Rank Lowering Linear Maps and Multiple Dirichlet Series Associated to GL(n, R)

Transcription

1 Pure and Applied Mathematics Quarterly Volume, Number Special Issue: In honor o John H Coates, Part o 6 65, 6 Ran Lowering Linear Maps and Multiple Dirichlet Series Associated to GLn, R Introduction Dorian Goldeld and Meera Thillainatesan Dedicated to John Coates The Godement-Jacquet L-unction associated to the discrete subgroup SLn, Z with n acting on h n : GLn, R/On, R R was irst introduced in [Godement-Jacquet, 97] where its analytic properties: holomorphic continuation, unctional equation, Euler product, etc were obtained by a generalization o Tate s thesis [Tate, 95] Later, [Jacquet-Piatetsi- Shapiro-Shalia, 979] obtained a dierent derivation o the construction o the Godement-Jacquet L-unction by the use o Whittaer models Following [Goldeld, 6], we review a classical construction o the Godement- Jacquet L-unction In order to do this, it is necessary to introduce the ollowing notation Let z n xn yn h n with x, x,3 x,n y y y n x,3 x,n xn, yn x n,n y y y n y, where x i,j R or i < j n and y i > or i n be in Iwasawa reduced orm Deine U n R to be the subgroup o SLn, R consisting o upper Received September 4, 5 The irst author s research is supported in part by NSF grant DMS The second author was partially supported by the NSF-AWM Mentoring Travel Grant

2 6 Dorian Goldeld and Meera Thillainatesan triangular unipotent matrices and let u, u,3 u,n u,3 u,n u u n,n denote a generic element o U n R, where the superdiagonal elements are relabeled as u u n,n, u u n,n,, u n u, For m m, m,, m n Z n, deine ψ m to be the character o U n R deined by [ ] ψ m u : e πi m u +m u ++m n u n For ν ν, ν,, ν n C n, and z n xn yn h n, as above, deine the unction, I ν : h n C, by the condition: where b i,j I ν z n : n n i j y b i,jν j i, { ij i i + j n, n in j i i + j n Set w n n/ to be the long element o the Weyl group SLn, Z, We may now deine Jacquet s Whittaer unction W Jacquet z n ; ν, ψ m by the integral ormula W Jacquet z n ; ν, ψ m : I ν w n u z n ψ m u d u, U nr when Rv i or i n and by meromorphic continuation to all v C n The Whittaer unction is characterized by the ollowing properties First, it satisies the relation W Jacquet uz n ; ν, ψ m ψ m u W Jacquet z n ; ν, ψ m

3 Ran Lowering Linear Maps and Multiple 63 or all u U n R Second, it is a square integrable unction or the Haar measure on a Siegel set Σ where { Σ z h n } 3 y i > i n, xi,j i < j n Finally, the Whittaer unction is also an eigenunction o all the GLn, R invariant dierential operators with the same eigenvalues as I ν Deinition Maass orm Let n A Maass orm or SLn, Z o type ν C n is a smooth unction : h n C which satisies the ollowing conditions: γz n z n, γ SLn, Z, z n h n, L SLn, Z\h n, is an eigenunction o all the GLn, R invariant dierential operators having the same eigenvalues as I ν z n, uz n du, SLn,Z U\U or all upper triangular groups U o the orm U I r Ir, I rb with r + r + r b n Here I r denotes the r r identity matrix, and denotes arbitrary real entries The Fourier expansion o a Maass orm or SLn, Z was irst derived in [Piatetsi-Shapiro, 975], [Shalia, 974] The Fourier expansion taes the orm see [Goldeld, 6] z n γ U n Z\SLn,Z m m n m n Am,, m n n m n γ W Jacquet M z n, ν, ψ m,,, n, m n

4 64 Dorian Goldeld and Meera Thillainatesan where M m m n m n m m m C A, Am,, m n C We may now give the deinition o the Godement-Jacquet L-unction Deinition 3 Godement-Jacquet L-unction Let s C with Rs > n+, and let z n be a Maass orm or SLn, Z, with n as in deinition, which is an eigenunction o all the Hece operators We deine the Godement- Jacquet L-unction, L s, by the absolutely convergent series L s φ p s, where 4 φ p s m Am,,, m s p Ap,, p s A, p,, p s + + n A,, pp n+s + n p ns It is natural to consider multiple Dirichlet series associated to o the type Am,, m n 5 L s,, s n : m s, ms n n m m m m s ms m n where s, s,, s n C with Rs i i n Such a series was irst considered by [Bump, 984] in the case n 3 Bump proved the identity Am, m L s L s, ζs + s where Am, m are the Fourier coeicients o a Maass orm normalized Hece eigenorm or SL3, Z and denotes the dual o In [Bump-Friedberg, 99], a generalization o this double Dirichlet series, and above identity, was obtained by considering the product o the standard and exterior square L-unctions, ie, a Ranin-Selberg construction involving two complex variables, one rom the Eisenstein series, and one o Hece type The multiple Dirichlet series o the type 5 are not, in general, expected to have meromorphic continuation to all C n when n 4 see [Brubaer-Bump- Chinta-Friedberg-Hostein, 5] It is expected that they have natural boundaries

5 Ran Lowering Linear Maps and Multiple 65 Question 6 One may as i there exist multiple Dirichlet series associated to Maass orms or SLn, Z with n 4 beyond the series o Bump-Friedberg type [Bump-Friedberg, 99]? A multiple Dirichlet series is said to be perect i it satisies a inite group o unctional equations and has meromorphic continuation in all its complex variables The main object o this paper is to construct, by the employment o a simple integral operator, classes o perect multiple Dirichlet series associated to Maass orms on SLn, Z Our construction maes use o a ran lowering map which maps automorphic orms or SLn, Z with n 3 into automorphic orms or SLn, Z, ie, the ran is lowered by one The ran lowering map is given by a single Mellin transorm and a restriction o variables It has the very interesting property that the cuspidal image o a Maass orm or SLn, Z is an an ininite sum o Maass orms or SLn, Z weighted by twists o L by the Godement-Jacquet L-unctions o the Maass orms on SLn, Z In theorem 4, the cuspidal projection o the ran lowering map is precisely computed The ran lowering linear map also satisies natural unctional equations so iterations o this map can be used to construct perect multiple Dirichlet series Ran lowering linear maps acting on automorphic orms Fix n and s C with Rs suiciently large We shall construct a ran lowering linear map denoted P n s acting on automorphic orms or the group SLn +, Z and mapping them to automorphic orms or SLn, Z which is a discrete group o lower ran Deinition ran lowering linear map Fix n and s C with Rs suiciently large For an automorphic orm or SLn +, Z, deine P n s y I n s y n dy Detz n, y where I n is the n n identity matrix It is assumed that has suicient decay properties so that the above integral converges absolutely and uniormly on compact subsets with Rs suiciently large This will be the case, or example, i is a Maass orm as in deinition Fix a constant D An automorphic orm z n or SLn, Z is said to be strongly L i or any ixed subset Y {y l, y l, y lr } with l < < l r n,

6 66 Dorian Goldeld and Meera Thillainatesan or Y the empty set, we have z n y i Y or all N,, 3,, and all z n h n satisying the condition that y i > D or y i Y Here, the constant in depends at most on, N, D, and the values o the ixed variables < y li Y Note that this constant may blow up as a particular y li Since a undamental domain or SLn, Z\h n is contained in a Siegel set see [Goldeld, 6], it easily ollows that strongly L implies L in the usual sense y N i The ey properties o the ran lowering map are given in the next proposition Proposition 3 Fix n and s C with Rs suiciently large Then the ran lowering linear map Ps n as given in deinition maps strongly L automorphic orms or SLn +, Z to strongly L automorphic orms or SLn, Z Proo: First we show that Ps n is well deined on h n, ie, that Ps n ri n Ps n or all On, R and all r R Since On +, R and is right invariant by On +, R it immediately ollows that Ps n Ps n It remains to show that Ps n is right invariant by rin or r R Clearly, we may assume r > The invariance ollows rom the computation: P n s r Ps n, ater maing the transormation y y r ry I n r ns s y n dy Detz n y I γ SLn, Z then γ SLn +, Z Since is automorphic or SLn +, Z and Det γ, it immediately ollows rom deinition that Ps n γ Ps n or all γ SLn, Z This shows that Ps n is automorphic or SLn, Z Next, we establish the strongly L property Fix a set Y {y l,, y lr } It ollows rom the deinition and the bounds that or ixed s with Rs

7 Ran Lowering Linear Maps and Multiple 67 suiciently large and N nrs that P n s z n D y I n Rs y n dy Detz n + D y I n Rs y n dy Detz n y y D n Detz n Rs + y i Y y N i D y nrs N Detz n Rs dy yi N y y i Y or N N nrs In the estimation o the irst integral above, we have used the act that since is automorphic and is bounded on a undamental domain it is bounded everywhere Next, we will apply the ran lowering operator to Maass orms It is necessary to show that Maass orms are strongly L This is a long tedious computation which we omit but is based on the act that the bound holds or Jacquet s Whittaer unction This was irst established in [Jacquet-Piatetsi- Shapiro-Shalia, 979] It is easy to show that holds on GL by classical estimates or the K-Bessel unction In theorem [Stade, 99], it is proved that Jacquet s Whittaer unction on GLn can be written as an integral o the classical K-Bessel unction multiplied by another Jacquet Whittaer unction or GLn The bound can then be obtained by induction Alternatively, it is also possible to obtain sharp bounds by the methods in 7 in [Jacquet, 4] The ollowing theorem gives the contribution o the discrete spectrum to the ran lowering map deined on Maass orms It shows that this contribution is a Ranin-Selberg convolution Theorem 4 Fix n and s C, suiciently large Let be an even Maass orm o type ν C n or SLn +, Z For every m, m, m n Z n, let Am,, m n be the corresponding Fourier coeicient o Let φ be a Maass orm o type λ C n or SLn, Z For every r, r n Z n, let Br,, r n be the corresponding Fourier coeicient o φ Let, denote the

8 68 Dorian Goldeld and Meera Thillainatesan Petersson inner product on SLn, Z\h n Then Ps n, φ m m n Am,, m n Bm,, m n G n ν,λ s m n+ s+ where G ν,λ s y y y y n y y y n W Jacquet, ν, ψ,, y y y y n y y n n W Jacquet, λ, ψ,, y n s y Note that a similar theorem can be obtained or odd Maass orms Proo: To compute the inner product, we will use the Ranin-Selberg unolding technique The irst step will be to write the Fourier expansion o P n s as a sum over the coset representatives o U n Z\SLn, Z With this expansion, we can unold the integral over SLn, Z\h n to obtain an integral over U n Z\h n The computation goes as ollows n i dy i y i P n s z n m,,m n where M is as in deinition γ U nz\sln,z Am,, m n n m n+ γ y I s W Jacquet M n, ν, ψ,, y n dy Detz n, y Now let z n γ be the Iwasawa orm o γz n There exists γ On, R and dγ diagn, R such that γz n z n γγdγ Note that taing determinants implies Detz n dγ n Det z n γ

9 Ran Lowering Linear Maps and Multiple 69 Hence P n s z n W Jacquet M m,,m n γ U nz\sln,z Am,, m n n γ y dγ I n m n+ s, ν, ψ,, y n dy Detz n y As beore, we mae the change o variables y dγ y Consequently P n s z n m,,m n W Jacquet M γ U nz\sln,z Am,, m n n m n+ γ y I s n, ν, ψ,, y n dy Detz n γ y When is a Maass orm or SL3, Z, or example, the above calculation gives P s as ollows: P s x + iy W Jacquet m,m γ U Z\SL,Z m m y y cz+d m y Am, m m m, ν, ψ,, em Reγz y s ys cz + d s dy y The above calculations give another proo o the invariance o P n s under SLn, Z

10 6 Dorian Goldeld and Meera Thillainatesan The inner product can now be evaluated by unolding We have Ps n, φ Ps n z n φz n d z n SLn,Z\h n m,,m n W Jacquet M γ U nz\sln,z γ y I n Am,, m n n m n+ SLn,Z\h n φz n, ν, ψ,, y n Det z n γ s d z n dy y Consequently Ps n, φ m,,m n W Jacquet M m,,m n W Jacquet M Am,, m n n m n+ U nz\h n φz n y I n, ν, ψ,, y n Detz n Am,, m n n m n+ y yn, ν, ψ,, s d dy z n y n sφ y n em x + + m n x n d xn d yn dy y m,,m n W Jacquet M Am,, m n Bm,, m n n y yn m n+ n, ν, ψ,, s y n m m n y y n W Jacquet m y, ν, ψ,, n y n n dy i y i i

11 Ran Lowering Linear Maps and Multiple 6 Now, in the above integral, mae the change o variables m i+ y i y i or each i rom to n The theorem ollows We now show that the ran lowering map applied to a Maass orm satisies natural unctional equations Recall the long element w o the Weyl group o SLn +, Z Theorem 5 Functional Equations Fix n and let s C with Rs suiciently large Let be a Maass orm o type ν C n or SLn +, Z Deine the contragredient Maass orm by z w t z w Then Ps n has analytic continuation to all s C and satisies the ollowing unctional equations: i P n s z n P n s t z n ii P n s Ps n P n Pv n, v where v n s + n+ n s and v n s + n n s By iterating the ran lowering map, perect multiple Dirichlet series may be created Proo: We compute, or Rs, P n s y I n y I n + ± C A s y n dy Detz n y s y n dy Detz n y y I n y n Detz n s dy y

12 6 Dorian Goldeld and Meera Thillainatesan y I s w n y n dy Detz n y + y I n s y n dy Detz n y t y I s n y n dy Detz n y + y I n s y n dy Detz n y In the irst integral above, mae the change o variables y y Note that the second integral converges absolutely or any s C since the Maass cusp orm has rapid decay as y It ollows that 7 Ps n t z n y I n y n Det t s dy y + y I n Let τ n denote the Iwasawa orm o t, so that τ n t z n dz n, or some and z n On, R dz n diagn, R s y n dy Detz n y By the Iwasawa decomposition, we have the ollowing equivalence: t y I n τn y dz n I n mod On +, Rdiagn +, R It is then clear rom this representation that as y, the contragrediant Maass orm will have rapid decay Consequently, both the irst and second integrals in 7 converge absolutely or any s C and deine holomorphic unctions on all o C The unction

13 Ran Lowering Linear Maps and Multiple 63 P n s can then be deined or every s C by analytic continuation It ollows that or any s C, P s n t + y I n s y n dy Detz n y t z n y I n y n Det t s dy, y Ps n The last line gives the unctional equation P n s P s n t The second unctional equation in theorem 5 does not involve the contragredient We shall deduce it by a double iteration o the ran lowering linear map Let v, v be as in the statement o theorem 5 We compute, or Rs, Rs, Rv, Rv all simultaneously suiciently large: P n s Ps n z n Ps n y I n y n s dy Detz n y z n y I n y I n y n y n s Detz n y n s dy Detz n z n y y I n y y ns yn s +s y dy y Detz n s +s dy y dy y

14 64 Dorian Goldeld and Meera Thillainatesan I n z n y y I n y y ns yn s +s s +s dy dy Detz n y y z n I n y y I n y y ns yn s +s s +s dy dy Detz n y y z n y y I n I n y y ns yn s +s s +s dy Detz n z n y y I n y ns yn s +s y Detz n s +s dy y dy y y dy y It ollows that P n s P n s z n n I n y y ns yn s +s y s +s dy dy Detz n y y In the above integral, let s mae the successive transormations: y then y y y We obtain y and

15 P n s P n s z n Ran Lowering Linear Maps and Multiple 65 z n y y I n y y ns y y n s s +s +s dy Detz n y dy y z n y I n y I n y n y n v Detz n y n v dy Detz n where v and v are as given in the statement o the theorem Reerences y dy y, [] B Brubaer, D Bump, G Chinta, S Friedberg, J Hostein, Weyl group multiple Dirichlet series I, to appear [] D Bump, Automorphic Forms on GL3, R, Lecture Notes in Math , Springer- Verlag [3] D Bump and S Friedberg, The exterior square automorphic L-unctions on GLn, Festschrit in honor o I I Piatetsi-Shapiro on the occasion o his sixtieth birthday, Part II Ramat Aviv, 989, Israel Math Con Proc, 3, Weizmann, Jerusalem, 99, [4] R Godement and H Jacquet, Zeta unctions o simple algebras, Lecture Notes in Mathematics, Vol 6 Springer-Verlag, Berlin-New Yor, 97 [5] D Goldeld, Automorphic orms and L-unctions or the group GLn, R, to appear 6 [6] H Jacquet, Integral representation o Whittaer unctions, Contributions to automorphic orms, geometry, and number theory, Johns Hopins Univ Press, Baltimore, MD, 4, [7] H Jacquet, II Piatetsi-Shapiro, J Shalia, Automorphic orms on GL3, I, II, Annals o Math 9 979, [8] II Piatetsi-Shapiro, Euler subgroups Lie groups and their representations, Proc Summer School, Bolyai Jnos Math Soc, Budapest, 97, Halsted, New Yor, 975, [9] JA Shalia, The multiplicity one theorem or GLn, Annals o Math 974, 7 93 [] E Stade, Integral ormulas or GLn, R Whittaer unctions, Due Math J, Vol 6, No 99, [] J Tate, Fourier analysis in number ields and Hece s zeta unction, Thesis, Princeton 95, also appears in Algebraic Number Theory, edited by JW Cassels and A, Frohlich, 968, Academic Press, New Yor, Meera Thillainatesan Dorian Goldeld UCLA, Mathematics Department Columbia University Los Angeles, CA Department o Mathematics thill@mathuclaedu New Yor, NY 7 goldeld@mathcolumbiaedu