Introduction to zeta integrals and L-functions for GL n

(June 6, 0) Introduction to zeta integrals and L-functions for GL n Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/. Fourier-Whittaer expansions of cuspforms on GL r. The Hece-type case GL n GL n 3. The Ranin-Selberg case GL n GL n 4. Comments on GL m GL n with m n We give a quic introduction to Fourier-Whittaer expansions of cuspforms on GL n, and integral representations of associated L-functions, following a part of Jacquet, Piatetsi-Shapiro, and Shalia s extensions of Hece s wor for GL. ll nown ways to analytically continue automorphic L-functions involve integral representations using the corresponding automorphic forms. The simplest cases, extending Hece s treatment of GL, need no further analytic devices and very little manipulation beyond Fourier-Whittaer expansions. [] Poisson summation is a sufficient device for several accessible classes of examples, as in Riemann, [Hece 98,0], [Tate 950], [Iwasawa 95], and [Godement-Jacquet 97], and including treatment of the degenerate Eisenstein series needed for the GL n GL n Ranin-Selberg convolutions. [] For f a cuspform on GL n the most natural L-function obtained by an integral representation is the Hece-type integral representation, also involving a cuspform F on GL n, Λ(s, f F ) = det h s f F (h) dh GL n ()\GL n () More properly, the integral is a zeta integral Z(s, f F ), since within a given automorphic representation there is an essentially unique choice of automorphic form giving the correct local factors everywhere locally in that zeta integral: see [Jacquet-PS-Shalia 98], [Jacquet-Shalia 990], [Cogdell-PS 003]. t good primes this is not an issue, but the general case must subsume the theory of newforms, as well as coping with complications at archimedean places. [3] nother simple natural case, m = n, is the Ranin-Selberg integral using an auxiliary (degenerate) Eisenstein series E s (g) = ϕ s (γ g) where P n, has Levi component GL n GL, and γ P n, \GL n() ϕ s = v ϕ s,v with ϕ s,v in a (degenerate) induced representation from Pv n,. The zeta integral attached to two cuspforms f, F on GL n is Λ(s, f F ) = E s (h) f(h) F (h) dh GL n()\gl n() [] See [Hece 937a,b]. pparently the extension to GL n GL n was considered so apparent that it was not explicitly mentioned in [Jacquet-PS-Shalia 979], which was concerned with GL GL 3 as prototype for GL m GL n. [] We do not discuss examples relying on meromorphic continuation of non-trivial Eisenstein series, as in [Langlands 967/976, 97] and [Shahidi 978,985] have other requirements. See [Shahidi 00] for a recent survey. [3] In contrast to [Godement-Jacquet 97], the standard L-function for f is not produced by this Hece-type integral representation, except for n =. This was understood by Jacquet, Piatetsi-Shapiro, and Shalia in the late 970 s, who developed the desired integral representations of a family of L-functions including the standard ones in the papers in the bibliography below. See also Cogdell s lecture notes in the bibliography.

. Fourier-Whittaer expansions of cuspforms on GL r Some non-trivial aspects of the group structure of GL r enters in the derivation of the Fourier expansion. The outcome is not obvious for r >. Let G = GL r, reserving the character n for elements of unipotent subgroups. Let... N min. =. = unipotent radical of standard minimal parabolic.. Fix a non-trivial additive character ψ o on \, and let ψ std be the corresponding standard character on the unipotent radical of the standard minimal parabolic, namely, ψ std (u) = ψ o (sum super-diagonal entries) = ψ o (u + u 3 +... + u r,r ) (non-trivial ψ o on \) We obtain the Fourier expansion of a cuspform by an induction. First, a cuspform f has a Fourier expansion along the abelian unipotent radical N = N r, r x = {n x = : x = (r )-by-} of the form ψ N \N ψ(n) f(ng) dn (ψ summed over characters on N \N ) The cuspform condition implies that the component for the trivial character on N \N is 0. The fragment H = H r GLr 0 = { } of the Levi component of the parabolic P r, acts transitively on the non-trivial characters on N \N. Letting ψ (n x ) = ψ o (x r ) the isotropy subgroup Θ = Θ r of ψ in H is Θ = {m H : ψ (mnm ) = ψ (n) for all n N } = { GL r 0 0 } 0 Thus, for a cuspform f, γ Θ \H N \N ψ (γnγ ) f(ng) dn Replacing n by γ nγ and using the left G -invariance of f, this is ψ (n) f(nγg) dn N \N γ Θ \H For the induction step, let N = {u x = r x 0 0 } H r

Note that N normalizes N, and Letting ψ (unu ) = ψ (n) (for all n N and u N ) H r = { GL r 0 0 0 } 0 we have Θ = N H r = H r N. For each γ, the function h ψ (n) f(nhγg) dn (for h H r ) N \N is left N -invariant, because ψ (n) f(nαhγg) dn = N \N N \N ψ (n) f(αnhγg) dn = N \N ψ (n) f(nhγg) dn (for α U ) by replacing n by αnα and using the left G -invariance of f. Thus, for each γ, there is a Fourier expansion along N, namely, ψ (n) f(nhγg) dn = ψ (u) ψ (n) f(nuhγg) dn du (characters ψ of N \N ) N \N ψ N \N N \N In fact, we only need h = : N \N ψ (n) f(nγg) dn = ψ N \N ψ (u) ψ (n) f(nuγg) dn du N \N (characters ψ of N \N ) The ψ = summand is 0, because f is cuspidal, since N ( er ψ on N ) { r 0 } = unipotent radical of (r, ) parabolic 0 The action of H r on non-trivial characters ψ on N is transitive. Let ψ r x 0 0 = ψ o (x r ) (with x (r )-by-) 0 The isotropy group of ψ is Thus, ψ N \N ψ (u) ψ (n) f(nuγg) dn du = N \N = δ Θ r \H r GL r 3 0 0 Θ r 0 0 = { } 0 0 0 0 N \N δ Θ r \H r N \N ψ (u) ψ (n) f(nδ uδγg) dn du N \N 3 ψ (δuδ ) ψ (n) f(nuγg) dn du N \N

by replacing u by δ uδ. We can also replace n by δ nδ without affecting ψ, so this becomes ψ (u) ψ (n) f(nuδγg) dn du N \N δ Θ r \H r N \N ltogether, γ Θ r \H r δ Θ r \H r N \N ψ (u) ψ (n) f(nuδγg) dn du N \N Since Θ r = H r N, the elements δγ with γ Θ r \H r and δ Θ r \H r are in natural bijection with Θ r N \Hr. Certainly nu ψ (u)ψ (n) gives a character ψ on NN, which is the unipotent radical of the (r,, ) parabolic. Thus, so far, ψ (n) f(nγg) dn γ Θ r N \H \N We need a separate notation for unipotent radicals inside H = H r GL r : let U b,...,bm be the unipotent radical of the standard parabolic of H with blocs of size b,..., b m along the diagonal. Then Θ r N = H r 3 U r 3,, Thus, γ H r 3 U r 3,, \H \N We repeat the induction step once more. For each γ H r 3 h is left invariant under N, where \N U r 3,, ψ (n) f(nγg) dn \H, the function ψ (n) f(nhγg) dn (for h H r ) r 3 x 0 0 N 0 0 = {u x =, with x = (r 3)-by-} 0 0 0 0 For each γ, there is a Fourier expansion along N, ψ (n) f(nhγg) dn = ψ \N N \N ψ (u) The summand for trivial ψ is 0, because f is a cuspform, and \N ψ (n) f(nuhγg) dn du N ( r 3 er ψ on ) 0 0 { } = unipotent radical of (r 3, 3) parabolic 0 0 0 0 The rational points of GL r 3 0 0 0 H r 3 0 0 = { } 0 0 0 0 4

act transitively on non-trivial characters ψ of U \U. Let ψ (u x ) = ψ o (x r 3 ). The isotropy group of ψ is GL r 4 0 0 0 0 0 0 Θ r 3 = { 0 0 0 } 0 0 0 0 0 0 Note that Setting h =, γ H r 3 U r 3,, \H Θ r 3 U r 3,, δ Θ r 3 \H r 3 N \N = H r 4 U r 4,,, ψ (δuδ ) \N ψ (n) f(nuγg) dn du Replace u by δ uδ, and n by δ nδ, noting that conjugation by N leaves ψ invariant: γ H r 3 U r 3,, \H δ Θ r 3 \H r 3 N \N ψ (u) \N The double sum over δγ can be regrouped into a single sum of γ H r 4 Then ψ 3 r 4 0 x r 3,r 0 x r,r 0 0 x r,r 0 0 0 γ H r 4 U r 4,,, \H ψ (n) f(nuδγg) dn du U r 4,,, \H. Let = ψ o(x r 3,r + x r,r + x r,r ) N r 3,,, \N, ψ 3 (n) f(nγg) dn By induction, with U min the unipotent radical of the standard minimal parabolic in H, and N min the unipotent radical of the standard minimal parabolic in G, γ U min \H Letting the Whittaer function attached to f be W f (g) = N min \N min N min \N min ψ std (n) f(nγg) dn ψ std (n) f(ng) dn the Fourier expansion is W f (γg) γ U min \H 5

. The Hece-type integral representation: GL n GL n Still let H denote the copy of GL r in the standard Levi component of the standard (r, ) parabolic of G = GL r. For cuspform f on GL r and cuspform F on H GL r, the Hece-type integral representation det h s f F (h) dh H \H produces the L-function Λ(s, f F ), up to normalization, as follows. Expressing f in its Fourier expansion, as above, unwind: H \H det h s W f γ U min \H U min \H ( ) γ F (h) dh = det h s U min \H The function det h s W f is left ψ std -equivariant under U min, so rewrite det h s Wf F (h) dh U min \H ( = det h s Wf = U min \H U min \U min det h s Wf ( ) ψ std (u) F (uh) du dh ) W F (h) dh Wf F (h) dh where we note that the Whittaer function for F is formed with the complex-conjugated character ψ std restricted from N min to U min. Under various hypotheses on f, F, the Whittaer functions factor over primes, and, then the zeta integral factors over primes, giving an Euler product Z(s, f F ) = v ( ) det h v s hv 0 Wf,v W Uv min \H v F,v (h v ) dh v Further, at places v where f and F are spherical, via an Iwasawa decomposition H = U v Mv min K v with M min the standard Levi component of the minimal parabolic in H, the v th local integral becomes a much smaller, (r )-dimensional integral: M min v det m v s m 0 Wf,v W F,v (m) dm δ(m) m 0 (with m =... 0 m r GL r ) where δ(m) is the modular function of M min v on U min v. [.0.] Remar: The most important normalization constants are ρ f () and ρ F (), which are the (higherran analogues of) leading Fourier coefficients of f and F, when f and F are normalized to have L -norm. Further, it is less clear that the archimedean local zeta integral is the correct gamma factor. Thus, even with everywhere-spherical f and F, zeta-integral Z(s, f F ) = ρ f () ρ F () (archimedean integrals) L(s, f F ) 6

[.0.] Remar: The analytic continuation of this Euler product follows from the original integral representation, with relatively straightforward estimates on the cuspforms f, F. The functional equation comes essentially from replacing h by h-transpose-inverse in the integral. The effect of transpose-inverse on the local representations, hence, on the Euler factors, requires some further attention. 3. The Ranin-Selberg case GL n GL n Now we need an auxiliary (degenerate) Eisenstein series E s (g) = γ P n, \GL n() where P n, has Levi component M GL n GL, and ϕ s (γ g) ϕ s = v ϕ s,v with ϕ s,v in a (degenerate) induced representation from Pv n,. Specifically, the representation induced from M should be of the form 0 m = det s (det ) χ 0 d d n d n (for GL n and d GL ) where s C and χ is a Hece character. [4] We suppress reference to other data specifying the vector ϕ s in the induced representation, although in practice this data must be chosen to accommodate bad-prime aspects of f, F, for example. The character χ must be chosen so that the central character of E s f F is trivial, or the following integral is not well-defined. The Ranin-Selberg zeta integral attached to two cuspforms f, F on GL n is [5] Z(s, f F ) = E s (g) f(g) F (g) dg Z GL n()\gl n() Let H be the GL n factor of the Levi component M. In the region of convergence, unwind the zeta integral by unwinding the Eisenstein series: Z(s, f F ) = ϕ s (g) f(g) F (g) dh Z H N n, \GL n() Let U min be the unipotent radical of the standard minimal parabolic in H. Expanding f in its Fourier- Whittaer expansion and unwinding, using the H -invariance of ϕ s, Z(s, f F ) = ϕ s (g) W f (γg) F (g) dg Z H N n, \GL n() γ U \H = ϕ s (g) W f (g) F (g) dh = ϕ s (g) W f (g) F (g) dh Z U min N n, \GL n() Z N min \GL n() [4] The case of trivial χ is already useful. On the other hand, the character s can be incorporated into χ, if desired. Nevertheless, for analytical purposes, it is often convenient to separate the continuous parameter s from a parametrization of the compact part of the idele-class group: J / where J is ideles of idele-norm. [5] The complex conjugation on F avoids certain uninteresting technicalities, as will become apparent. 7

where N min = U min N n, is the unipotent radical of the standard minimal parabolic P min in GL n. Since ϕ s is left N -invariant, and the Whittaer function W f is left N, ψ std -equivariant, with the standard character ψ std on N, Z N min \GL n() ϕ s (g) W f (g) F (g) dh = = ϕ s (g) W f (g) Z N min \GLn() ( ϕ s (g) W f (g) W F (g) dg Z N min \GLn() N min \N min ) ψ std (u) F (ug) du dh It is here that the pre-emptive complex-conjugation of F gives the Whittaer function of F with respect to ψ std, rather than with respect to its complex conjugate. Under various hypotheses, the Whittaer functions factor over primes. When we tae ϕ s to be a monomial tensor, we have an Euler product Z(s, f F ) = ϕ s (g) W f,v (g) W F,v (g) dg (with inducing data suppressed) v Z vnv min \GL n( v) [3.0.] Remar: The most important normalization constants are ρ f () and ρ F (), the (higher-ran analogues of) leading Fourier coefficients of f and F, when f and F are normalized to have L -norm. It is less clear that the archimedean local zeta integral is the correct gamma factor. Thus, even with everywhere-spherical f and F, zeta-integral Z(s, f F ) = ρ f () ρ F () (archimedean integrals) L(s, f F ) [3.0.] Remar: The analytic continuation of the zeta integral follows from the original integral representation, with relatively straightforward estimates on the cuspforms f, F, and from the meromorphic continuation and function equation of E s. For this very degenerate Eisenstein series, the analytic continuation and functional equation follow from Poisson summation. 4. Comments on GL m GL n with m n s Jacquet, Piatetsi-Shapiro, and Shalie found, for m < n an intermediate integration is necessary. In the literature, such auxiliary integrations are often called unipotent integrations, and occur in other situations, as well. In the end, one has a zeta integral Z(s, f F ) = det h s (proj n m f) F (h) dh n m GL m()\gl m() where the projection operator proj n m is the identity map for m = n, but non-trivial otherwise, described as follows. Let m+... N = Nm r =. = unipotent radical of (m +,,,..., ) parabolic... The proper definition of the projection turns out to be (proj r mf)(g) = det h r (m+) 8 N \N ψ(n) f(ng) dn

For example, for r = 3 and m =, with F trivial on GL, the standard L-function attached to f is essentially the zeta integral Z(s, f) = h s + 3 (+) proj 3 f h dh GL ()\GL () = GL ()\GL () h s \ \ ψ o (y) f ( x y h See [Cogdell 003,07,08] for many further details about this general case. ) dx dy dh Bibliography [Cogdell 003] J. Cogdell, nalytic theory of L-functions for GL n ; Langlands conjectures for GL n ; Dual Groups and Langlands Functoriality, in n introduction to the Langlands program, ed. J. Bernstein, S. Gelbart, Birhauser, Boston, 003, 97-68. [Cogdell 007] J.W. Cogdell, L-functions and converse theorems for GL n, in utomorphic forms and applications, ed. P. Sarna, F. Shahidi, Par City/IS Math. Series, MS, 007, 97-77. [Cogdell 008] J. Cogdell, Notes on L-functions for GL n, in School on utomorphic Forms on GL n, ICTP Lecture Notes Series, vol., June 008, 75-57. [Cogdell-PS 003] J. Cogdell, I. Piatetsi-Shapiro, Remars on Ranin-Selberg convolutions, in Contributions to utomorphic Forms, Geometry, and Number Theory (Shaliafest 00), H. Hida, D. Ramarishnan, and F. Shahidi eds., Johns Hopins University Press, Baltimore, 005. [Godement-Jacquet 97] R. Godement, H. Jacquet, Zeta functions of simple algebras, Lecture Notes in Math. 60, Springer-Verlag, Berlin-New Yor, 97. [Hece 98] E. Hece, Eine neue rt von Zetafuntionen und ihre Beziehungen der Verteilung der Primzahlen, Math. Z. no. 4 (98), 357-376. [Hece 90] E. Hece, Eine neue rt von Zetafuntionen und ihre Beziehungen der Verteilung der Primzahlen, Math. Z. 6 no. - (90), -5. [Hece 937a] E. Hece, Über Modulfunctionen und die Dirichletschen Reihen mit Eulerscher Produtentwiclung, I, Math. nn. 4 (937) no., -8. [Hece 937b] E. Hece, Über Modulfunctionen und die Dirichletschen Reihen mit Eulerscher Produtentwiclung, II, Math. nn. 4 (937) no., 36-35. [Iwasawa 95] K. Iwasawa, Letter to J. Dieudonné, dated pril 8, 95, in Zeta Functions in Geometry, ed. N. Kuroawa and T. Sunada, dvanced Studies in Pure Mathematics (99), 445-450. [Jacquet-PS-Shalia 979] H. Jacquet, I. Piatetsi-Shapiro, and J. Shalia, utomorphic forms on GL(3), nnals of Math. 09 (979), 69-58. [Jacquet-PS-Shalia 98] H. Jacquet, I. Piatetsi-Shapiro, J. Shalia, Conducteur des représentations du groupe linéaire, Math. nn. 56 (98),99-4. [Jacquet-PS-Shalia 983] H. Jacquet, I. Piatetsi-Shapiro, and J. Shalia, Ranin-Selberg convolutions, mer. J. Math. 05 (983), 367-464. [Jacquet-Shalia 98] H. Jacquet and J. Shalia, On Euler products and the classification of automorphic representations, I, mer. J. Math. 03 (98), 499-558. 9

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