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The local zeta integrals for GL(2, C)x GL(2, C).

1. Introduction. Let F = R or C. Let [PI] and [PI]' be irreducible admissible infinite-dimensional representations of GL(2,F). We consider here the local zeta integral Z(s, W, W', f) for GL(2,F)x GL(2,F), which is defined from Whittaker functions W for ([PI], [psi]), W' for ([PI]',[[psi].sub.-1]) and a standard Schwartz function f on [F.sup.2] with the standard character [psi] of F. In Theorems 17.2 (3) (for F = R) and 18.1 (3) (for F = C) of the lecture note [Ja], Jacquet asserts only that the associated local L-factor L(s, [PI] x [PI]') can be expressed as a finite sum of the local zeta integrals for GL(2, F)x GL(2,F), that is,

[m.summation over (i=1)] (s, [W.sub.i], [W'.sub.i], [f.sub.i]) = L(s, [PI] x [PI]')

with some [W.sub.i], [W'.sub.i] and [f.sub.i]. However, in the proof of Theorem 17.2 of [Ja], he shows a stronger result for F = R. He gives Whittaker functions [W.sub.0], [W'.sub.0] and a standard Schwartz function [f.sub.0] satisfying

(1.1) Z(s, [W.sub.0], [W'.sub.0], [f.sub.0]) = L(s, [PI] x[PI]')

for F = R, explicitly. (See Proposition 2.5.2 in [Zh] for the case omitted in [Ja].) On the other hand, the proof of Theorem 18.1 in [Ja] is written with the modified zeta integrals defined from vector valued functions, and it is not clear whether the stronger assertion (1.1) for F = C is true or not. In this article, we give Whittaker functions [W.sub.0], [W'.sub.0] and a standard Schwartz function [f.sub.0] satisfying (1.1) for F = C, explicitly, rewriting Jacquet's calculation in [Ja] using Schur's orthogonality and explicit formulas of Whittaker functions.

2. Whittaker functions on GL(2, C). Let G = GL(2, C) be the complex general linear group of degree 2, and we fix an Iwasawa decomposition G = NAK with

[mathematical expression not reproducible]

and the unitary group K = U(2) of degree 2. Here [R.sub.+] is the set of positive real numbers. We denote by [g.sub.C] the complexification g [[cross product].sub.R] C of the associated Lie algebra g of G.

For c [member of] [C.sup.x], we define a character [[psi].sub.c] of C by

[[psi].sub.c] (x) = [e.sup.2[pi] [square root of (1)](cx + [bar.cx])] (x [member of] C),

and let [C.sup.[infinity]] (N\G; [[psi].sub.c]) be the space of smooth functions f on G satisfying

[mathematical expression not reproducible],

on which G acts by the right translation. Here we note that there is a G-isomorphism

(2.1) [[XI].sub.c]: [C.sup.[infinity]] (N\G; [[psi].sub.1]) [right arrow] [C.sup.[infinity]] (N\G; [[psi].sub.c]),

defined by [[XI].sub.c] (f)(g) = f (diag (c, 1) g) (g [member of] G).

Let ([PI], [H.sub.[PI]]) be an irreducible admissible infinite-dimensional representation of G. We denote by [H.sub.[PI],K] the subspace of [H.sub.[PI]] consisting of all K-finite vectors. We define the space [mathematical expression not reproducible] of homomorphisms [PHI]: [H.sub.[PI],K] [right arrow] [C.sup.[infinity]] [(N\G; [[psi].sub.c]).sub.K] of ([g.sub.C], K)-modules such that [PHI](f) is of moderate growth for any f [member of] [H.sub.[PI],K]. Theorem 6.3 in [JL] tells that the space [mathematical expression not reproducible] is one dimensional. We define the space W([PI], [[psi].sub.c]) of Whittaker functions for ([PI], [[psi].sub.c]) by

[mathematical expression not reproducible].

3. Irreducible representations of K. Let A be the set {[lambda] = ([[lambda].sub.1], [[lambda].sub.2]) [member of] [Z.sup.2] | [[lambda].sub.1] > [[lambda].sub.2]} of dominant weights. Let [V.sub.[lambda]] be the C-vector space of degree ([[lambda].sub.1] - [[lambda].sub.2]) homogeneous polynomials in [z.sub.1], [z.sub.2], for [lambda] = ([[lambda].sub.1], [[lambda].sub.2]) [member of] [LAMBDA]. The group K acts on [V.sub.[lambda]] by

[mathematical expression not reproducible]

for k [member of] K and p([z.sub.1], [z.sub.2]) [member pf] [V.sub.[lambda]]. Then the representations ([[tau].sub.[lambda]], [V.sub.[lambda]]) ([lambda] [member of] [LAMBDA]) of K are irreducible, and exhaust the equivalence classes of irreducible representations of K.

Let [lambda] = ([[lambda].sub.1], [[lambda].sub.2]) [member of] [LAMBDA]. We define [mathematical expression not reproducible] as a basis of [V.sub.[lambda]] by [mathematical expression not reproducible]. We set

[??] = (-[[lambda].sub.1], -[[lambda].sub.2]).

Then there is a K-invariant C-bilinear pairing <*, *> on [V.sub.[??]] [[cross product].sub.C] [V.sub.[lambda]], which is determined by

[mathematical expression not reproducible]

and [mathematical expression not reproducible]. Here we denote by [mathematical expression not reproducible] the binomial coefficient n!/i(n - i)! for n, i [member of] Z such that n [greater than or equal to] i [greater than or equal to] 0.

4. Principal series representations. For v = ([v.sub.1], [v.sub.2]) [member of] [C.sup.2] and d = ([d.sub.1], [d.sub.2]) [member of] [Z.sup.2], let [H.sup.[infinity].sub.(v,d)] be the space of smooth functions f on G such that, for [t.sub.1], [t.sub.2] [member of] [C.sup.x], x [member of] C and g [member of] G,

[mathematical expression not reproducible].

The group G acts on [H.sup.[infinity].sub.(v,d)] by the right translation

([[PI].sub.(v,d)] (g)f)(h) = f(hg) (g, h [member of] G, f [member of] [H.sup.[infinity].sub.(v,d)]).

Let ([[PI].sub.(v,d)], [H.sub.(v,d)]) be a Hilbert representation of G, which is the completion of ([[PI].sub.(v,d)], [H.sup.[infinity].sub.(v,d)]) relative to the [L.sup.2]-inner product on K with respect to the Haar measure. We call ([[PI].sub.(v,d)], [H.sub.(v,d)) a principal series representation of G.

Theorem 6.2 in [JL] tells that any irreducible admissible infinite-dimensional representation of G is isomorphic to some irreducible principal series representation of G as ([g.sub.C], K)-modules. Moreover, when [[PI].sub.(v,d)] is irreducible, we may assume d [member of] [LAMBDA] without loss of generalities, since

(4.1) [mathematical expression not reproducible]

as ([g.sub.C], K)-modules. Here [H.sub.(v, d), K] is the subspace of [H.sub.(v,d)] consisting of all K-finite vectors. From Lemma 6.1 (ii) in [JL], we obtain the following

Proposition 4.1. As K-modules,

[mathematical expression not reproducible]

holds for v = ([v.sub.1], [v.sub.2]) [member of] [C.sup.2] and d = ([d.sub.1], [d.sub.2]) [member of] [LAMBDA].

5. Explicit formulas of Whittaker functions. For s [member of] C and i [member of] [Z.sub.[greater than or equal to]0], we set

[(s).sub.i] = [GAMMA] (s + 1)/[GAMMA] (s) = s(s + 1) ... (s + i - 1), [[GAMMA].sub.C] (s) = 2[(2[pi]).sup.-s] [GAMMA](s)

as usual, where [GAMMA](s) is the Gamma function and [Z.sub.[greater than or equal to]0] is the set of non-negative integers. For [a.sub.1], [a.sub.2] [member of] C, we define a function K([a.sub.1], [a.sub.2]) on [R.sub.+] by

[mathematical expression not reproducible]

for [y.sub.1] [member of] [R.sub.+]. Here [K.sub.[mu]] (z) is the modified Bessel function of the second kind ([section] 17.71 in [WW], (6.5) in [Bu]), and [alpha] is a real number such that

[alpha] > max{-Re([a.sub.1]), -Re([a.sub.2])}.

Let ([[PI].sub.(v,d)], [H.sub.(v,d)]) be an irreducible principal series representation with v =([v.sub.1], [v.sub.2]) [member of] [C.sup.2], d = ([d.sub.1], [d.sub.2]) [member of] [LAMBDA]. Let c [member of] [C.sup.x]. Let m [member of] [Z.sub.[greater than or equal to]0] and we set [lambda] = ([[lambda].sub.1], [[lambda].sub.2]) = ([d.sub.1] + m , [[lambda].sub.2] - m). By Proposition 4.1, there is a non-zero K-homomorphism [psi]: [V.sub.[lambda]] [right arrow] W([[PI].sub.(v,d)], [[psi].sub.c]), which is unique up to scalar multiple. For g [member of] G with the Iwasawa decomposition

[mathematical expression not reproducible]

and v [member of] [V.sub.[lambda]], we have

[mathematical expression not reproducible].

Hence we note that [phi] is characterized by the functions [phi] ([v.sup.[lambda]].sub.q]) (diag([y.sub.1], 1)) (0 [less than or equal to] q [less than or equal to] [[lambda].sub.1] - [[lambda].sub.2]) on [R.sub.+]. We will give explicit formulas of these functions.

Because of (2.1), it suffices to consider the case of c = 1. We set c = 1 and

[mathematical expression not reproducible].

Translating Eq. (15) and Eq. (16) in [Po] into our notation, for 0 [less than or equal to] q [less than or equal to] [[lambda].sub.1] - [[lambda].sub.2], we have

(5.1) {([[partial derivative].sub.1] - 2[v.sub.1] - m + q)([[partial derivative].sub.1] - 2[v.sub.2] + m - [??]) -[(4[pi][y.sub.1]).sup.2]}[[phi].sub.q] = -8[pi]q[y.sub.1][[phi].sub.q-1],

(5.2) [mathematical expression not reproducible]

with [??] = [[lambda].sub.1] - [[lambda].sub.2] - q and [[partial derivative].sub.1] = [y.sub.1] d/d[y.sub.1]. Here we set [[phi].sub.q] = 0 if q < 0 or q > [[lambda].sub.1] - [[lambda].sub.2].

Since the functions [[phi].sub.q] (0 [less than or equal to] q [less than or equal to] [[lambda].sub.1] - [[lambda].sub.2]) are determined from [[phi].sub.0] by the equation (5.2), we note that [phi] [not equal to] 0 implies [[phi].sub.0] [not equal to] 0. Taking q = 0 in the equation (5.1) and comparing with the Bessel differential equation, Popa shows that [[phi].sub.0] is a nonzero constant multiple of K([v.sub.1] + m/2, [v.sub.2] + [[lambda].sub.1] - [[lambda].sub.2] - m/2) in [Po].

If we assume [[phi].sub.0] = K([v.sub.1] + m/2, [v.sub.2] + [[lambda].sub.1] - [[lambda].sub.2] - m/2, we obtain the formula

[mathematical expression not reproducible]

for 0 [less than or equal to] q [less than or equal to] [[lambda].sub.1] - [[lambda].sub.2], recursively, by (5.2) with

[mathematical expression not reproducible]

for [a.sub.1], [a.sub.2], [b.sub.1], [b.sub.2] [member of] C and [y.sub.1] [member of] [R.sub.+]. From the above arguments, we obtain the following proposition.

Proposition 5.1. Let ([[PI].sub.(v,d)], [H.sub.(v,d)]) be an irreducible principal series representation of G with v = ([v.sub.1], [v.sub.2]) [member of] [C.sup.2], d = ([d.sub.1], [d.sub.2]) [member of] [LAMBDA]. Let c [member of] [C.sup.x] and m [member of] [Z.sub.[greater than or equal to]0]. Set [lambda] =([[lambda].sub.1], [[lambda].sub.2]) = ([d.sub.1] + m, [d.sub.2] - m). There is a K-homomorphism [[phi].sup.(c).sub.[v,d;m]] [V.sub.[lambda]] [right arrow] W([[PI].sub.(v,d)], [[psi].sub.c]) such that, for [y.sub.1] [member of] [R.sub.+] and 0 [greater than or equal to] q [greater than or equal to] [[lambda].sub.1] - [[lambda].sub.2],

[mathematical expression not reproducible]

(5.3) [mathematical expression not reproducible]

(5.4) [mathematical expression not reproducible]

with [??] = [[lambda].sub.1] - [[lambda].sub.2] - q.

Here the second expression (5.4) is derived from the formula (7.4.4.1 in [PBM])

[mathematical expression not reproducible]

(m is a non-negative integer)

of the generalized hypergeometric series (the both sides of this equality are rational functions of [a.sub.1], [a.sub.2], [b.sub.1], [b.sub.2]), and the expression

[mathematical expression not reproducible]

obtained from (5.3).

Remark 5.2. From the explicit formulas of [[phi].sup.(c).sub.[v,d;m]] ([v.sup.[lambda].sub.q])(diag([y.sub.1], 1)) in Proposition 5.1, we have

[mathematical expression not reproducible].

6. The local zeta integrals for G x G.

Let S([C.sup.2]) be the space of Schwartz functions on [C.sup.2]. Let S[([C.sup.2]).sup.std] be the subspace of S([C.sup.2]) consisting of all functions f of the form

(6.1) [mathematical expression not reproducible]

for [z.sub.1], [z.sub.2] [member of] C, with polynomial functions p on [C.sup.4]. We call functions in S[([C.sup.2]).sup.std] standard Schwartz functions on [C.sup.2].

Let ([[PI].sub.(v,d)], [H.sub.(v,d)]) and ([[PI].sub.(v',d')], [H.sub.(v',d')]) be irreducible principal series representations of G with v =([v.sub.1], [v.sub.2]) [member of] [C.sup.2], d = ([d.sub.1], [d.sub.2]) [member of] [LAMBDA], v' = ([v'.sub.1], [v'.sub.2]) [member of] [C.sup.2] and d' = ([d'.sub.1], [d'.sub.2]) [member of] [LAMBDA]. From the Langlands parameters of [[PI].sub.(v,d)] and [[PI].sub.(v', d')], we define the local L-factor

[mathematical expression not reproducible].

For W [member of] W([[PI].sub.(v,d)], [[psi].sub.[epsilon]]), W' [member of] W([[PI].sub.(v', d'), [psi] - [epsilon]) ([epsilon] [member of] {[+ or -] 1}) and f [member of] S([C.sup.2]), we define the local zeta integral Z(s, W, W', f) for G x G by

[mathematical expression not reproducible],

where d[??] is the right invariant measure on N\G. In this article, we normalize d[??] so that, for any compactly supported continuous function F on N \G,

[mathematical expression not reproducible]

with y = diag([y.sub.1] [y.sub.2], [y.sub.2]) [member of] A and dk is the Haar measure on K such that [[integral].sub.K] dk = 1. The local zeta integral Z(s, W, W', f) converges for Re(s) [much greater than] 0. The group K acts on S[([C.sup.2]).sup.std] by

([tau] (k)f)([z.sub.1], [z.sub.2]) = f (([z.sub.1], [z.sub.2])k)

for k [member of] K and f [member of] S[([C.sup.2]).sup.std]. For non-negative integers l, r, let S([C.sup.2]).sup.std.sub.l,r] be the subspace of S([C.sup.2]).sup.std] consisting of all functions of the form (6.1) with polynomial functions p([w.sub.1], [w.sub.2], [w.sub.3], [w.sub.4]) which are degree l homogeneous with respect to [w.sub.1], [w.sub.2], and degree r homogeneous with respect to [w.sub.3], [w.sub.4]. Then it is easy to see that S([C.sup.2]).sup.std] = [[direct sum].sub.l,r[greater than or equal to]0] S([C.sup.2]).sup.std.sub.l,r] and

(6.2) S([C.sup.2]).sup.std.sub.l,r] [??] [V.sub.(1,0)] [[cross product].sub.C] [V.sub.(0,-r)].

For n = ([n.sub.1], [n.sub.2], [n.sub.3], [n.sub.4]) [member fof] [([Z.sub.[greater than or equal to]0]).sup.4], we define a function [mathematical expression not reproducible] on [C.sup.2] by

[mathematical expression not reproducible].

Then we note that [mathematical expression not reproducible].

Let [epsilon] [member of] {[+ or -] 1}. We take K-homomorphisms [[phi].sup.([epsilon]).sub.[v,d;m]] and [[phi].sup.(-[epsilon]).sub.[v',d';m']] as in Proposition 5.1. Let m and m' be non-negative integers. Set

[mathematical expression not reproducible].

For each s [member of] C such that Re(s) [much greater than] 0, we note that

v [cross product] v'[cross product] f [right arrow] Z(s, [[phi].sup.([epsilon]).sub.[v,d;m]] (v), [[phi].sup.(-[epsilon]).sub.[v',d';m']] (v'), f)

defines a K-homomorphism from the tensor product [V.sub.[lambda]] [[cross product].sub.C] [V.sub.[lambda]]' [[cross product].sub.C] S([C.sup.2]).sup.std.sub.l,r] to [V.sub.(0,0)] = C, and this homomorphism vanishes unless

(6.3) [Hom.sub.K] ([V.sub.[lambda]] [[cross product].sub.C] [V.sub.[lambda]]' [[cross product].sub.C] S([C.sup.2]).sup.std.sub.l,r], [V.sub.(0,0)])) = {0}.

By (6.2) and the decomposition law

[mathematical expression not reproducible]

of K-modules for [mu] = ([[mu].sub.1], [[mu].sub.2]), [mu]' = ([[mu]'.sub.1], [[mu]'.sub.2]) [member of] [LAMBDA], we know that (6.3) holds if and only if the non-negative integers m, m', l and r satisfy

(6.4) [[lambda].sub.1] + [[lambda]'.sub.1] [greater than or equal to] 0 [greater than or equal to] [[lambda].sub.2] + [[lambda]'.sub.2],

(6.5) r = [[lambda].sub.1] + [[lambda]'.sub.1] + [[lambda].sub.2] + [[lambda]'.sub.2] + l,

(6.6) l [greater than or equal to] max{-[[lambda].sub.1] - [[lambda]'.sub.2], - [[lambda].sub.2] + [[lambda]'.sub.1]}.

The inequality (6.4) implies that m + m' [greater than or equal to] [m.sub.0] with

(6.7) [m.sub.0] = max{0, -[d.sub.1] - [d'.sub.1], [d.sub.2] + [d'.sub.2]}.

Assume m = [m.sub.0] and m' = 0. Then the smallest non-negative integers l, r satisfying (6.5) and (6.6) are given by l = [l.sub.1] + [l.sub.2] and r = [r.sub.1] + [r.sub.2] with

(68) [mathematical expression not reproducible].

Theorem 6.1. Retain the notation. Then

(6.9) Z(s, [W.sub.0], [W'.sub.0], [f.sub.0]) = L(s, [[PI].sub.(v,d)] x [[PI].sub.(v',d')])

holds for s [member of] C such that Re(s) [much greater than] 0, where

[mathematical expression not reproducible].

7. The proof of Theorem 6.1. First, we prepare two lemmas.

Lemma 7.1. (i) Let [a.sub.1], [a.sub.2], [b.sub.1], [b.sub.2] [member of] C. Then for s [member of] C such that Re(s) >> 0, it holds that

[mathematical expression not reproducible].

(ii) For s [member of] C such that Re(s) > 0, it holds that

[mathematical expression not reproducible].

(iii) For [z.sub.1], [z.sup.2] [member of] C, n [member of] [Z.sub.[greater than or equal to]0] such that Re ([z.sub.1]) > 0, Re([z.sub.2]) > n, it holds that

[mathematical expression not reproducible].

Proof. The statement (i) is derived from Barnes' lemma ([section]14.52 in [WW]) and the Mellin inversion formula (see for example, [section]1.5 in [Bu]). The statement (ii) is immediately follows from Euler's integral form of the Gamma function. The statement (iii) is derived from the formula of the value of the Gaussian hypergeometric series at 1 ([section]14.11 in [WW]).

Lemma 7.2. Let y = diag([y.sub.1], [y.sub.2], [y.sub.2]) [member of] A. For [lambda] = ([[lambda].sub.1], [[lambda].sub.2]), [lambda]' = ([[lambda]'.sub.1], [[lambda]'.sub.2]) [member ;of] [LAMBDA], 0 < q < [[lambda].sub.1] - [[lambda].sub.2], 0 < q' < [[lambda]'.sub.1] - [[lambda]'.sub.2] and n = ([n.sub.1], [n.sub.2], [n.sub.3], [n.sub.4]) [member [of] [([Z.sub.[greater than or equal to]0]).sup.4], the integral

[mathematical expression not reproducible]

is equal to

[mathematical expression not reproducible]

if [[lambda].sub.1] + [[lambda]'.sub.2] + [n.sub.1] - [n.sub.3] = [[lambda].sub.2] + [[lambda]'.sub.1] + [n.sub.2] - [n.sub.4] = [[lambda].sub.1] + [[lambda]'.sub.1] - q - q' = 0, and is equal to 0 if otherwise. Here we set [??] = [[lambda].sub.1] - [[lambda].sub.2] - q and [??] = [[lambda]'.sub.1] - [[lambda]'.sub.2] - q'.

Proof. By direct computation, we have

[mathematical expression not reproducible]

for [mathematical expression not reproducible]. Hence, we have

[mathematical expression not reproducible].

Applying Schur's orthogonality relations (Proposition 4.4 in [BD])

[mathematical expression not reproducible]

to the right-hand side of this equality, we obtain the assertion.

Let us prove Theorem 6.1. Let [epsilon] [member of] {[+ or -] 1}. Let ([[PI].sub.(v,d)], [H.sub.(v,d)]) and ([[PI]'.sub.(v,d)], [H'.sub.(v,d)])) be irreducible principal series representations of G with v = ([v.sub.1], [v.sub.2]) [member of] [C.sup.2], d = ([d.sub.1], [d.sub.2]) [member of] [LAMBDA], v' = ([v'.sub.1], [v'.sub.2]) [member of] [C.sup.2] and d' = ([d'.sub.1], [d'.sub.2]) [member of] [LAMBDA]. Let [m.sub.0] be the integer defined by (6.7), and put [lambda] = ([[lambda].sub.1], [[lambda].sub.2]) = ([d.sub.1] + [m.sub.0], [d.sub.2] - [m.sub.0]). Let [l.sub.1], [l.sub.2], [r.sub.1] and [r.sub.2] be the integers defined by (6.8). In order to simplify the notation, hereafter, we set

[mathematical expression not reproducible].

For s [member of] C such that Re(s) >> 0, we have

[mathematical expression not reproducible]

with y = diag([y.sub.1], [y.sub.2], [y.sub.2]) [member of] A. Since

[mathematical expression not reproducible]

with [mathematical expression not reproducible], we have

[mathematical expression not reproducible].

Applying Lemma 7.2 and Lemma 7.1 (ii), successively, we find that

[mathematical expression not reproducible]

with y' = diag([y.sub.1], 1) [member of] A.

Let us consider the case of [m.sub.0] = 0, that is, [lambda] = d. By Remark 5.2, Lemma 7.1 (i) and the above equality, we have

[mathematical expression not reproducible].

Replacing q [right arrow] j + [r.sub.1] and applying Lemma 7.1 (iii), we obtain (6.9) in this case.

Let us consider the case of [m.sub.0] = -[d.sub.1] - [d'.sub.1]. In this case, we note that

[l.sub.1] = -[[lambda].sub.1] - [d'.sub.2], [l.sub.2] = -[[lambda].sub.2] - [d'.sub.1], [r.sub.1] = [r.sub.2] = 0

and [[lambda].sub.1] = --[d'.sub.1]. Hence, we have

[mathematical expression not reproducible].

By Remark 5.2 and Lemma 7.1 (i), we obtain (6.9) in this case.

Let us consider the case of [m.sub.0] = [d.sub.2] + [d'.sub.2]. In this case, we note that

[l.sub.1] = [l.sub.2] = 0, [r.sub.1] = [[lambda].sub.1] + [d'.sub.2], [r.sub.2] = [[lambda].sub.2] + [d'.sub.1]

and [[lambda].sub.2] = -[d'.sub.2]. Hence, we have

[mathematical expression not reproducible].

By Remark 5.2 and Lemma 7.1 (i), we obtain (6.9) in this case.

doi: 10.3792/pjaa.94.1

Acknowledgements. This work is supported by JSPS KAKENHI Grant Number 15K04800. The author would like to express his gratitude to Profs. Miki Hirano and Taku Ishii for valuable advice on this work.

References

[BD] T. Brocker and T. tom Dieck, Representations of compact Lie groups, translated from the German manuscript, corrected reprint of the 1985 translation, Graduate Texts in Mathematics, 98, Springer-Verlag, New York, 1995.

[Bu] D. Bump, Automorphic forms and representations, Cambridge Studies in Advanced Mathematics, 55, Cambridge University Press, Cambridge, 1997.

[JL] H. Jacquet and R. P. Langlands, Automorphic forms on GL(2), Lecture Notes in Mathematics, Vol. 114, Springer-Verlag, Berlin, 1970.

[Ja] H. Jacquet, Automorphic forms on GL(2). Part II, Lecture Notes in Mathematics, Vol. 278, Springer-Verlag, Berlin, 1972.

[Po] A. A. Popa, Whittaker newforms for Archimedean representations, J. Number Theory 128 (2008), no. 6, 1637 1645.

[PBM] A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and series. Vol. 3, translated from the Russian by G. G. Gould, Gordon and Breach Science Publishers, New York, 1990.

[WW] E. T. Whittaker and G. N. Watson, A course of modern analysis, reprint of the 4th (1927) ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996.

[Zh] S.-W. Zhang, Gross-Zagier formula for GL2, Asian J. Math. 5 (2001), no. 2, 183 290.

By Tadashi MIYAZAKI

Department of Mathematics, College of Liberal Arts and Sciences, Kitasato University, 1-15-1 Kitasato, Minami-ku, Sagamihara, Kanagawa 252-0373, Japan

(Communicated by Shigefumi MORI, M.J.A., Dec. 12, 2017)

2010 Mathematics Subject Classification. Primary 11F70; Secondary 11F30, 22E46.
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Author:Miyazaki, Tadashi
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Geographic Code:1USA
Date:Jan 1, 2018
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