(September 23, 2007)
Standard archimedean integrals for GL(2) Paul Garrett
[email protected] http://www.math.umn.edu/˜garrett/
The integrals evaluated here have been understood for more than 100 years, and constantly recur everywhere. The computations shown here not only derive the standard facts, but also show that these facts are manifestations of a few simple ideas. To see the usual connections to the Special Functions literature, one might compare N.N. Lebedev, Special functions and their applications, trans. R.A. Silverman, Prentice-Hall, 1965; reprinted Dover, 1972. E.T. Whittaker, G.N. Watson, A Course of Modern Analysis, Cambridge University Press, 1927, 4th edition, 1952.
1. 2. 3. 4. 5. 6. 7. 8. 9.
Basics involving Γ(s), B(α, β), etc. Holomorphic discrete series Spherical Whittaker functions for GL2 ( ) Spherical Mellin transforms for GL2 ( ) Spherical Rankin-Selberg integrals for GL2 ( ) Moment integrals for discrete series of GL2 ( ) Spherical Whittaker functions for GL2 ( ) Spherical Mellin transforms for GL2 ( ) Non-spherical principal series • Appendix: explicit Iwasawa decompositions • Appendix: Γ(s) · Γ(1 − s) = π/ sin πs √ • Appendix: Duplication: Γ(s) · Γ(s + 12 ) = 21−2s · π · Γ(2s)
R C
R C
R R
1. Basic identities involving Γ(s), B(α, β), etc. We take the Gamma function Γ(s) to be defined by Euler’s integral Γ(s) =
Z
∞
ts e−t
0
dt t
(for Re (s) > 0)
Integration by parts proves the functional equation Γ(s + 1) = s · Γ(s)
Z
For 0 < s ∈ , this relation and induction show the connection to factorials, Γ(n) = (n − 1)!
(for n = 1, 2, . . .)
From the functional equation, we get a meromorphic continuation of Γ(s) to the entire complex plane, except for poles at non-positive integers −n. The poles are simple, with residue (−1)n /n! at −n. The identity Z
∞
ts e−ty
0
Γ(s) dt = t ys
(for y > 0 and Re(s) > 0)
for y > 0 first follows for Re(s) > 0 by replacing t by t/y in the integral. Then Z
0
∞
ts e−tz
Γ(s) dt = t zs
(for Re(z) > 0 and Re(s) > 0) 1
Paul Garrett: Standard archimedean integrals for GL(2) (September 23, 2007) by complex analysis, since both sides are holomorphic in s and agree on the positive reals. This identity allows non-obvious evaluation of a Fourier transform. Namely, let α −x x ·e (for x > 0) f (x) = 0 (for x < 0) For Re(α) > −1 this function is locally integrable at 0, and in any case is of rapid decay at infinity. We can compute its Fourier transform: Z ∞ Z Z ∞ dx dx Γ(α + 1) e−2πiξx f (x) dx = e−2πiξx xα+1 e−x = xα+1 e−x(1+2πiξ) = x x (1 + 2πiξ)α+1 R 0 0 Further, Fourier inversion gives the non-obvious α −x Z 1 1 x ·e e2πiξx dξ = · 0 (1 + 2πiξ)α+1 Γ(α + 1) R
(for x > 0) (for x < 0)
Z
For α ∈ , the same conclusion can be reached by evaluation by residues. Next, we recall the argument that expresses Euler’s beta integral in terms of gamma functions, as B(a, b) =
Z
1
xa−1 (1 − x)b−1 dx =
0
t t+1
Indeed, replacing x by Z
1
a−1
x
0
b−1
(1 − x)
=1−
1 t+1
dx =
Z
in the integral gives
∞
0
Γ(a) Γ(b) Γ(a + b)
Z ∞ t a−1 t b−1 dt 1 a+b dt a 1− = t t+1 t+1 (t + 1)2 t+1 t 0
Use the gamma identity in the form Z ∞ 1 s 1 du = e−u(t+1) us t+1 Γ(s) 0 u to rewrite the beta integral further as Z ∞Z ∞ Z ∞Z ∞ 1 Γ(a) Γ(b) 1 du dt dt du = = ua+b ta e−u(t+1) ub ta e−u e−t Γ(a + b) 0 u t Γ(a + b) t u Γ(a + b) 0 0 0 as claimed.
///
A similar integral, with one more factor, is Euler’s integral integral for hypergeometric functions, F (α, β, γ; z) =
1 B(β, γ − β)
Z
0
1
xβ−1 (1 − x)γ−β−1 (1 − xz)−α dx
where B is the beta function B(a, b) =
Z
0
1
xa−1 (1 − x)b−1 dx =
Γ(a) Γ(b) Γ(a + b)
This F is the 2 F1 hypergeometric function, whose series definition is F (α, β, γ; z) = 1 +
∞ X (a)n (b)n z n a(a + 1) b(b + 1) z 2 ab z + + ... = c 1! c(c + 1) 2! (c)n n! n=0
The notation (a)n is the Pockhammer symbol. 2
Paul Garrett: Standard archimedean integrals for GL(2) (September 23, 2007)
2. Holomorphic discrete series R
Whittaker functions W (g) for holomorphic discrete series for GL2 ( ) are completely elementary. For the holomorphic discrete series of weight 0 < κ ∈ , the Whittaker function corresponding to the lowest K-type is essentially an exponential function, namely, κ y 0 (for y > 0) W = y 2 e−2πy 0 1
Z
The extension of this to the whole group is z 0 1 x y 0 cos θ W( · · · 0 z 0 1 0 1 − sin θ
κ sin θ ) = ω(z)e2πix · y 2 e−2πy · eiκθ cos θ
(for y > 0, z ∈
R× )
where the central character ω must be compatible with κ, in the sense that −1 0 ω = eπiκ 0 −1 That is, the central character must have the same parity, as a character, as the weight κ has, as an integer. Mellin transforms of holomorphic discrete series Whittaker functions are easy to evaluate, with the exponent s − 12 rather than s, as usual, to have the functional equations s −→ 1 − s rather than s −→ −s. Namely, regardless of the parity of κ, Z Z ∞ y 0 dy y 0 dy s− 21 κ s− 12 W W = 2· |y| sgn(y) · |y| 0 1 |y| 0 1 y R× 0 Z ∞ κ κ 1 1 dy κ = 2 · (2π)−(s− 2 + 2 ) Γ(s − 12 + ) = 2· |y|s− 2 + 2 e−2πy y 2 0 The archimedean integral arising in the Rankin-Selberg convolution L-function for two holomorphic cuspforms of the same weight κ is similarly easy to evaluate explicitly: Z ∞ Z ∞ dy y 0 y 0 dy |y|s−1 W W = |y|s−1+κ e−4πy = (4π)−(s−1+κ) · Γ(s − 1 + κ) 0 1 0 1 y y 0 0
3. Spherical Whittaker functions for GL2(R) First, we want integral representations of Whittaker functions for spherical principal series with trivial central character for G = GL2 ( ). As usual, define subgroups ∗ ∗ 1 ∗ P ={ } N ={ } 0 ∗ 0 1
R
C
For µ ∈ , let Iµ be the µth (naively normalized) spherical principal series of G with trivial central character, by definition consisting of the collection of all smooth functions f on G such that a ∗ f (p · g) = χ(p) · f (g) (for χ(p) = |a/d|µ with p = ∈ P , and g ∈ G) 0 d 3
Paul Garrett: Standard archimedean integrals for GL(2) (September 23, 2007) Computation of the Fourier expansion of Eisenstein series suggests presenting the Whittaker function as an image under an intertwining operator from Iµ to the Whittaker space
R
Wh = {f on GL2 ( ) : f (ng) = ψ(n) · f (g) for all n ∈ N, g ∈ G}
(where ψ
1 0
x 1
= e2πix )
Such an intertwining amounts to taking a Fourier transform, as follows. Let the spherical function be y ∗ cos θ sin θ ϕ( ) = |y|µ (for y > 0) 0 1 − sin θ cos θ with trivial central character. This spherical function gets mapped to the Whittaker space by an intertwining Z nf e−2πix ϕ(wo nx g) dx Wµ (g) =
R
For diagonal elements my =
y 0
y/r 0
∗ r
=
Z
0 , we can explicitly determine the Iwasawa decomposition of wo nx my , 1
namely (see the appendix) wo nx my =
y/r −x/r
x/r y/r
wo
(where r =
p x2 + y 2 )
Therefore, Wµnf
y 0
0 1
y/r e−2πix ϕ( 0 R
∗ r
y/r −x/r
x/r y/r
wo ) dx
The spherical-ness is simply right K-invariance, and we are assuming trivial central character, so this simplifies to µ Z |y| y 0 −2πix nf e Wµ = dx (for Re (µ) > 21 ) 0 1 y 2 + x2 R Replace x by xy to obtain the useful integral representation Z 1 y 0 nf 1−µ e−2πixy dx Wµ = |y| 0 1 (1 + x2 )µ R
(for Re (µ) > 21 )
It is awkward that Re(µ) > 21 is necessary for absolute convergence, since the unitary principal series have Re(µ) = 21 , in the naive normalization. However, we can rearrange the integral to a form that shows the meromorphic continuation, and explains the normalization constant at the same time. Namely, using the identity involving Γ(s), 1−µ
|y| Replace x by x ·
√ √π t
Z
R
e
−2πixy
1 |y|1−µ dx = (1 + x2 )µ Γ(µ)
Z Z
R
∞
0
to obtain √ Z Z √ π |y|1−µ ∞ dt −2πixy· √πt µ− 12 −t −πx2 e t e e dx Γ(µ) t 0 R
Taking the Fourier transform of the Gaussian in x, this is √ Z π |y|1−µ ∞ −πy2 · π µ− 1 −t dt t t 2 e e Γ(µ) t 0 4
2
e−2πixy tµ e−t(1+x ) dx
dt t
Paul Garrett: Standard archimedean integrals for GL(2) (September 23, 2007) Replacing t by π|y| · t gives Wµnf
y 0
0 1
π µ |y|1/2 Γ(µ)
=
Z
∞
dt t
1
1
e−π|y|( t +t) tµ− 2
0
This last integral converges nicely for any complex µ. A less naive normalization of the parametrization of the principal series Iµnf replaces µ by µ + 12 , taking nf Iµ = Iµ+ 1 2
And a less naive normalization of the Whittaker function not only shifts µ, but also gets rid of the Γ(µ) and π µ , by taking Γ(µ + 21 ) nf · Wµ+ Wµ = 1 2 πµ That is, Wµ
y 0
Γ(µ + 12 ) = πµ
0 1
Z
R
e
−2πix
|y| x2 + y 2
µ+ 12
1/2
dx = |y|
Z
∞
1
e−π|y|( t +t) tµ
0
dt t
Note that, in this normalization, W−µ = Wµ One might further make a comparison to classical special functions.
4. Spherical Mellin transforms for GL2(R) The Hecke-Maaß integral representation of L-functions attached to waveforms already requires that we understand archimedean local integrals, namely, Mellin transforms of Whittaker functions, given by Z ∞ y 0 dy y s Wµnf 0 1 y 0 where Wµnf is the Whittaker function of the µth (naively normalized) spherical principal series Iµnf . As with holomorphic discrete series, we will later renormalize the exponent of y to s − 21 , as a global normalization to give L-functions functional equation s −→ 1 − s rather than s −→ −s. The integral representation obtained above by consideration of an intertwining operator from Iµnf to the Whittaker space gives us what we need: Z
∞
0
y s Wµnf
y 0
0 1
dy = y
Z
∞
ys
0
Z
R
e−2πix
y x2 + y 2
µ
dx
dy y
The obvious thing to do is to interchange the order of integration, and apply the gamma identity, obtaining µ Z ∞Z Z ∞ Z Z ∞ 2 2 dy dy 1 dt 1 dx = dx e−2πix y s+µ e−2πix y s+µ tµ e−t(x +y ) 2 2 x +y y Γ(µ) 0 y t R 0 R 0 √ Replace y by y/ t Z ∞Z Z ∞ µ−s 2 2 dy dt 1 dx e−2πix y s+µ t 2 e−tx e−y Γ(µ) 0 y t R 0 √ Replace y by y 1 2 Γ(µ)
Z
0
∞
Z Z
R
0
∞
e−2πix y
s+µ 2
t
µ−s 2
2
e−tx e−y
Γ( s+µ dt dy 2 ) dx = y t 2 Γ(µ) 5
Z
0
∞
Z
R
e−2πix t
µ−s 2
2
e−tx dx
dt t
Paul Garrett: Standard archimedean integrals for GL(2) (September 23, 2007) Then replace x by x ·
√ √π t
and take the Fourier transform of the Gaussian
√ √ Z ∞Z Z ∞ √ µ−s−1 dt π Γ( s+µ π Γ( s+µ 2 dt π −2πix· √πt µ−s−1 2 ) 2 ) e = e−π· t t 2 t 2 e−πx dx 2 Γ(µ) t 2 Γ(µ) t R 0 0 Replace t by 1/t and then by t · π 2
√ 1 −(s+1−µ) Z ∞ π Γ( s+µ π µ−s− 2 Γ( s+µ s+1−µ dt 2 )π 2 ) = · Γ( s+1−µ e−t t 2 ) 2 2 Γ(µ) t 2 Γ(µ) 0
That is, in a naive normalization, ∞
Z
0
|y|s Wµnf
y 0
0 1
1
π µ−s− 2 dy s+1−µ ) = · Γ( s+µ 2 ) Γ( 2 y 2 Γ(µ)
The less naive normalization replaces µ by µ + 12 , s by s − 21 , and uses Wµ = This gives
∞
Z
1
|y|s− 2 Wµ
0
y 0
0 1
Γ(µ + 12 ) 1
π µ+ 2
dy = y
nf · Wµ+ 1
2
1 2
s−µ · π −s Γ( s+µ 2 ) · Γ( 2 )
Note the nature of the product of gammas, specifically, the symmetry in ±µ.
5. Spherical Rankin-Selberg integrals for GL2(R) R
Let Wµnf and Wνnf be spherical Whittaker functions for the µth and ν th spherical principal series for GL2 ( ) in the naive normalization. The local Rankin-Selberg integral is Z ∞ y 0 dy y 0 Wνnf y s−1 Wµnf 0 1 0 1 y 0 The exponent s−1 arises naturally from unwinding an Eisenstein series with a functional equation s −→ 1−s. Also, one might complex conjugate the second Whittaker function, but this would only complex-conjugate the parameter ν (and replace y by −y, but the function is even). Replace the Whittaker functions by their integral representations, letting ψ(x) = e2πix , and apply the gamma identity: ∞
Z Z Z
R R
=
1 Γ(µ) Γ(ν)
Replace x by x ·
√ √π t
Z
0
∞Z
0
0
∞Z Z Z
R R
Z
∞
0
Z
∞ 0
∞
Z
ψ(xy) 1−ν ψ(uy) dy y dx du (1 + x2 )µ (1 + u2 )ν y 2
2
y s+1−µ−ν ψ(xy) tµ e−t(1+x ) ψ(uy) τ ν e−τ (1+u
0
√ √π τ
and u by u ·
π Γ(µ) Γ(ν)
y s−1 y 1−µ
)
dy dt dτ dx du y t τ
and take the Fourier transforms of the Gaussians, giving
∞
1
y s+1−µ−ν tµ− 2 e−t e−πy
2 π (t)
1
τ ν− 2 e−τ e−πy
2 π (τ )
0
dy dt dτ y t τ
√ Replace y by y and then replace y by y/π 2 , to obtain π µ+ν−s 2 Γ(µ) Γ(ν)
Z
∞ 0
Z
0
∞
Z
∞
y
s+1−µ−ν 2
t
s+µ−ν 2
0
6
e−t e−y(τ +t) τ
s−µ+ν 2
e−τ
dy dt dτ y t τ
Paul Garrett: Standard archimedean integrals for GL(2) (September 23, 2007) Replace t by t/(1 + y) and τ by τ /(1 + y) π µ+ν−s Γ( s+µ−ν )Γ( s−µ+ν ) 2 2 2 Γ(µ) Γ(ν) =
) Γ( s−µ+ν ) π µ+ν−s Γ( s+µ−ν 2 2 2 Γ(µ) Γ(ν) Γ(s)
Z
y
s+1−µ−ν 2
(1 + y)−s
0
∞
Z
∞
0
Z
∞
y
s+1−µ−ν 2
dy y
xs e−x(1+y)
0
dy dx y x
by the gamma identity. Replace y by y/x, to obtain π µ+ν−s Γ( s+µ−ν )Γ( s−µ+ν ) 2 2 2 Γ(µ) Γ(ν) Γ(s) =
Z
∞
0
Z
∞
y
s+1−µ−ν 2
x
s−1+µ+ν 2
e−x e−y
0
dy dx y x
) Γ( s−µ+ν ) Γ( s+1−µ−ν ) Γ( s−1+µ+ν ) Γ( s+µ−ν π µ+ν−s 2 2 2 2 · 2 Γ(µ) Γ(ν) Γ(s)
That is, in the naive normalization, Z ∞ ) Γ( s−µ+ν ) Γ( s+1−µ−ν ) Γ( s−1+µ+ν ) Γ( s+µ−ν π µ+ν−s y 0 y 0 dy s−1 nf nf 2 2 2 2 y Wµ Wν = · 0 1 0 1 y 2 Γ(µ) Γ(ν) Γ(s) 0 For the less naive normalization, replace µ by µ + Wµ =
1 2
and ν by ν + 21 , and recall the renormalization
Γ(µ + 12 ) 1
π µ+ 2
nf · Wµ+ 1
2
Then the local Rankin-Selberg integral is Z ∞ Γ( s+µ−ν ) Γ( s−µ+ν ) Γ( s−µ−ν ) Γ( s+µ+ν ) y 0 dy y 0 2 2 2 2 = π 1−s · Wν y s−1 Wµ 0 1 0 1 y Γ(s) 0 Note the pleasing symmetry in the parameters ±µ and ±ν.
6. Moment integrals for discrete series of GL2(R) In the study of moments of L-functions, we encounter local moment integrals such as ′ Z Z ∞Z ∞ dx dy dy ′ y 0 y 0 W ψ((y − y ′ ) · x) y it y ′−it W 2 w/2 0 1 0 1 y y′ (1 + x ) R 0 0
R
C
where the W ’s are Whittaker functions for irreducibles of GL2 ( ) or GL2 ( ), with ψ(x) = e2πix as usual, and where we have made the simplest useful choice 1/(1 + x2 )w/2 of archimedean data. By Diaconu’s computations, these integrals are not generally elementary, except for holomorphic discrete series of GL2 ( ), where we can evaluate them as follows. For holomorphic discrete series with lowest weights both equal to κ, the local moment integral is Z ∞Z ∞ Z ′ dx dy dy ′ κ/2 ψ((y − y ′ ) · x) y it y ′−it y κ/2 e−2πy y ′ e−2πy (1 + x2 )w/2 y y ′ R 0 0 Z ∞Z ∞Z ′ dx dy dy ′ κ −it+ κ 2 −(2π+2πix)y −(2π−2πix)y e e y it+ 2 y ′ = (1 + x2 )w/2 y y ′ 0 0 R
R
Using the gamma identity in both y and y ′ , this gives Z 1 κ 1 dx κ (2π)−(1+κ) Γ(it + ) Γ(−it + ) κ · κ · 2 2 R (1 + ix)it+ 2 (1 − ix)−it+ 2 (1 + x2 )w/2 7
Paul Garrett: Standard archimedean integrals for GL(2) (September 23, 2007) Because of the simple choice of data, this is (2π)−(1+κ) Γ(it +
κ κ ) Γ(−it + ) 2 2
R
Z
R
1 1 w · w dx κ κ + it+ (1 + ix) 2 2 (1 − ix)−it+ 2 + 2
This is the inner product on of two functions whose Fourier transforms we essentially know from the gamma identity. That is, we know Z
R
e
2πiξx
1 1 dξ = · α+1 (1 + 2πiξ) Γ(α + 1)
xα · e−x 0
(for x > 0) (for x < 0)
Thus, replace x by 2πx in the given integral, to have Z 1 κ 1 κ (2π)−κ Γ(it + ) Γ(−it + ) κ κ w · w dx 2 2 R (1 + 2πix)it+ 2 + 2 (1 − 2πix)−it+ 2 + 2 By the Plancherel identity, Z
R
1 1 1 dx = (1 + 2πix)α+1 (1 − 2πix)β+1 Γ(α + 1) Γ(β + 1)
In the case at hand, α + 1 = it +
κ w + 2 2
and
Z
R
xα e−x · xβ e−x dx =
β + 1 = −it +
2−(α+β+1) Γ(α + β + 1) Γ(α + 1) Γ(β + 1)
κ w + 2 2
Thus, the whole integral is (2π)−κ Γ(it +
κ 2−(κ+w) Γ(κ + w) κ ) Γ(−it + ) 2 2 Γ(it + κ2 + w2 ) Γ(−it + κ2 +
w 2)
R
Thus, for weight κ holomorphic discrete series on GL2 ( ), and with the simplest choice of data, (moment integral) = (2π)−κ · 2−(κ+w) ·
Γ(it + κ2 ) Γ(−it + κ2 ) Γ(κ + w) Γ(it + κ2 + w2 ) Γ(−it + κ2 + w2 )
From the asymptotic result Γ(s) 1 = s−a · (1 + O( )) Γ(s + a) |s| for fixed w and for s =
1 2
(for a fixed)
+ it this moment integral has simple asymptotics, namely (moment integral) ∼ (constant) · t−w
To clarify the dependence upon w, use the duplication formula Γ(s) · Γ(s + 12 ) = 21−2s ·
√
π · Γ(2s)
to rewrite
√ κ+w κ+w+1 κ+w ) = 2κ+w−1 · π · Γ( ) · Γ( )· 2 2 2 Thus, the exponential depending on w disappears, and Γ(κ + w) = Γ(2 ·
(moment integral) = (2π)−κ ·
√ π Γ(it + κ2 ) Γ(−it + κ2 ) Γ( κ2 + w2 ) · Γ( κ2 + · 2 Γ(it + κ2 + w2 ) Γ(−it + κ2 + w2 ) 8
w 2
+ 21 )·
Paul Garrett: Standard archimedean integrals for GL(2) (September 23, 2007)
7. Spherical Whittaker functions for GL2(C) First, we want integral representations of Whittaker functions for spherical principal series with trivial central character for G = GL2 ( ). As usual, let
C
P ={
∗ 0
∗ } ∗
N ={
1 0
∗ } 1
For compatibility with global normalizations, take additive character and norm on ψ
1 0
x 1
C to be
|α|C = |α · α|
= e2πi(x+x)
R
C
where the norm on the right-hand side is the usual norm on . For µ ∈ , let Iµ be the µth (naively normalized) spherical principal series of G with trivial central character, by definition consisting of the collection of all smooth functions f on G such that µ
(for χ(p) = |a/d|C with p =
f (p · g) = χ(p) · f (g)
a 0
∗ d
∈ P , and g ∈ G)
R
As over , computation of the Fourier expansion of Eisenstein series suggests presenting the Whittaker function as an image under an intertwining operator from Iµ to the Whittaker space
C
Wh = {f on GL2 ( ) : f (ng) = ψ(n) · f (g) for all n ∈ N, g ∈ G} Such an intertwining amounts to taking a Fourier transform, as follows. Let the spherical function be y ϕ( 0
∗ 1
µ = y 2µ · k) = |y|C
(for y > 0, k ∈ U (2))
with trivial central character. This spherical function gets mapped to the Whittaker space by an intertwining Wµnf (g) =
Z
C
ψ(x) ϕ(wo nx g) dx
C
where the measure is twice theusual measure on , for compatibility with the normalization of the character. y 0 For diagonal elements my = with y > 0, we can explicitly determine the Iwasawa decomposition 0 1 of wo nx my , namely (see the appendix) wo nx my =
y/r 0
y 0
0 1
∗ r
y/r −x/r
x/r y/r
wo
(where r =
p xx + y 2 )
Therefore, Wµnf
=
y/r ψ(x) ϕ( 0 C
Z
∗ r
y/r −x/r
x/r y/r
wo ) dx
The spherical-ness is right U (2)-invariance, and we are assuming trivial central character, so this simplifies to Wµnf
y 0
0 1
=
Z
C
e−2πi(x+x)
2µ 2µ Z y y −2πi(x+x) dx = dx e 2 2 y + xx y + xx C 9
(for Re(µ) > 21 )
Paul Garrett: Standard archimedean integrals for GL(2) (September 23, 2007)
R
where we reverted to the usual absolute value on , causing the exponent µ to effectively double. Replace x by xy to obtain a useful integral representation, valid for y ∈ × , Z 1 y 0 1−µ ψ(x · y) dx (for Re(µ) > 21 ) = |y|C Wµnf 0 1 (1 + xx)2µ C
C
R
As in the case of GL2 ( ), the integral can be rearranged to show the meromorphic continuation, and explain subsequent renormalization. Namely, using the identity involving Γ(s), for y > 0, y
2−2µ
Replace x by x ·
Z
C
√ √π t
e
−2πi(x+x)y
1 y 2−2µ dx = (1 + xx)2µ Γ(2µ)
Z Z
C
∞
e−2πi(x+x)y t2µ e−t(1+xx) dx
0
dt t
to obtain π |y|1−µ Γ(µ)
∞
Z
Z
C
0
e
√
−2πi(x+x)y· √πt
Taking the Fourier transform of the Gaussian in x ∈ π |y|2−2µ Γ(2µ)
∞
Z
t2µ−1 e−t e−πxx dx
dt t
C, this is
e−πy
2 π ·t
t2µ−1 e−t
0
dt t
Replacing t by π|y| · t gives Wµnf
y 0
0 1
=
π 2µ |y| Γ(2µ)
Z
∞
1
e−π|y|( t +t) t2µ−1
0
dt t
This last integral converges nicely for any complex µ. A less naive normalization of the parametrization of the principal series Iµnf replaces µ by µ + 12 , taking nf Iµ = Iµ+ 1 2
and gets rid of the Γ(2µ) and π 2µ , by taking Γ(2µ + 1) nf · Wµ+ 1 2 π 2µ+1
Wµ = That is, Wµ
y 0
0 1
=
Γ(2µ + 1) π 2µ+1
Z
C
e−2πi(x+x)
|y| xx + y 2
2µ+1
dx = |y|
Z
∞
1
e−π|y|( t +t) t2µ
0
dt t
Thus, in this normalization, visibly W−µ = Wµ
8. Spherical Mellin transforms for GL2(C) C
For GL2 ( ), the Mellin transform of the spherical Whittaker function gives an integral of the form Z
0
∞
|y|sC Wµnf
y 0
0 1
dy = y 10
Z
0
∞
y 2s Wµnf
y 0
0 1
dy y
Paul Garrett: Standard archimedean integrals for GL(2) (September 23, 2007) where Wµnf is the Whittaker function of the µth (naively normalized) spherical principal series Iµnf . As for
R
s− 1
GL2 ( ), later we will rewrite this with |y|sC replaced by |y|C 2 , in order to give L-functions functional equation s −→ 1 − s rather than s −→ −s. To compute this integral in terms of gamma functions, use the integral representation obtained above from an intertwining operator from Iµnf to the Whittaker space: Z
∞
y
2s
0
Wµnf
y 0
0 1
dy = y
Z
∞
y
2s
0
R
Z
C
ψ(x)
y xx + y 2
2µ
dx
dy y
As over , the obvious thing to do is to interchange the order of integration, and apply the gamma identity, obtaining Z Z
C
∞
ψ(x) y
2s+2µ
0
1 xx + y 2
2µ
√ Replace y by y/ t 1 Γ(2µ) Replace y by 1 2 Γ(2µ)
√ y Z ∞Z Z 0
C
Then replace x by x · π Γ(s + µ) 2 Γ(2µ)
∞
Z
∞ 0
dy 1 dx = y Γ(2µ) ∞
Z Z
C
Z
0
∞
0
∞
Z Z
C
∞
ψ(x) y 2s+2µ t2µ e−t(xx+y
2
0
ψ(x) y s+µ tµ−s e−txx e−y
2
0
ψ(x) y 2s+2µ tµ−s e−txx e−y
0
√ √π t
Z
dy dt Γ(s + µ) dx = y t 2 Γ(2µ)
dy dt dx y t
dy dt dx y t
Z
0
∞
Z
C
ψ(x) tµ−s e−txx dx
and take the Fourier transform of the Gaussian √ Z dt π Γ(s + µ) ∞ −π· π µ−s−1 dt π µ−s−1 −πxx t t ψ x· √ dx = e t e t 2 Γ(2µ) 0 t t C
Z
Replace t by 1/t and then by t/π 2 π Γ(s + µ) π −2(s+1−µ) 2 Γ(µ)
Z
∞
e−t ts+1−µ
0
That is, in a naive normalization, Z ∞ y nf s |y|C Wµ 0 0
0 1
π 2µ−2s−1 Γ(s + µ) dt = · Γ(s + 1 − µ) t 2 Γ(2µ)
dy π 2µ−2s−1 = · Γ(s + µ) Γ(s + 1 − µ) y 2 Γ(2µ)
The less naive normalization replaces µ by µ + 21 , s by s − 21 , and uses Wµ =
Γ(2µ + 1) nf · Wµ+ 1 2 π 2µ+1
C
In this normalization, the Mellin transform of the spherical GL2 ( ) Whittaker function is Z ∞ y 0 dy s− 1 = 21 · π −2s Γ(s + µ) · Γ(s − µ) |y|C 2 Wµ 0 1 y 0 As usual, note the nature of the product of gammas, especially, the symmetry in ±µ.
9. Non-spherical principal series For kv either
)
R or C, the unramified characters of kv× are the characters α −→ |α|sv (with s ∈ C) 11
dt t
Paul Garrett: Standard archimedean integrals for GL(2) (September 23, 2007)
R
where the norm on kv is that compatible with the product formula. For kv ≈ , there are also the odd characters α −→ ǫ(α) · |α|sv (with s ∈ , where ǫ(α) = sgn(α) = α/|α|) For kv ≈
C
C, there are also characters α −→
α ℓ · |α|sv |α|
(with s ∈
C and ℓ ∈ Z)
R
Thus, over , in addition to the spherical (or even) principal series, there are the odd principal series Iε,s . For GL2 ( ), there is only one type of non-spherical principal
R
C, in addition to the spherical principal series, there are principal series Iℓ,s indexed by 0 6= ℓ ∈ Z. Further, over both R and C we should allow arbitrary central characters.
Over
Identifying the corresponding integrals and evaluating them is left as a (non-trivial) exercise.
10. Appendix: explicit Iwasawa decompositions To compute the images of various vectors under natural intertwinings from principal series to Whittaker spaces, as well as for other purposes, we need explicit Iwasawa decompositions of group elements. Let 0 −1 1 x y 0 wo = nx = my = 1 0 0 1 0 1 Given x in
R or C and y > 0, let r =
p y 2 + xx. We claim that we have an Iwasawa decomposition
wo nx my =
y/r 0
−x/r r
y/r · −x/r
x/r y/r
· wo
Of course, once asserted, this can be verified by direct computation. However, instead, we should see how the decomposition could be found in the first place. Over either
R or C, the maximal compact K can be defined as
K = {g : g ∗ g = 12 } = {g : g · g ∗ = 12 } Of course, over
(where g ∗ is g-conjugate-transpose)
R the conjugation does nothing. Assuming that g = pk with p upper triangular and k in K, g g ∗ = (pk) (pk)∗ = p(kk ∗ )p = pp∗
Letting p =
a 0
b d
g =
with a > 0 and d > 0, we can solve for a, b, d. Here,
wo nx my wo−1
=
Then gg
∗
=
0 −1 1 0
1 0 −x y
1 0
x 1
1 −x 0 y
a 0
b d
y 0
0 1
=
1 −x x y 2 + xx
From an equality
a2 + bb bd
bd d2
=
12
a b 0 d
∗
=
0 1 −1 0
=
1 −x x y 2 + xx
1 0 −x y
Paul Garrett: Standard archimedean integrals for GL(2) (September 23, 2007) immediately d = r, and b = −x/d = −x/r, and then y 2 + xx − xx y2 = r2 r2
a2 = 1 − bb = 1 − xx/r2 = from which a = y/r. Thus, g =
wo nx my wo−1
=
y/r 0
−x/r r
·k
for some k ∈ K. Then k = p−1 g =
r/y 0
x/ry 1/r
=
=
y/r 0
r/y − xx/ry −x/r
x/ry y/r
=
y/r −x/r
x/ry y/r
Thus, we have the Iwasawa decomposition wo nx my wo−1
−x/r r
y/r · −x/r
x/ry y/r
This can be slightly rewritten as wo nx my =
y/r 0
−x/r r
y/r · −x/r
11. Appendix: Γ(s) Γ(1
x/ry y/r
· wo
s) = π/ sin πs
Take 0 < Re(s) < 1 for convergence of both integrals, and compute Z ∞Z ∞ Z ∞Z ∞ du dv du dv = u e−u(1+v) v 1−s Γ(s) · Γ(1 − s) = us e−u · v 1−s e−v u v u v 0 0 0 0 by replacing v by uv. Replacing u by u/(1 + v) (another instance of the basic gamma identity) and noting that Γ(1) = 1 gives Z ∞ −s v dv 1 +v 0 Replace the path from 0 to ∞ by the Hankel contour Hε described as follows. Far to the right on the real line, start with the branch of v −s given by (e2πi v)−s = e−2πis v −s , integrate from +∞ to ε > 0 along the real axis, clockwise around a circle of radius ε at 0, then back out to +∞, now with the standard branch of v −s . For Re(−s) > −1 the integral around the little circle goes to 0 as ε → 0. Thus, Z Z ∞ −s v −s 1 v dv = lim dv −2πis ε→0 1 − e 1+v Hε 1 + v 0 The integral of this integrand over a large circle goes to 0 as the radius goes to +∞, for Re(−s) < 0. Thus, this integral is equal to the limit as R → +∞ and ε → 0 of the integral from from from from
R to ε ε clockwise back to ε ε to R R counterclockwise to R
This integral is 2πi times the sum of the residues inside it, namely, that at v = −1 = eπi . Thus, Z ∞ −s 2πi π 2πi v · (eπi )−s = πis = dv = Γ(s) · Γ(1 − s) = −2πis −πis 1 + v 1 − e e − e sin πs 0 13
Paul Garrett: Standard archimedean integrals for GL(2) (September 23, 2007)
12. Appendix: Duplication: Γ(s) Γ(s + 12 ) = 21−2s
pπ Γ(2s)
In many regards, this well-known identity is the most conceptually obscure in this whole discussion, at least insofar as the integral representation of the gamma function does not suggest the truth of the relation. From the Eulerian integral definition, Z ∞ Z ∞Z ∞ Z ∞ 1 du 1 dt du dt · e−u us+ 2 = e−t−u (tu)s u 2 e−t ts Γ(s) · Γ(s + 21 ) = t u t u 0 0 0 0 Replace t by t2 and u by u2 , giving 4
Z
∞
0
Z
∞
2
e−t
−u2
(tu)2s−1 u dt du
0
To symmetrize this expression, add it to itself, with the roles of t and u interchanged, and divide by 2, obtaining Z ∞Z ∞ 2 2 2 e−t −u (tu)2s−1 (t + u) dt du 0
0
We will make the change of variables x = t2 + u2 and y = 2tu. Thus, √ √ t−u = x−y t+u = x+y and t =
1 2
·
√ √ x+y+ x−y
u =
1 2
·
√ √ x+y− x−y
which suggests that we should be sure to have x − y, and, to avoid ambiguity, have t − u > 0 as well. Thus, take twice the integral over 0 < u < t, namely, Z ∞Z t 2 2 4 e−t −u (tu)2s−1 (t + u) du dt 0
0
The change of measure is ux vx
1 1 √ + √ 4 x+y 4 x−y uy = 1 vy √1 4 x + y − 4√x − y
1 1 √ − √ 4 x+y 4 x−y 1 1 √ + √ 4 x+y 4 x−y
= 1·p 1 4 x2 − y 2
Thus, the integral becomes Z ∞ Z ∞Z x Z 1 2s−1 1 1 1−2s −x 2s− 21 −x y √ dy dx = 2 e x e dx · y 2s−1 (1 − y)− 2 dy x − y 2 0 0 0 0 Z 1 1 = 21−2s Γ(2s + 21 ) · y 2s−1 (1 − y)− 2 dy 0
by replacing y by xy. The integral in y is a beta integral Z 1 Z 1 Γ(2s) Γ( 12 ) 1 1 y 2s−1 (1 − y) 2 −1 dy = B(2s, 21 ) = y 2s−1 (1 − y)− 2 dy = Γ(2s + 12 ) 0 0 Thus, the whole is 21−2s Γ(2s + 21 ) ·
√ Γ(2s) Γ( 12 ) = 21−2s Γ(2s) · π 1 Γ(2s + 2 )
In summary, we have Legendre’s duplication formula, Γ(s) · Γ(s + 21 ) =
14
√
π 21−2s Γ(2s)
