Transition to the Adele Group

1 Transition to the Adele Group This lecture transfers functions on the complex upper half plane that satisfy classical conditions to functions on a Lie group that satisfy more natural conditions, and then transfers these functions further to functions on an adele group where we can naturally gather them. Enlarging the Lie group domain gathers all the weights together, and then enlarging the adele group domain gathers all the levels. In particular, all of this applies to Hecke operators and to Fourier coefficients. 1.1 Classical Domain Consider the classical domain H = {z C : Im(z) > 0 and the classical group {[ ] G + a b Q = GL+ 2 (Q) = c d M 2 (Q) : ad bc > 0. The group G + Q acts on the domain H via fractional linear transformations, The standard factor of automorphy is This satisfies the cocycle condition γz = az + b cz + d. j : G + Q H C, j(γ, z) = cz + d. j(γγ, z) = j(γ, γ z)j(γ, z).

2 1 Transition to the Adele Group For any γ G + Q and any κ Z, the weight κ operator associated to γ is [γ] κ : {functions f from H to C {functions f from H to C, taking each f to f[γ] κ, where (f[γ] κ )(z) = f(γz)j(γ, z) κ (det γ) κ/2. For each positive integer L, the classical group G + Q contains the principal congruence subgroup of level L, Γ L = {γ SL 2 (Z) : γ [ 1 0 0 1 ] (mod LZ). Let p be prime, and let p (L) = {δ M 2 (Z) : det(δ) = p, δ [ ] 1 0 0 p (mod LZ). The Hecke operator T p,l acts on [Γ L ] κ -invariant functions f : H C as T p,l f = δ Γ L \ p(l) which is again [Γ L ] κ -invariant because Γ L normalizes p (L). Suppose that the function f : H C satisfies f(z + L) = f(z). f[δ] κ, (1.1) For example, this holds if f is [Γ L ] κ -invariant for any κ. Let z = x + iy and think of f as an LZ-periodic function of the variable x with parameter y. For any integer l let ψ l/l (x) = e 2πilx/L, x R. The set of such characters is a complete orthonormal set for R/LZ, and so the Fourier series of f is f(z) = l Z ψ l/l (x)c l/l (f, y), where the (l/l)th Fourier coefficient is c l/l (f, y) = L 1 R/LZ ψ l/l (ξ)f(ξ + iy) dξ. If f is C 2 then the Fourier series of f converges pointwise to f. If L L, so that also f is L -periodic, then the Fourier series of f at the two levels agree. We will show this as part of a more general calculation in the adele group environment.

1.2 Classical Domain to Lie Group Next consider the Lie group associated to G + Q, 1.2 Classical Domain to Lie Group 3 G + R = GL+ 2 (R). This group acts on the domain H via fractional linear transformations. Let z 0 = i. To transfer functions on H to functions on G + R, fix any κ Z and define a map by φ κ : {functions from H to C {functions from G + R to C ( ) ( f H : z f H (z) f R,κ : g f H (gz 0 )j(g, z 0 ) κ (det g) κ/2). The point z 0 is fixed by a compact subgroup of G + R, K + R = SO(2). The entire fixing subgroup is R + K + R but we want the compact group. The cocycle condition and the fact that K + R fixes z 0 combine to show that there is a character χ κ : K + R C, χ κ (k) = j(k, z 0 ) κ. For any function f H : H C and any k K + R, the corresponding function φ κ f = f R,κ : G + R C satisfies f R,κ (gk) = f R,κ (g)χ κ (k). That is, f R,κ is right K + R -equivariant of type κ. In fact φ κ is a bijection from the functions on H to the type κ right K + R -equivariant functions on G+ R. However, since we soon will be considering a larger class of functions in the Lie group environment and then a larger one yet in the adele group environment, this isn t particularly important. In particular, [ ] κ -invariance on H transfers to a more natural invariance on G + R. For any γ G+ R define the composition operator by γ : {functions from G + R to C {functions from G+ R to C (f γ)(g) = f(γg). Then for any γ G + Q and any κ Z, the following diagram commutes: { φ κ type κ right K + R {functions from H to C -equivariant functions from G + R to C [γ] κ {functions from H to C φ κ γ { type κ right K + R -equivariant functions from G + R to C.

4 1 Transition to the Adele Group If f H : H C is [Γ L ] κ -invariant then the diagram shows that the corresponding function f R,κ : G + R C is left Γ L-invariant, i.e., f(γg) = f(g) for all γ Γ L. In sum, φ κ restricts to a bijection for any L, { { [ΓL ] κ -invariant left ΓL -invariant, type κ right φ κ,l : functions on H K + R -equivariant functions on. G+ R The conditions on the right side are considerably tidier, defined on the Lie group G + R rather than the domain H, so that now Γ L is a subgroup (as LZ is a subgroup of R in the one-dimensional situation), and separating the awkward condition of [ ] κ -invariance under Γ L as a group that acts on H into the natural conditions of left invariance under Γ L and type κ right equivariance under K + R as subgroups of G+ R. The Hecke operator T p,l on left Γ L -invariant functions f R : G + R C is defined to make the following diagram commute: T p,l f H φ κ,l f R T p,l T p,l f H φ κ,l T p,l f R. That is, T p,l f R = (T p,l f H ) R. For any γ G + Q and g G+ R, since (f H [γ] κ ) R (g) = f R (γg) by the earlier commutative diagram, it follows from (1.1) that the Lie group Hecke operator is T p,l f R = f R δ. (1.2) δ Γ L \ p(l) We next enlarge the Lie group environment. Rather than continue keeping track of each weight κ separately, consider left Γ L -invariant functions on G + R, or equivalently, functions on the quotient space Γ L \G + R. For each κ the map φ κ,l can be viewed as is an injection into this κ-independent space, φ κ,l : {[Γ L ] κ -invariant functions on H C(Γ L \G + R ). In particular, the Lie group definition of T p,l in (1.2) makes no reference to a weight or to right equivariance. Now we consider Fourier series on G + R. This is facilitated by the Iwasawa decomposition, G + R = N RM + R K+ R where the first two subgroups are {[ ] {[ ] 1 x N R = : x R, M + R = y1 0 : y 0 y 1, y 2 R +, 2

1.2 Classical Domain to Lie Group 5 and as before, K + R = SO(2). Indeed, recall that z 0 = i, and for any g G + R let gz 0 = z = x + iy. Then also z = n x m y z 0 where [ ] [ ] 1 x y 0 n x =, m y =. So (n x m y ) 1 g takes the form rk R + K + R, and thus g = n x rm y k as desired. It is easy to see that the decomposition is unique. Suppose that the function f : G + R C satisfies ([ ] ) 1 L f g = f(g). Consider the group N Z,L = {[ ] 1 LZ N R. Let g = nmk and think of f as a left N Z,L -invariant function of the variable n with parameter mk. For any integer l, let [ ] ψ l/l (n x ) = e 2πilx/L 1 x, n x = N R. The Fourier series of f is f(g) = l Z ψ l/l (n)c l/l (f, mk), where the (l/l)th Fourier coefficient is c l/l (f, mk) = L 1 N Z,L \N R ψ l/l (ν)f(νmk) dν. (1.3) This discussion is set in C(N Z,L \G + R ), a larger space than C(Γ L\G + R ) earlier in the section. For any weight κ the [Γ L ] κ -invariant functions on H are LZperiodic, and there is a commutative diagram of injections { [ΓL ] κ -invariant φ κ,l C(Γ L \G + R functions on H ) { LZ-periodic functions on H C(N Z,L \G + R ), but the map across the bottom row depends on κ. That is, no map from the LZ-periodic functions on H to C(N Z,L \G + R ) is compatible with different weights. The simplest, most natural map across the bottom is the one compatible with weight 0.

6 1 Transition to the Adele Group 1.3 Lie Group to Adele Group Introduce the global notation G = GL 2, so that G Q denotes GL 2 (Q) and G R denotes GL 2 (R). Similarly the associated adele group is G A = GL 2 (A). This group contains a copy of G Q embedded at each component and therefore a copy of G Q embedded diagonally. The group G Q is identified with its diagonally embedded image in G A, and the group G R is naturally identified with the infinite component of G A. The group G A contains a compact subgroup associated to any positive integer L, as follows. For each finite place v of Q, the local group is G v = GL 2 (Q v ). Define a compact subgroup of the local group, K v = GL 2 (Z v ), and define an L-dependent compact subgroup of the local group as well, K v,l = {g K v : g 1 (mod LZ v ). (Here 1 denotes the 2 2 identity matrix.) Thus K v,l = K v if v L. The product K f,l = v< K v,l is the finite global compact subgroup of G A associated to L. Recall the group K + R = SO(2). This group embeds in G A at the infinite place, and so it makes sense to enlarge the subgroup K f,l of G A to K + A,L = K f,lk + R. The same applies to the group K R = O(2), giving K A,L = K f,l K R. Proposition 1.3.1. The embedding of G + R in G A induces a map ι L : Γ L \G + R G Q\G A /K f,l, Γ L g G Q gk f,l. (1.4) This map is an injection.

1.3 Lie Group to Adele Group 7 Proof. To see that ι L is well defined, replace g by γg for any γ Γ L. Let γ f γ R be the diagonally embedded image of γ in G A. Compute that G Q γ R gk f,l = G Q γ 1 γ R gγ f K f,l = G Q gk f,l. since γ 1 G Q and γ f K f,l To see that ι L is injective, suppose that ι L (Γ L g ) = ι L (Γ L g), i.e., g = γgk, g, g G + R, γ G Q, k K f,l. Then γ f = k 1 f, and so γ GL 2 (Z) and γ 1 (mod L). The relation at the -place (i.e., in G + R ) is g = γg, so that γ Γ L and thus Γ L g = Γ L g. The fact that ι L is an injection automatically gives a surjection π L : C(G Q \G A /K f,l ) C(Γ L \G + R ). Letting suitably invariant functions act on elements rather than on cosets, the inverse image of a function f R : G + R C is the set of all functions f A : G A C satisfying f A (g R ) = f R (g R ), g G + R, (1.5) where the first g R in the display is the embedded image of the second one. If f R is type κ right K + R -equivariant and f A(g f g ) = f A (g f )f A (g ) then f A is type κ right K + A,L-equivariant, meaning f A (gk) = f A (g)χ κ (k ), k K + A,L. But this condition does not hold for all f A in π 1 L f R. The Hecke operator T p,l on left G Q -invariant, right K f -invariant functions f A : G A C is defined to make the following diagram to commute: T p,l f A π L f R T p,l T p,l f A π L T p,l f R. So compute, using (1.2) and then (1.5), that going across the top of the diagram and then down the right side gives for any g R G + R, ( ) (T p,l π L f A )(g R ) = π L f A δ (g R ) = (π L f A )(δg R ) = f A (δ R g R ). δ δ δ Since δ G Q it follows that G Q δ R = G Q δ 1 f, and so each summand is f A (δ R g R ) = f A (δ 1 f g R ) = f A (g R δ 1 f ).

8 1 Transition to the Adele Group Let v be the finite place corresponding to p, and identify δv 1 G v with its image in G A. Since δw 1 K w,l for all other finite places w, it follows that δ 1 f K f,l = δv 1 K f,l and thus f A (g R δ 1 f ) = f A (g R δv 1 ). In sum, the Hecke operator on C(G Q \G A /K f,l ) is suitably defined as (T p,l f)(g) = f(gδv 1 ), g G A. (1.6) δ Γ L \ p(l) This makes it obvious that T p,l T q,l = T q,l T p,l for distinct primes p and q. Just as enlarging the Lie group environment gathers all the weights together, enlarging the adele group environment gathers all the levels. Take the (inverse) limit lim L (G Q \G A /K f,l ) = G Q \G A. (Email from Paul: Let K be a compact totally disconnected group acting effectively from the right on a topological space X, meaning that xk x unless k = 1. Let K i be a basis at the identity in K. Then lim i (X/K i ) = X.) In the Lie group environment, lim L (Γ L \G + R ) has no convenient description. A left G Q -invariant function f : G A C that is defined on some quotient G Q \G A /K f,l is called uniformly locally constant on the right. The transition from the classical domain to the Lie group and then the adele group is summarized in the following diagram, in which φ κ,l and ϕ L are injections and π L is a surjection: C(G Q \G A /K f,l ) ϕ L C(G Q \G A ) { [ΓL ] κ -invariant functions on H π L φ κ,l C(Γ L \G + R ) As mentioned, C(Γ L \G + R ) incorporates all weights and C(G Q\G A ) incorporates all levels. If we had used SL 2 as our group throughout the discussion then the strong approximation theorem would make the vertical arrow a bijection, but we need the larger group for the Hecke operators and for pending calculations with the Fourier coefficients. A left G Q -invariant function f : G A C is called an automorphic form. An automorphic form is called a cusp form if N Q \N A f(νg) dν = 0, g G A. This will be explained soon. The level L Hecke operator T p,l on C(G Q \G A /K f,l ), defined by (1.6), does not obviously extend to C(G Q \G A ) since the right K f,l -invariance of f is required to make the summand well defined. However, T p,l can be rewritten

1.3 Lie Group to Adele Group 9 as an integral that does extend. Let v be the finite place corresponding to p. Recall the groups G v and K v,l. Define v (L) = {δ M 2 (Z v ) : det δ pz v, δ [ ] 1 0 0 p (mod LZ v ). Let η L be the characteristic function of 1 v (L) on G v. Since K v,l is compact and open in G v, the quotient G v /K v,l is discrete. Also view η L as defined on G v /K v,l. Let µ L = µ(k v,l ). If µ 1 is normalized to 1 and L = v< pev v then although the value of µ L is not particularly important, we can note that µ 1 L = K v/k v,l = GL 2 (Z v /LZ v ) = GL 2 (Z/p ev v Z). For any f C(G Q \G A /K f,l ) and any g G A, compute G v f(gh)η L (h) dh = µ L h G v/k v,l f(gh)η L (h) = µ L δ K v,l \ v(l) f(gδ 1 ). The embedding G Q G v induces a bijection Γ L \ p (L) K v,l \ v (L). (Check surjectivity.) Consequently, for any f C(G Q \G A /K f,l ) the integral gives the Hecke operator, f(gh)η L (h) dh = µ L f(gδv 1 ), g G A. G v δ Γ L \ p(l) But the integral makes sense for any f C(G Q \G Q ). Therefore it is the definition of the level L Hecke operator T p,l on C(G Q \G A ), (T p,l f)(g) = µ 1 L f(gh)η L (h) dh, g G A. G v And once we stop thinking about the classical environment, we can drop the artifact-constant µ 1 L from this definition. More generally, the local Hecke algebra is a convolution algebra. Let Cc (G v ) denote the set of locally constant functions on G v with compact support. If η 1 and η 2 belong to Cc (G v ) then so does their convolution η 1 η 2, defined as (η 1 η 2 )(g) = η 1 (gh 1 )η 2 (h) dh, g G v. G v For any η Cc (G v ) and any integrable function f : G v C, define the integral operator (η f)(g) = f(gh)η(h) dh, g G v. G v Then for all η 1, η 2, and f, (η 1 η 2 ) f = η 1 (η 2 f).

1 Transition to the Adele Group The spherical Hecke algebra (or the Iwahori Hecke algebra) is the subalgebra of functions η that are left and right K v -invariant. Gelfand showed that the spherical Hecke algebra is commutative, using the Borel Matsumoto theorem. We return to Fourier coefficients. Recall the definition N Z,L = {[ 1 LZ ]. There is a diagram similar to the previous one, C(N Q \G A /K f,l ) ϕ L C(N Q \G A ) π L C(N Z,L \G + R ) Since now the group is GL 2 rather than GL + 2, the Iwasawa decomposition at the infinite component changes to G R = N R M R K R where the new subgroups are {[ ] y1 0 M R = : y 0 y 1, y 2 R, K R = O(2). 2 The Iwasawa decomposition of each finite local component G v = GL 2 (Q v ) is G v = N v M v K v where the first two subgroups are {[ ] {[ ] 1 x y1 0 N v = : x Q v, M v = : y 0 y 1, y 2 Q v, 2 and as before, K v = GL 2 (Z v ). To see this, compute that for any g = [ ] a b c d G v, any n = [ 1 0 x 1 ] N v, and any m = [ y 1 0 ] 0 y 2 Mv, [ ] [ ] [ ] [ ] m 1 n 1 y 1 g = 1 x a b y 1 0 y2 1 = 1 (a xc) y 1 1 (b xd) c d y2 1 c y 1 2 d. Since c and d can t both be zero, at least one of c/d, d/c is defined and integral. If c/d is integral then set y 1 = /d, y 2 = d, and x = b/d to make the right side [ ] 1 0 c/d 1. And if d/c is integral then set y1 = /c, y 2 = c, and x = a/c to make the right side [ ] 1 d/c. The Iwasawa decomposition is no longer unique, but n and mk are still unique at the infinite component. Suppose that the function f : G A C is left N Q -invariant. Even though the Iwasawa decomposition is not unique, think of f as a function of the variable n N A with parameter mk M A K A. Then the Fourier expansion of f is a sum over characters ψ on N Q \N A. At least in some L 2 sense,

1.3 Lie Group to Adele Group 11 f(g) = ψ ψ(n)c ψ (f, mk) where the Fourier coefficients are c ψ (f, mk) = ψ 1 (ν)f(νmk) dν. (1.7) N Q \N A The measure here is normalized so that µ(n Q \N A ) = 1. When ψ is the trivial character, call this the 0th Fourier coefficient or the constant term along N. This explains the definition of cusp form earlier in the section. Since N Q \N A is compact, if f is continuous then the integral exists. Also if f is locally constant at each finite place and is C at the infinite place then the Fourier series converges to f pointwise. Despite the nonunique decomposition, it is straightforward to verify that the Fourier series is well defined, i.e., if nmk = n m k then ψ(n)c ψ (f, mk) = ψ(n )c ψ (f, m k ). The adele group Fourier series is indeed a generalization from the Lie group. Suppose f A C(N Q \G A ) takes the form f A = ϕ L f A,L for some f A,L C(N Q \G A /K f,l ). Define N A,L = { N v K v,l v <, N v,l, N v,l = N v R v =. Then N Q N A,L \N A = NZ,L \N R. Thus for any character ψ of N Q \N A and for any element g R of G + R embedded in G A, we may take an Iwasawa decomposition of g R in G + R and then the ψth Fourier coefficient of f A(g R ) works out to c ψ (f A, m R k R ) = ψ 1 (ν)f A,L (νm R k R ) dν N Q N A,L \N A ψ 1 (κ) dκ. N Q \N Q N A,L The second integral is 0 unless ψ is a character of N Q N A,L \N A, in which case it is µ(n Q \N A )/µ(n Q N A,L \N A ) = L 1. Let f R = π L f A,L, and view ψ as a character of N Z,L \N R. Then the Fourier coefficient is c ψ (f A, m R k R ) = L 1 N Z,L \N R ψ 1 (ν)f R (νm R k R ) dν. Comparing this to (1.3) shows that the adele group Fourier series of f A (g R ) is the Lie group Fourier series of f R (g R ), as claimed. Let f : N Q \G A C be left M A -equivariant, meaning that f(mg) = χ(m)f(g) for some character χ of M A. Such a function gives rise to essentially only one non-0th Fourier coefficient. Since N Q \N A = A/Q, again blur the

12 1 Transition to the Adele Group notation by writing ψ ([ 1 0 x 1 ]) = ψ(x). Fix any nontrivial character ψ on A/Q. Then all other characters take the form ψ α (x) = ψ(αx), α Q. So ψ α ranges through the nontrivial characters as α ranges through Q. For any such α, the ψ α th Fourier coefficient of f on G Q \G A at g is c ψα (f, mk) = ψα 1 (x)f ([ 1 0 x 1 ] mk) dx = ψ 1 (αx)f ([ 1 0 x 1 ] mk) dx A/Q A/Q = ψ 1 (x)f ([ ] ) 1 x/α mk dx. [ α 1 0 A/Q The last equality here uses the fact that d(x/α) = dx/ α = dx. But [ ] 1 x/α = ] [ 1 x ] [ α 0 0 1 ] and f is left M A-equivariant, so this is c ψα (f, mk) = χ( [ ] α 1 0 )cψ (f, [ α 0 0 1 ] mk). The calculation here makes essential use of GL 2 rather than SL 2.