h } C h. {W n,i A,ν (y =yρ W n,i A,ν (y A = {a,...,a h } n C h }, h =0, W n,,ν (y =W ν n (y (= Whittaker, ( W n,i A,ν (y = W,I 0 t t B<A R B, ν y,...,y t + t n exp { (πy p t p } t p p= (πy p n p p= ν i +α(p n ( t ν i + (α(p β(p p (n, h., A = {a,...,a h }, B<A B = {b,...,b h } C h ( a b <a <a b <a n ( n., α, β, h t= [b t,a t ], h t= [a t,b t ]., W ξa = P i I a iw n,i A,ν. p= dt p t p..5 GL(n GL(m (n m L. π, π GL(n, A, GL(m, A, ϕ i π i (i =, 5

. n>m Z(s, ϕ,ϕ = GL(m,Q\GL(m,A (Pϕ ( g 0 ϕ (g det(g s / dg. 0 n m, P GL(m + Pϕ (h = det(h (m n+/ X n,m (Q\X n,m (A ϕ (x ( h n m ψ (x dx., X n,m (m+,,..., GL(n. Z(s, ϕ,ϕ = Z( s, ϕ,ϕ. ϕ i (g =ϕ i ( t g = ϕ(g ι πi (, ( g 0 Z(s, ϕ,ϕ = ( Pϕ ϕ (g det(g s / dg. 0 n m GL(m,Q\GL(m,A P = ι P ι. Shalika Fourier unfold, Basic idenity: ( g 0 Z(s, ϕ,ϕ = W ϕ W ϕ (g det(g s (n m/ dg 0 n m N m (A\GL(m,A. ϕ i = v ξ i,v (decomposable, Whittaker, Whittaker Whittaker, Euler Z(s, ϕ,ϕ = v Z v (s, W ξ,v,w ξ,v, Z(s, ϕ,ϕ = v Z v (s, W ξ,v,w ξ,v., Z v (s, W,W = Z v (s, W,W = N m (Q v \GL m (Q v M n m,m (Q v W ( g 0 W W (g det(g s (n m/ dg, 0 n m N m (Q v \GL m (Q v g x n m W (g det g s (n m/ dxdg., L L v (s, π,v,π,v ε ε(s, π,v,π,v,ψ v, Z v ( s, W,W L v ( s, π,v,π,v = ε(s, π,v,π,v,ψ v Zv(s, W,W L v (s, π,v,π,v 6

,, L(s, π,π =ε(s, π,π L( s, π,π., L(s, π,π := v L v(s, π,v,π,v, ε(s, π,π := v ε(s, π,v,π,v,ψ v., n = m, ϕ, ϕ GL(n Eisenstein GL(n, Q\GL(n, A, Eisenstein., A π,v GL(n, C, A π,v GL(m, C π i,v Satake, Shintani [8], L L v (s, π,v,π,v := det( A π,v A π,v qv s., v = R, π i,. π, = π,ν=(ν,...,ν n, π, = π,µ=(µ,...,µ m. ξ 0,i π i, Whittaker (= Whittaker W i, Iν,µ n,m (s := Z (s, W,W m n>m W ν n (y,...,y m,,..., W µ m (y,...,y m yi is dy i, y i n = m R m + m Γ R (ms W ν n (y,...,y m W µ m (y,...,y m y is dy i i. R m y + i Γ R (s =π s/ Γ(s/. Stade [0], []. Theorem.4. m = n, n, I n,m ν,µ (s =L (s, π,,π, := i n, j m Γ R (s + ν i + µ j m = n, n, Z (s, W,W, ε =. Stade GL(n, R GL(n, R Whittaker Mellin-Barnes, (.,,. Mellin-Barnes Key Lemma Barnes (=Iν,µ(s, : πi i i Γ(a + sγ(b + sγ(c sγ(d s ds = Γ(a + cγ(a + dγ(b + cγ(b + d Γ(a + b + c + d. (. Barnes. Lemma.5. z 0 =0, z,...,z,λ C, ( ( js sj z j js sj z j + λ Γ Γ V n (πi ν (s,...,s ds j s,...,s j= n = j= Γ( s+ν j+λ V Γ( ns z +λ ν n (s z, s z,...,(n s z. 7 j=

, I n,n ν,µ (s = Γ( ns n ( s + µ + ν j Γ j= ( (n s Iν,µ n, ν (s =Γ ( I n, ν, µ s ( s + ν + µ j Γ j= µ (n ( I, ν,µ s ν (n. µ =(µ,...,µ m, µ =(µ + µ /(m,...,µ m + µ /(m., GL(n GL(m (m = n, n, GL( GL(. Remark. n m>, L. m = n,, Iν,µ n,n (s = n Γ R (s + ν i + µ j Γ(w + ν i i n j n i πi i n j= Γ(s + w µ j π(n s/ w dw. Z (s, W,W unipotet (x,, m = n,,. Remark.,. K-type, Whittaker. Whittaker,,.. SO(n +, R =SO(n +,n,r SL(n, R Jacquet, Whittaker,. ν =(ν,...,ν n C n, y =(y,...,y n R n + Whittaker M n ν (y M n ν (y =y ν+ρ m=(m,...,m n N n C n m(ν n (πy i m i 8

, C n m (ν C 0(ν = m i + m n m i m i+ + (ν i ν i+ m i + ν n m n }Cm(ν n { = Cm e n i (ν+ Cn m e n (ν,. Theorem.. ν =(ν,...,ν C, n C n m(ν = {l,...,l } {k,...,k } C (k,...,k ( ν (m i l i! (m n k! (l i k i! n (ν i + ν n + mi l i (ν i ν n + li k i., {k i,l i } 0 k i l i m i ( i n, 0 k m n, k 0 = l 0 =0. Theorem.. Wν n (y =y ρ W n ν (y W ν (y =K ν (πy, n W ν n (y = exp { (πy i t i } t i R n + = c R + R + W ν {( n ( y t u y i ( { exp (πy i u i tn u,...,y,y n t u t u n } νn n dt i du i ti u i t n t i u i n ( (+ui ( K νn πy i +u W ν i t u t i t i+ ( u un du i y,...,y,y n u. u u u i, W n ν (y SO(n +, R Whittaker. u i }. Sp(n, R, SO(n, R Ginzburg, Rallis, Soudry A 0 (SO(n A 0 (Sp(n A 0 (SO(n+ Fourier-Whittaker., Sp(n, R SO(n, R Whittaker. (ν,...,ν n C n Sp(n, R, SO(n, R, (ν,...,ν n+ C n+ SO(n +, R, a = (a,...,a n R n +, b = 9

(b,...,b n+ R n+ +, t = (t,...,t n R n + Sp(n, R. (a =a ρ W SO n ν (a Whit- Theorem.3. W Sp n ν taker, (t =t ρ W Sp n ν W Sp n (ν,...,ν n (t = W SO n+ (ν,...,ν n,0 (b = (t, W SO n ν R n + R n + SO(n, R, SO(n +, R, W SO n (ν,...,ν n (a LC n D n (a, t W Sp n (ν,...,ν n (t LD n+ C n (t, b n n da i a i, (. dt i t i. (., [ {( L C n t ( D n (a, t = exp π + a t ( + + + a t }] + n + t a t a t n a na n, n [ { L D n+ b ( C n (t, b = exp π t ( + + b t ( + + + b n t }] + n + b t b t b n t n b n+ t n. n+ Remark 3. SO(n +, R, Whittaker,., (. n = Niwa [7] (., (., W SO n+ (ν,...,ν n+ (b W Sp n (ν,...,ν n (t., W SO n+ (ν,...,ν n+ (b W SO n (ν,...,ν n (a, Whittaker..3 G (R G (R Whittaker M G ν=(ν,ν (y =yν+ρ, C G (m,m (ν (m,m N C G (m,m (ν(πy m (πy m (m +3m 3m m + ν m + ν m C G m,m (ν =C G m,m (ν+3c G m,m (ν... Theorem.4. ([3] C G (m,m (ν = 0 n +n m 0 n 4 n 3 n m (m n n! (m n! n!(n n 3! (n 3 n 4! n 4! (ν + ν + m n 3 (ν + m n (ν + n n 4. (ν +ν + n (ν +3ν + n3 (ν +3ν + n4 0

, M G (ν,ν (y SL(3, R Whittaker. M G (ν,ν (y =y4 y k,k =0 (π 3 y y (k +k +ν +3ν /3 C SL 3 (k,k (ν + ν,ν, ν ν M SL 3 ((k +k +ν +3ν /3, ( k +k ν /3, (k k ν 3ν /3 (y., Whittaker,. Theorem.5. ([3] W G ν (y =yy 5 W 3 G ν (y, W SL 3 SL ν (y =y y W 3 ν (y Whittaker, { W G (ν,ν (y = exp (πy t (πy t (πy t 3 t } 0 0 0 t t t 3 t ( W SL 3 t dt dt dt 3 (ν +ν,ν, ν ν y y t t 3,y. t 3 t t t 3.4 SO(n + GL(m L SO(n + GL(m L Gelbart, Piatetski-Shapiro. GL(n GL(m, unipotent n = m, m., SO I n (s = W n+ (ν,...,ν n (y,...,y n W SL n (µ,...,µ n (y,...,y (y y yn n s (R n SO J n (s = W n+ (ν,...,ν n (y,...,y n W SL n+ (µ,...,µ n+ (y,...,y n (y y yn n s (R n. Conjecture.6. n n dy i y i, dy i y i. I n (s = J n (s = n n j= Γ R(s + ν i + µ j Γ R (s ν i + µ j i<j n Γ, R(s + µ i + µ j n n+ j= Γ R(s + ν i + µ j Γ R (s ν i + µ j i<j n+ Γ. R(s + µ i + µ j Remark 4. (., Lemma.5 I n (s J (s. I n, n = (Niwa [7], 3, 4, 5, J n, n =,3,4.

[] D. Bump, Automorphic forms on GL(3, R, Lect. Note Math. 083, Springer-Verlag, 984. [] M. Hashizume, Whittaker functions on semisimple Lie groups, Hiroshima Math. J. (98, 59 93. [3] T. Ishii, Whittaker functions on real semisimple Lie groups of rank two, Canad. J. Math. (to appear. [4] T. Ishii and E. Stade, New formulas for Whittaker functions on GL(n, R, J. Funct. Anal. 44 (007, 89 34. [5] T. Ishii and T. Oda, Calculus on principal sereis Whittaker functions on SL(n, R, preprint. [6] H. Manabe, T. Ishii and T. Oda, Principal series Whittaker functions on SL(3, R, Japan. J. Math. 30 (004, 83 6. [7] S. Niwa, Commutation relations of differential operators and Whittaker functions on Sp (R, Proc. Japan Acad. 7 Ser A.(995, 89 9. [8] T. Shintani, On an explicit formula for class- Whittaker functions on GL n over P -adic fields, Proc. Japan Acad. 5 (976, no. 4, 80 8. [9] E. Stade, On explicit integral formulas for GL(n, R-Whittaker functions, Duke Math. J. 60 (990, no., 33 36 [0] E. Stade, Mellin transforms of GL(n, R Whittaker functions, Amer. J. Math. 3 (00, 6. [] E. Stade, Archimedean L-factors on GL(n GL(n and generalized Barnes integrals, Israel J. Math. 7 (00, 0 0. [] I. Vinogradov and L. Tahtajan, Theory of the Eisenstein series for the group SL(3, R and its application to a binary problem, J. of Soviet Math. 8 (98, 93 34. Faculty of Science and Technology, Seikei University, 3-3- Kichijoji-Kitamachi, Musashino, Tokyo, 80-8633, Japan E-mail address: ishii@st.seikei.ac.jp

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