May 8, 26 Transition: Eisenstein series on adele groups Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/mfms/notes 23-4/2 2 transition Eis.pdf]. Moving automorphic forms from domains to groups 2. Iwasawa decompositions of GL 2 R and GL 2 Q p 3. Rewriting the GL 2 Eisenstein series 4. Bruhat decomposition for GL 2 5. Application: constant term of GL 2 Eisenstein series 6. Application: Hecke operators on GL 2 Eisenstein series A simple Eisenstein series, for z H, Γ SL 2 Z, is E s z Imγz s 2 Γ \Γ c,d coprime y s cz + d 2s with γ, Γ { Γ} This Eisenstein series can be rewritten to exhibit the role of p-adic groups SL 2 Q p, thereby exhibiting Hecke operators as integral operators, parallel to the relevance of the representation theory of SL 2 R to invariant differential operators. [] The explicit nature of Eisenstein series allows vivid illustration of some basic mechanisms. For SL 2 Q, one can survive without this viewpoint. However, for SL 2 over number fields with non-trivial class groups, for SL n with n > 2, and for more general groups, the neo-classical elementary ideas sufficient for SL 2 Q fall short. Even in the simplest case of SL 2 Z, understanding the role of the p-adic groups SL 2 Q p greatly simplifies and clarifies many classical issues. That is, this shift in viewpoint is helpful, not merely a stylistic choice.. Moving automorphic forms from domains to groups Automorphic forms on domains such as the upper half-plane H are usually introduced first because it is easier to discuss them. However, this picture hides important features about the representation theory of SL 2 R. It is easy to convert automorphic forms on H, both waveforms and holomorphic modular forms, to automorphic forms on SL 2 R. [2] [.] Γ-invariant functions and SO 2 R-invariance Choice of maximal compact subgroup SO2, R in SL 2 R amounts to choice of basepoint i H, as SO 2 R is the isotropy subgroup of i. Via the isomorphism SL R/SO 2 R H by gso 2 R gi, and function f on H can be converted to a function F on SL 2 R by F g fgi [] Despite contrary assertions in the literature, rewriting Eisenstein series, as opposed to more general automorphic forms, on adele groups does not use Strong Approximation. Strong Approximation does make precise the relation between general automorphic forms on adele groups and automorphic forms on SL 2 and even on SL n, but rewriting these Eisenstein series does not need this comparison. Indeed, Strong Approximation does not hold in the simplest form for general semi-simple or reductive groups, but this does harm anything. Strong approximation does show that there can be no other extension of the Eisenstein series to the adele group beyond that described here. [2] This is an example of conversion to automorphic forms on reductive or semi-simple real Lie groups, of which SL 2 R is a small example.
Paul Garrett: Transition: Eisenstein series on adele groups May 8, 26 The function F is right SO 2 R-invariant: F gk fgki fgi F g for g SL 2 R and k SO 2 R Oppositely, any such right-so R-invariant function F on SL 2 R descends to a function f on the quotient H by fz F g z for any g z SL 2 R with g z i z For f also left invariant by Γ SL 2 Z on H, the corrersponding function F on SL 2 R is left Γ-invariant, and vice-versa: F γg fγgi fgi F g for γ Γ and g SL 2 R [.2] Automorphy condition and SO 2 R-equivariance To motivate extension of the treatment of left Γ-invariant function on H, of course holomorphic modular forms f on H are not quite left Γ-invariant, but satisfy an automorphy condition fγz jγ, z fz where jαβ, z jα, βz jβ, z Apart from the trivial cocycle jg, z, the simplest example of cocycle is jg, z cz + d 2k for g. This extends to be a cocycle defined not merely on Γ H but on SL 2 R H. Associate the function F on SL 2 R given by F g jg, i fgi [.2.] Claim: The function F is left Γ-invariant on SL 2 R, and right SO 2 R-equivariant by F gk jk, i F g and k jk, i is a group homomorphism SO 2 R C. Proof: This is a direct computation. For γ Γ, F γg jγg, i fγgi jγ, gi jg, i jγ, gi fgi jg, i jγ, gi jγ, gi fgi jg, i fgi F g For k SO 2 R, F gk jgk, i fgki jg, ki jk, i fgi jg, i jk, i fgi jk, i jg, i fgi jk, i F g Finally, for k, h SO 2 R, jhk, i jh, ki jk, i jh, i jk, i proving that k jk, i is a group homomorphism on SO 2 R. /// [.2.2] Remark: Half-integral weight automorphic forms have a cocycle on Γ H which does not extend to SL 2 R H. The obstruction to this extension defines a two-fold covering group of SL 2 R, the metaplectic group, the group where half-integral weight automorphic forms live. This and other complications account for our emphasis on integral-weight holomorphic modular forms, and on waveforms, in these notes. 2
Paul Garrett: Transition: Eisenstein series on adele groups May 8, 26 [.2.3] Remark: The seemingly different analytic conditions, holomorphy and being an eigenfunction for the invariant Laplacian, both become eigenfunction conditions on the group SL 2 R. We will return to this a little later. [.3] Conversion to automorphic forms on GL 2 R To eventually accommodate Hecke operators, and for many other reasons, the group GL 2 is better than SL 2 R. Already, the subgroup preserves the upper half-plane H, and GL + 2 R {g GL 2R : det g > } GL + 2 Z {g GL 2Z : det g > } SL 2 Z since g GL 2 Z has determinant ±. Now the isotropy group of i H is As with SL 2 R, we have a Z { : a R } a H GL + 2 R/isotropy subgroup of i GL+ 2 R/Z+ R GL+ 2 Z For left GL + 2 Z-invariant functions f on H, the associated function F on GL+ 2 R is F g fgi and F is left GL + 2 Z-invariant, right SO 2R-invariant, and Z-invariant. Since Z is the center of GL 2 R, the function F is both right and left Z-invariant. For f not left GL + 2 Z-invariant, but only meeting a cocycle condition fγz jγ, z fz with a cocycle extending to GL + 2 R H, such as jg, z cz + d 2k det g k with g GL + 2 R for weight 2k holomorphic modular forms, the corresponding function F on GL + 2 R is again F g jg, i fgi as for SL 2 R. Some cocycles are trivial on Z, some are not. Further, the positive-determinant condition can be removed when the function f on H is extended to be a function on H H, since GL 2 R stabilizes the union of both half-planes, producing functions F on GL 2 R that are left GL 2 Z-invariant, right O 2 R-equivariant by some representation of O 2 R, and Z-equivariant by some character Z C. 3
Paul Garrett: Transition: Eisenstein series on adele groups May 8, 26 2. Iwasawa decompositions of GL 2 R and GL 2 Q p That g GL 2 R can be written as a product g pk of upper-triangular p and orthogonal matrices k was known for a long time before K. Iwasawa s work in the 94s, but after Iwasawa s work this and other related classical facts are understood as instances of a very general pattern that applies, not only to real or complex matrix groups, but also to related p-adic matrix groups. The pattern of an Iwasawa decomposition in a real or complex or p-adic matrix group is essentially [3] whole group upper-triangular elements maximal compact subgroup The subgroup P of upper-triangular matrices is a parabolic subgroup. The general definition of parabolic is not essential here, nor is determination or certification that various subgroups are maximal compact. Rather, we are acknowledging that our present examples fit into a larger pattern. [2.] Iwasawa decomposition for GL 2 R Let G GL 2 R. Given g G, choose the element k K O 2 R to right multiply by to put gk into the group P of upper-triangular real matrices, by rotating the lower half of g in R 2 into the form. Indeed, realizing that matrices in K are of the form cos θ sin θ for θ R sin θ cos θ in particular with the squares of the left column summing to, choose k d c c 2 +d 2 with the right column of course completely determined by the left, so that gk d c 2 +d 2 c d c + d c c 2 +d 2 c 2 +d 2 as desired. For application to Eisenstein series, we only need the diagonal entries of gk, and we easily further compute the lower-right entry gk d c c 2 +d 2 c d c 2 +d 2 c2 + d 2 For g SL 2 R, the determinant-one condition gives K c2 + d 2 P K [3] Probably the group should be reductive or semi-simple, whose formal definitions do not concern us for the moment. Rather, we note that this class of groups includes important examples: GL nr, GL nr, GL nc, GL nc, as well as orthogonal groups, unitary groups, and other matrix groups defined by preservation of additional structure. The class does not include upper-triangular matrices in GL nr, although this subgroup is important in its own right. This class of groups does also include p-adic versions of groups over R and C. 4
Paul Garrett: Transition: Eisenstein series on adele groups May 8, 26 For general g G, det g K c2 + d 2 P K [2.2] Iwasawa decomposition for GL 2 Q p Even though the substance of a p-adic Iwasawa decomposition is quite different from that of archimedean Iwasawa decompositions, the commonalities are very useful. The standard maximal compact subgroup K v of G v GL 2 Q v with v a finite prime, in sharp contrast to the orthogonal group as maximal compact subgroup of GL 2 R, is K v GL 2 Z v {p-adic integral matrices with determinants in Z v } v is finite prime p It is not so hard to prove that this is compact, but a little harder to prove that it is maximal compact. But, for the moment, we don t use either property, especially not the maximality. The essential aspect of Q p is that, for any two x, y Q p, either x/y Z p or y/x Z p, since Z p {z Q : z p } This contrasts violently with the classical situation of rational numbers x, y, where the classical archimedean! notion of size is very far from a sufficient determiner of divisibility. Thus, given g GL 2 Q p, d c for c v d v for c v d v In both cases, the right-multiplying matrix is in K v GL 2 Z v : in the first case c/d v implies c/d Z v, and in the second d/c v implies d/c Z v. In the first case, the resulting product is already in the group P v of upper-triangular matrices in G v. In the second, further right-multiplication by the long Weyl element w puts the product into P v. This proves that the p-adic Iwasawa decomposition succeeds. Specifics about the diagonal entries in the Iwasawa decomposition will be needed in rewriting Eisenstein series: a bc c d for c d d v d v d c ad c + b for c c v d v That is, a bc d K d v for c v d v ad c + b K c v for c v d v [2.2.] Remark: It is very convenient that the shape of the parabolic P v is the same for both archimedean and non-archimedean v, while the shape of the maximal compact subgroup K v is significantly different. 5
Paul Garrett: Transition: Eisenstein series on adele groups May 8, 26 3. Rewriting the GL 2 Eisenstein series With Γ SL 2 Z and Γ { Γ}, the simplest waveform-type Eisenstein series on the domain H is the usual E s z Im γz s for z H and Res > Γ \Γ This Eisenstein series on the domain H is easily converted to a left Γ-invariant, right SO 2 R-invariant function on the group GL + 2 R, still called an Eisenstein series, by E group s g E s gi for g GL + 2 R The basepoint i H is chosen since K SO 2 R is its isotropy group, which immediately explains the right SO 2 R-invariance. In fact, E s can be extended to an automorphic form on the union of upper and lower half-planes by E s z E s z, noting that z z Thus, now letting Γ GL 2 Z and Γ { Γ}, we have an expression for the extended Eisenstein series that is of the same form: E s z Im γz s for z H H Γ \Γ The Eisenstein series on the whole group G GL 2 R is E group s g E s gi for g GL 2 R [3.] The localized rewrite As above, let v be an index for completions [4] of Q, with Q v the v th completion, and Z v the v-adic integers for v a prime, with Q R, and the usual real absolute value. The latter is distinguished from genuine primes by the convention of calling it the infinite prime, in analogy with a more legitimate use of this in the function-field setting. For each place v, finite and infinite, using the v th Iwasawa decomposition G v P v K v, define a function ϕ v on G v by ϕ v d k a d s v for k GL 2 Z v cos θ sin θ for k O2 sin θ cos θ for v non-archimedean for v archimedean where in all cases a, d Q v and b Q v. By design, each ϕ v is invariant under the center Z v of G v. Define a function on the adele group G A GL 2 A by ϕ v ϕ v meaning ϕ{g v } v ϕ vg v, where g v G v [4] Another tradition for refering to p-adic completions, as well as to R, is as places of Q. Also, sometimes is called the infinite prime, while the actual primes p are finite primes. 6
Paul Garrett: Transition: Eisenstein series on adele groups May 8, 26 [3..] Claim: The product formula For x Q, x v product over all p-adic norms as well as R v Proof: Since the assertion is multiplicative, it suffices to prove the product formula for units and for primes. All norms of ± are, and all norms but the archimedean and p th of a prime p are. Since p p and p p p, we have the product formula. /// The product formula shows shows that ϕ is left P Q -invariant: using the Iwasawa decomposition locally everywhere, with k v K v, A a ϕ k Aa/Dd s A/D s a/d s A/D s a ϕ k D d d Let a Z A { a : a, a A} [3..2] Theorem: For g GL 2 R, acting as usual on H H, E s g i ϕγ g This expression gives a left G Q -invariant, Z A -invariant function still denoted E s on the adele group G A GL 2 A: Es adelic g Proof of this will occupy the rest of this section. ϕγ g [3..3] Remark: The notations Es group and Es adelic are not standard, but have the obvious descriptive utility. In fact, we will revert to writing E s for the Eisenstein series on G A GL 2 A. [3.2] Disambiguation The immediate question arises of evaluation of ϕγ g. One point is that G Q GL 2 Q should not be considered as only a subgroup of G GL 2 R, but also a subgroup of every G v GL 2 Q v. Thus, G Q is best considered as imbedded on the diagonal in G A. Adding a temporary notational burden for clarity, for each place v let j v : GL 2 Q GL 2 Q v be the natural injective map, and let j j v : GL 2 Q GL 2 Q v v v be the natural diagonal map to the product. Then ϕγ g ϕ j γ g ϕ j v γ That is, in the definition of ϕ, the archimedean g G does not interact with the non-archimedean groups G v GL 2 Q v. [3.3] Comparison to classical formulation The familiar Eisenstein series E s z can be obtained from the above by reverting to a form that does not refer to anything p-adic or adelic. That is, we claim that v< ϕ g Img i s for g GL 2 R 7
Paul Garrett: Transition: Eisenstein series on adele groups May 8, 26 with GL 2 R acting on H H. The argument is about the Iwasawa decomposition, namely, that any element of GL 2 R can be written as a product of upper-triangular and orthogonal matrices. Indeed, given a matrix in GL 2 R, right multiplication by an orthogonal matrix can be viewed as rotating the bottom row. This suggests the appropriate orthogonal group element: formulaically, Thus, a ϕ c d c 2 +d 2 c b d c c 2 +d 2 d ϕ On the other hand, a familiar computation gives Im i 2i ad bc ac+bd c 2 +d 2 c2 + d 2 ai + b ci + d ai + b ci + d Since γ GL 2 Z maps to GL 2 Z v at all finite places v, c2 + d 2 / c 2 + d 2 c2 + d 2 s c 2 + d 2 s ad bc c 2 + d 2 c 2 + d 2 ϕ v γ for γ GL 2 Z and finite place v Thus, ϕγ g ϕ γ g s Imγg i γ P Z \GL 2Z γ P Z \GL 2Z γ P Z \GL 2Z Taking the popular choice g x y y with x R and y > produces E s x + iy on H, as claimed. [3.4] Well-definedness on P Q \G Q We should show that ϕγg depends only upon the coset P Q γ. Any γ GL 2 Q is in GL 2 Z v for almost all v, since the entries are in Z v for almost all v, and the determinant is a v-adic unit for almost all v, so the inverse is v-integral also. Thus, in an infinite product v< ϕ vγ, all but finitely-many factors are. Let χ v be the character on upper-triangular v-adic matrices P v given by χ v d a d s v with a, d Q v and b Q v The usual maximal compact [5] subgroups K v of the groups GL 2 Q v are K v GL 2 Q v O2 The description of ϕ v can be rewritten more succinctly as for v finite for v real ϕ v pk χ v p for p P v and k K v [5] We will not use the fact that these are maximal among compact subgroups, despite refering to them as such. 8
Paul Garrett: Transition: Eisenstein series on adele groups May 8, 26 For g G R, γ G Q, and β P Q, keeping in mind that G Q maps to all groups G v, not just to G, ϕβ γ g ϕ β γ g ϕ v β γ At the archimedean place, let γg pk be an Iwasawa decomposition in G v, with p P v and k K v. We see the left equivariance of ϕ v by χ v, namely, ϕ v βγg ϕ v βpk χ v β p χ v β χ v p χ v β ϕ v pk χ v β ϕ v γg Similarly, but now without g playing any role, at a finite place v, let γ pk be an Iwasawa decomposition in G v, with p P v and k K v. We see the left equivariance of ϕ v by χ v : v< ϕ v βγ ϕ v βpk χ v β p χ v β χ v p χ v β ϕ v γ Putting all these local equivariances together, ϕβ γ g v χ v β ϕ v γ g for β P Q, γ G Q, and g G By the product formula, χ v d v v a d s v for a/d Q That is, we have the left invariance ϕβ γ g ϕγ g for β P Q, γ G Q, and g G [3.5] Bijection of cosets Changing from the Γ and Γ notation, let G Z GL 2 Z and P Z { G Z }, we claim that P Q \G Q P Z \G Z More precisely, we claim that the obvious map P Z \G Z P Q \G Q by P Z g P Q g is a surjection. It is an injection because P Z G Z P Q. That is, we claim that that every coset P Q h with h G Q has a representative in G Z GL 2 Z. The argument attaches meaning to both these coset spaces, and will thereby give the bijection. The coset space P Q \G Q is in bijection with the set of lines in Q 2, respecting the right multiplication by G Q, because G Q is transitive on these lines, and P Q is the stabilizer of the line { }. Next, each line in Q 2 meets Z 2 in a free rank-one Z-module generated by a primitive vector x, y, meaning that gcdx, y. Call such a Z-module a primitive Z-line in Q 2. The collection of lines in Q 2 is thus in bijection with primitive Z-lines in Z 2, by sending a line to its intersection with Z 2. The group SL 2 Z GL 2 Z is already transitive on primitive Z-lines: for gcdx, y, let b, d Z be such that Then bx + dy gcdx, y y b x y x d That is, any primitive vector can be mapped to, so the action of SL 2 Z is transitive on primitive vectors, hence on primitive Z-lines. Thus, certainly the slightly larger group GL 2 Z is transitive on primitive vectors. The stabilizer subgroup of the primitive Z-line spanned by, in GL 2 Z is P Z { : a, d Z, b Z} d 9
Paul Garrett: Transition: Eisenstein series on adele groups May 8, 26 This proves the bijection of coset spaces. [3.6] The essential conclusion From E s g i ϕ γ g γ P Z \G Z with g GL 2 R from the bijection of coset spaces, and from the well-definedness of ϕ left modulo P Q, we have the desired re-expression of the Eisenstein series E s g i ϕγ g with g GL 2 R This is the first main point, and there are further advantages to the viewpoint. Computation of the constant term is the first illustration, after the preparation of the next section. 4. Bruhat decomposition for GL 2 Expressing the coset space P Z \G Z in terms of rational matrices P Q \G Q, rather than integral matrices, simplifies natural choices of representatives immediately, as we will see. Further, more significantly, this makes visible the unwinding and Euler factorization of the Bruhat-cell summands of the constant term, as we see in the next section. [4.] Bruhat decomposition The Bruhat decomposition for GL 2 k for any field k is [6] G k P k P k wn k with w and N This purely algebraic fact has natural extensions to GL n and the classical groups. For G GL 2, the Bruhat decomposition is easy to prove: of course, the little cell P Q consists of matrices with c, and we must claim that the big cell P k wn k is exactly Indeed, given g with c, g d/c P k wn k { GL 2 k : c } d/c That proves the Bruhat decomposition in this simple case. P k w This expression for G k exhibits a remarkably simple set of representatives for P k \G k when rational rather than only integral entries are allowed: P k \G k P k \P k P k \ P k wn k {} w w P k w N k \Nk {} wn k [6] The validity of this decomposition for GL 2 k or GL 2 k, and other specific groups, was known long before F. Bruhat s work in the 95s. Nevertheless, it is good practice to refer to the GL 2 case in terms indicated the extension to the general case.
Paul Garrett: Transition: Eisenstein series on adele groups May 8, 26 where the bijection of cosets is P k wn w w P k w n w w P k w N k n Indeed, with H, N subgroups of a larger group, H n H n for n N, if and only if n n H. Also, n n N, so H n H n H N n H N n for n, n N [4.2] Another comparison Another computation verifies our rewrite of the Eisenstein series, paying attention to the Bruhat-cell parametrization. In principle, this computation is unnecessary, but it is informative. Using the Bruhat decomposition, the sum defining the Eisenstein series is E s g ϕγ g ϕg + γ wn Q ϕγ g That is, the summands can be parametrized by {} and N Q Q. This will be important in computing the constant term in the next section. As ϕ is right O2-invariant and center-invariant, by the Iwasawa decomposition G P K we can take x y g with x R and y > With this g, the term γ is ϕg ϕ g ϕ v y s y s v< For the big cell contribution to the sum, we need to compute the archimedean part and the non-archimedean In the archimedean case, t ϕ w x y for t Q t ϕ v w for finite v, with t Q w t x y y x + t Right multiplying by a suitable orthogonal matrix rotates the bottom row to put the result in P R, namely, y x + t x+t x+t 2 +y 2 y x+t 2 +y 2 y x+t 2 +y 2 x+t x+t 2 +y 2 y x+t 2 +y 2 x + t2 + y 2 Thus, t ϕ w x y y x + t 2 + y s 2
Paul Garrett: Transition: Eisenstein series on adele groups May 8, 26 For finite v, adjust the given matrix by right multiplication by GL 2 Z v to make the result upper-triangular. For t Z v, the matrix is already in GL 2 Z v, so t ϕ v for finite v, with t Q Zv For t Z v, necessarily t Z v. Thus, the matrix can be multiplied by Thus, That is, t t in GL 2 Z v to obtain t t t t t t ϕ v t 2s v for finite v, with t Q Z v ϕ v t Combining the archimedean and non-archimedean, ϕ w t x for t v t 2s v for t v y y x + t 2 + y s 2 v< for t v t 2s v for t v > Since o is a principal ideal domain with units ±, we can easily parametrize t Q in a fashion conforming to evaluation of the displayed expression. Namely, write t d/c with c, d relatively prime, modulo ±. Note that for relatively prime integers c, d for d/c v d/c 2s v for d/c v > for p v not dividing c d/c 2s v for p v dividing c for p v not dividing c c 2s v for p v dividing c That is, by the product formula, v< for d/c v d/c 2s v for d/c v > v< c 2s v c 2s Then, once again, we recover the expected: y x + d c 2 + y s 2 c 2s y cx + d 2 + cy s 2 y s cz + d 2s Again, in principle the above computation is unnecessary, but it is informative to see the details of the reversion to a classical form. 2
Paul Garrett: Transition: Eisenstein series on adele groups May 8, 26 5. Application: constant term of GL 2 Eisenstein series An immediate use of the localized rewrite of the Eisenstein series is computation of the constant term presenting each Bruhat cell s contribution as an Euler product, by unwinding the integral defining the constant term. [5.] Transition for the constant term In these simplest situations, the constant term of any kind of modular/automorphic form is the zero-th Fourier component, separating variables in the x + iy coordinates: in most elementary, though far from optimally explanatory, terms, c P fiy constant term of fiy fx + iy dx Since fx + iy is periodic in x, we obtain the same outcome integrating over any interval [a, a + ] in place of [, ]. In fact, the integral is over the quotient R/Z c P fiy fx + iy dx R/Z Further, the integral can be written as an integral over a quotient of a subgroup of G, namely, with N { }, c P fiy fn iy dn N Z \N R where the Haar measure on N R is really just the usual measure on R. [5..] Claim: R + Q + Ẑ A. Proof: An adele x fails to be in Z v only for v in a finite set S of finite places v p. At such v, we can write x v a l p l +... + a p + a o + a p +... with a i Z The truncated sum y v a l p l +... + a p is a rational number, and is v -integral at all other finite v. Thus, + a o y v S y v is a rational number, and x y is everywhere locally integral, where y Q. That is, x y R + Ẑ. /// [5..2] Remark: The point of the corollary is not to get from A/Q back to R/Z, but to get from the classical quotient R/Z to A/Q. [5..3] Remark: In fact, additive approximation asserts that R + Q is dense in A, where Q is the diagonal copy. Equivalently, the diagonal copy of Q in the finite adeles A fin is dense. Thus, A/Q has representatives in R + Ẑ. Two elements r, r R have the same image in A/Q if and only if r r Q R + Ẑ. The latter intersection is the diagonal copy of Z, since a rational number that is in Z v for all v < is in Z. Thus, R/Z injects to A/Q. 3
Paul Garrett: Transition: Eisenstein series on adele groups May 8, 26 Applying this to the coordinate in the subgroup N of G A GL 2 A, N Z N R N A K fin N A and N Z \ N R N A K fin N Q \N A For right K fin -invariant f on the adele group, for n N and n o N fin, fn g fn g n o fn n o g because G and G fin commute as subgroups of G A. Thus, evaluating the constant term on g, c P fg fn g dn fn g n o dn o dn N Z \N R N Z \N R N A K fin fn n o g dn o dn fn g dn N Z \N R N A K fin N Q \N A [5.2] Unwinding With E s g ϕγ g the constant term of E s along P is the adelic integral c P E s g E s ng dn N Q \N A Parametrizing P Q \G Q via the Bruhat decomposition makes computation nearly trivial: E s ng dn ϕγng dn ϕγng dn N Q \N A N Q \N A w P Q \G Q /N N Q Q \N A γ P Q \P Q wn Q By the Bruhat decomposition, P Q \G Q /N Q has exactly two representatives,, w, and the constant term becomes, upon unwinding the second sum-and-integral, ϕng dn + ϕwγng dn ϕng dn + ϕwng dn N Q \N A N Q \N A γ N N Q Q \N A N A Because ϕ is left N A -invariant, the first of the two summands is ϕng dn ϕg vol N Q \N A the small Bruhat cell contribution N Q \N A Since the integral in the second summand unwound, it factors over primes ϕwng dn ϕ v wng v dn N A N v v [5.3] Essential features of p-adic integrals We do not need many formulaietails about integrals on Q p, since we will only consider Z p -invariant integrands fx, that is, fη x fx for all η Z p and x Q p. 4
Paul Garrett: Transition: Eisenstein series on adele groups May 8, 26 It is reasonable to normalize the additive Haar measure on Q p so that the compact, open subgroup Z p has total measure. The p n distinct compact, open subgroups a + p n Z p Z p are translates of each other, and mutually disjoint, so the total measure of a + p n Z p is p n for a Z p. By translation-invariance, the total measure of a + p n Z p is p n for any a Q p. For η Z p, η a + p n Z p ηa + p n Z p Thus, multiplication by units preserves Haar measure. [7] The expression Z p Z p pz p shows that the measure of Z p is p. Similarly, the measure of pn Z p is p p n. Thus, with continuous Z p -invariant f, Z p fx dx l p l Z p fx dx l fp l dx p l Z p fp l p p l l [5.4] Evaluation of local factors: non-archimedean case For g G, so that g v, the finiteprime local factors in the Euler product for the big Bruhat cell are readily evaluated, as follows. Above, we computed t for t v ϕ v w t 2s v for t v > With the v-adic factor corresponding to prime p, the v-adic local factor is t v dt + t v> + p t 2s v dt + p 2s l p l 2s v dt + p l Z p p l 2s p l p p 2s p 2s + p 2s p 2s p 2s p 2s p 2s ζ v2s ζ v 2s where ζ v s is the v th Euler factor of the zeta function. Thus, the finite-prime part of the big-cell summand is ζ2s /ζ2s. [5.5] Evaluation of local factors: archimedean case The archimedean factor of the big-cell summand of the constant term is y R x + t 2 + y s 2 dt R y s y s R t 2 dt + s Γs R y s π Γs 2 Γs y s x + t2 + y 2 s dt y s R y s π s 2 Γs 2 π s Γs e u+t2 u s du u y s ζ 2s ζ 2s l t2 + y 2 s dt y +s R y s dt Γs R ty2 + y 2 s dt e u+t2 u s 2 du u dt with ζ s π s/2 Γs/2 [7] Quite generally, a compact group of automorphisms of a topological group must preserve the Haar measure on the latter. 5
Paul Garrett: Transition: Eisenstein series on adele groups May 8, 26 [5.6] Conclusion of constant-term computations Thus, with ξs the completed zeta function ξs ζ s ζs, the constant term of E s is c P E s x + iy y s + ξ2s ξ2s y s The present point is that rewriting the Eisenstein series as an automorphization of a product of local data makes the computation of the constant term far more natural, and more genuinely representative of the corresponding computation for larger groups. 6. Application: Hecke operators on GL 2 Eisenstein series The rewritten Eisenstein series will show that the Hecke operators are not global things, but are local, just acting on the local components ϕ v. Indeed, the local components ϕ v are eigenfunctions for the local version of Hecke operators, with eigenvalues depending on the parameter s. [6.] Classical description of Hecke operators The p th Hecke operator T p on weight- automorphic forms f for Γ GL 2 Z is T p fz γ Γ\Θ p fγ z where Θ p integer matrices with det p with action [8] by linear fractional transformations : z az + b cz + d. [6.2] Hecke operators on rewritten Eisenstein series Directly computing, with g G, T p E s g δ Γ\Θ p E s j δ g δ Γ\Θ p ϕ γ j δ g Let j o v< j v. Replace γ by γ δ in G Q, to obtain δ Γ\Θ p ϕ γ j o δ g The finite-prime factor j o δ commutes with the archimedean-prime factor g, so this is δ Γ\Θ p ϕ γ g j o δ δ Γ\Θ p ϕ j γ g ϕ v j v γ δ At all finite places v but v p, δ is in the local maximal compact K v GL v Z v, so ϕ v γ δ ϕ v γ for v v. Thus, suppressing j v in the notation, T p E s g ϕγ g δ Γ\Θ p ϕ v γ δ ϕ v γ [8] That is, the weight- situation allows us to avoid worry over what to do with the determinant in GL 2. In the classical holomorphic case, the so-called slash operator is a normalization that accommodates this, usually without explanation or motivation. v< 6
Paul Garrett: Transition: Eisenstein series on adele groups May 8, 26 Thus, the p th Hecke operator s effect is local at v p. Further, this situation correctly suggests that we should hope that ϕ v is an eigenfunction for the effect of T p, with eigenvalue depending on the complex parameter s. [6.3] Hecke operators as integral operators Continue to let v correspond to prime p. The function ϕ v on G v GL 2 Q v is left P v -equivariant by χ v, and right K v -invariant. Each of the right translates g ϕ v g δ with g G v retains the left P v, χ v -equivariance, but cannot be expected to retain right K v -invariance. Nevertheless, we claim that the sum over δ Γ\Θ p recovers the right K v -invariance. That is, apparently, Γ\Θ p, or its image projected to G v, is stable under right multiplication by K v. As it stands, this doesn t make sense, since Θ p itself projected to G v is not literally stable under right multiplication by K v. As on other occasions, the necessary claim suggests itself: let Θ v be the v-adic analogue of Θ p, namely, elements of G v GL 2 Q v with entries in Z v and determinant of p-adic ord. Then we must claim that the natural map Θ p Θ v /K v induces ijection Θ p /Γ Θ v /K v Equivalently, inverting, Θ p K v \ Θ v induces ijection Γ\Θ p K v \ Θ v Indeed, K v \ Θ v has the same representatives as Γ\Θ p, namely, [9] b p p with b Z, b < p Thus, giving K v total measure, using the right K v -invariance of ϕ v, ϕ v g δ δ Γ\Θ p ϕ v g h h Θ /K v ϕ v g h dh Θ Letting η be the characteristic function of Θ v, this integral is an integral operator attached to the right translation action of G v on functions on G v : ϕ v g δ ηh ϕ v g h dh η ϕ v g δ Γ\Θ G v p The integral expression makes right K v -invariance clear, by changing variables in the integral: ηh ϕ v gk h dh G v ηk h ϕ v g h dh G v ηh ϕ v g h dh G v for k K v [9] The v-adic argument is easier than that over Z: given g in v, if gcda, c, then either a or c is in Z b v. Thus, left multiplication by either or puts g into the form c/a a/c d. Necessarily ord vd b, so further left multiplication gives the form. Since Z v/pz v Z/pZ, further left multiplication p by K v gives the indicated representatives parametrized by b. When gcda, c p, a similar argument p b gives the form, and now the b entry can be made. 7
Paul Garrett: Transition: Eisenstein series on adele groups May 8, 26 since η is left and right K v -invariant. [6.4] Hecke eigenvalues Now we can prove that ϕ v is an eigenvector for the localized version of T p with v p, and compute its eigenvalue. The v-adic Iwasawa decomposition is G v P v K v. Thus, up to constant multiples, there is a unique left P v, χ v -equivariant, right K v -invariant function on G v. Thus, every such is a multiple of ϕ v. In particular, with η the characteristic function of Θ v, necessarily η ϕ v λ s ϕ v for some λ s C. To determine λ s, it suffices to evaluate at g, using ϕ v. Thus, b λ s G v ηh ϕ v h dh Θ v /K v ϕ v h dh δ Γ\Θ p ϕ v δ b p χ v + χ p v p χ v p This is the p th Hecke eigenvalue of E s : p s p + v p s v p s + p s p + χ v T p E s p s + p s E s 8