Table Of ContentANALYTIC ISOLATION OF NEWFORMS OF GIVEN LEVEL
6
1 PAULD.NELSON
0
2
v
o
Abstract. Wedescribeamethodforunderstandingaverages overnewforms
N
on Γ0(q) in terms of averages over all forms of some level. The method is
simplestwhenq isdivisiblebythecubesofitsprimedivisors.
3
1
1. Introduction
]
T Fix apositiveintegerk. Foreachpositiveintegerq, letA(q)denote the spaceof
N weightk holomorphiccusp forms onΓ0(q). It is a finite-dimensional inner product
space. LetA∗(q)6A(q)denotetheAtkin–Lehnernewspace [2];itistheorthogonal
.
h complement of the oldspace, which is in turn the span of the forms
t
a ϕ| (z):=dk/2ϕ(dz)
m d
taken over all proper divisors ℓ6=q of q, all divisors d of q/ℓ, and all ϕ∈A(ℓ).
[
The newspace A∗(q) is of fundamental interest and importance because of its
2
strong interaction with the theory of Hecke operators. In analytic number theory,
v
it is of particular interest to understand averages(of Fourier coefficients, L-values,
0
3 ...) over the newspace. Unfortunately, the basic tools for studying such averages
5 (e.g., trace formulas) apply most directly to the larger spaces A(q).
2 There arises the problem of relating averages over A∗(q) to those over A(q).
0 Severalauthors1have addressed this problem via the Atkin–Lehner decomposition
.
1
A(q)= {ϕ| :ϕ∈A∗(ℓ)} (non-orthogonal direct sum) (1)
0 d
6 Mℓ|q Md|qℓ
1
followed by a computationally-involved Gram–Schmidt orthogonalization. In this
:
v article, we introduce an approach which is more direct for certain values of q.
i
X Theorem 1. Suppose q is divisible by the cubeof every prime that divides it. Then
r for z ,z in the upper half-plane,
a 1 2
ϕ(z )ϕ(z )= µ(d)µ(e) ϕ| (z )ϕ| (z ), (2)
1 2 d 1 d 2
ϕ∈XB∗(q) dX,e|q ϕ∈XB(dqe)
where B(ℓ),B∗(ℓ) denote arbitrary orthonormal bases for A(ℓ),A∗(ℓ) defined using
Petersson inner products with respect to normalized hyperbolic measures of volume
independent of ℓ, such as vol(Γ (ℓ)\H)−1 dxdy, so that ϕ7→ϕ| is unitary.
0 y2 d
The formulation of Theorem 1 requires only the definition of the newspace as
introduced by Atkin–Lehner [2] in 1970, but the simple identity (2) does not ap-
pear to have been anticipated prior to this work despite the considerable technical
effort expended on the problems that it addresses. The proof is short, simple and
2010 Mathematics Subject Classification. Primary11F03;Secondary 22E50,11F72.
1seeforinstance[1,§3],[9,§2],[14,§3],[4,§5],[3]
1
ANALYTIC ISOLATION OF NEWFORMS OF GIVEN LEVEL 2
independent of (1). The restriction on q will be discussed in due course, and is
irrelevant for our motivating applications in which it is a large power of a fixed
prime.
Theorem 1 applies to problems in which one knows how to average over all
forms of given level, but wants to average only over the newforms. For the sake of
illustration, we record here three such applications. Some of the consequences to
follow are new, some old. Theorem 1 itself is new.
We begin with an overly simple application which may nevertheless aid orien-
tation. Let n be a natural number coprime to q. By applying the Hecke operator
T to either variable and integrating over the diagonal z =z of (2), we obtain a
n 1 2
relationbetweentracesofHeckeoperatorsactingonthe newspaceandonallforms
of given level:
trace(T |A∗(q))= µ(d)µ(e)trace(T |A( q )). (3)
n n de
dX,e|q
Applying the Eichler–Selberg trace formula to each summand on the RHS of (3)
givesa trace formula for newforms, the casen=1ofwhichis a dimension formula
for newspaces. Such formulasarenot new: they follow (for generalq) from(1) and
Mo¨bius inversion (see [16, §5.1] or [8, §2] and [11]). The proof indicated here does
not use (1).
Second, write the Fourier expansion of ϕ∈A(q) as
ϕ(z)= nk/2ρ(n;ϕ)e2πinz,
n>1
X
so that ρ(n;ϕ| ) = 1 ρ(n;ϕ). Let m,n be positive integers. By taking the mth
d d|n d
(resp. nth) Fourier coefficient in the z (resp. z ) variable of (2), we obtain an
1 2
identity
∆∗(m,n;q)= µ(d)µ(e)∆(m,n; q ), (4)
d d de
d|gcdX(m,n,q)
e|q
expressing the averagesof Fourier coefficients
∆∗(m,n;q):= ρ(m;ϕ)ρ(n;ϕ)
ϕ∈XB∗(q)
over newforms in terms of the analogous averages ∆(m,n;q) over B(q). Applying
the classical Petersson formula to the RHS of (4) gives a Petersson formula for
newforms. Formulas of this general shape have already seen diverse applications
(see the references in footnote 1).
The special case of (4) in which the variables m,n satisfy the coprimality con-
straint(mn,q)=1andq isaprimepowerwasestablishedbyD.Rouymi[14,Prop.
9, Rmk. 4] in his work on newforms of level pν,ν →∞ after some involved calcu-
lationsalongthe lines indicatedfollowing(1) (seealso[4, §5],[3]). The proofgiven
here by way of Theorem 1 is simpler in that it avoids explicit orthogonalization of
the decomposition (1). The uniformity of (4) with respect to the variables m,n
appears to be relevant for applications such as those pursued recently in [13].
Third, let Ψ :Γ (q)\H →C be a measurable function of moderate growth, and
0
let n be a natural number coprime to q. The Peterssoninner products hϕ,Ψϕi are
of basic interest in many questions (quantum unique ergodicity, subconvexity, ...).
A formula relating their Hecke–twisted first moments over newforms and over all
ANALYTIC ISOLATION OF NEWFORMS OF GIVEN LEVEL 3
forms of given level follows from the proof of (3) by weighting the integrand by
Ψ(z) in the final step:
hϕ,Ψ·T ϕi= µ(d)µ(e) hϕ| ,Ψ·T ϕ| i. (5)
n d n d
ϕ∈XB∗(q) dX,e|q ϕ∈XB(dqe)
The summands on the RHS of (5) may be studied by integrating Ψ against the
holomorphickernelforT onconjugatesofΓ (q). Bythe multiplicityonetheorem,
n 0
the function of n given by the LHS of (5) determines the family of inner products
hϕ,ΨϕiarisingasϕtraversesanorthonormalbasisofHeckenewforms. Theoriginal
motivationfor the presentworkis that (5) and its variantsconstitute the firststep
in a method to study the quantum variance of newforms of large level (see [12,
§7.1]). Formula (5) is new in all cases.
For many problems involving modular forms, the case of squarefree (or even
prime) level is often the simplest and hence the natural first case to consider. Our
method, perhaps counterintuitively, applies most directly to cubefull levels. It ap-
plies alsoto levelsthat arenotnecessarilycubefull, butbecomes morecomplicated
to implement and is not clearly superior to existing approaches. The present gen-
erality suffices for the depth aspect in which levels are powers of a fixed prime and
hence for our motivating applications [12].
The cubefull levels often exhibit representative phenomena. Because of its di-
rectnessandsimplicity, ourmethod maybe usefulalsofor problemsinvolvingnon-
cubefulllevelsasafirststeptowardsunderstandingtheexpectedtruth. Thisisanal-
ogous(in severalrespects) to studying the asymptotics of smoothly weighted sums
f(n)W(n/x) before those of their sharply-truncated counterparts f(n).
n n6x
Toexplainthebasicideasbehindtheproofwithminimalnotation/prerequisites,
P P
werecordin§2adirectproofofarepresentativespecialcaseofTheorem1. Wethen
formulatein§3ourmainresult,whichmaybeunderstoodasalocalrepresentation-
theoreticformofTheorem1thatappliesalsotoMaassforms,onquotientsattached
to quaternionalgebras,overnumber fields, and (with minor modifications) in half-
integral weight; it consists of constructing an element of the Hecke algebra of GL
2
over a non-archimedean local field that projects onto the newvectors of given log-
conductor >3.
To elucidate that result from as many perspectives as possible, we then give
three short proofs. Each relies on a novel operator calculus for idempotents in the
Hecke algebra (§6) which we verify
(1) group-theoretically (§7),
(2) byreductiontoaprobabilisticassertionconcerningrandomnon-backtracking
walks on the Bruhat–Tits tree (§8), and
(3) usingtheKirillovmodelandrecurrencerelationsforHeckeeigenvalues(§9).
Inclosing,wenotethatitwouldbenaturalandinterestingtoextendthepresent
work to the setting of newvectors on GL [10].
N
2. The proof in a basic but representative case
WenowproveTheorem1intheprime-cubedcaseq =p3,whichalreadycaptures
the key ideas. The general case will be deduced in §5 from our main local results.
ANALYTIC ISOLATION OF NEWFORMS OF GIVEN LEVEL 4
The required identity (2) specializes to
ϕ(z )ϕ(z )= ϕ(z )ϕ(z )− ϕ(z )ϕ(z )
1 2 1 2 1 2
ϕ∈XB∗(p3) ϕ∈XB(p3) ϕ∈XB(p2) (6)
− ϕ| (z )ϕ| (z )+ ϕ| (z )ϕ| (z ).
p 1 p 2 p 1 p 2
ϕ∈XB(p2) ϕ∈XB(p)
For i,j ∈ {0,1,2,3} with i 6 j, let E : A(p3) → A(p3) denote the orthogonal
ij
projector onto the subspace A := {ϕ| : ϕ ∈ A(pj−i)}. One has A = {ϕ ∈
ij pi ij
A(p3): ϕ|γ =ϕ for all γ ∈Γ } where ϕ|γ is the slash operator used to define the
ij
automorphy of ϕ and Γ is the group
ij
p−i pi Z p−iZ
Γ := Γ (pj−i) = ∩SL (Q)
ij 1 0 1 pjZ Z 2
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
fitting into the lattice diagram
Γ Γ Γ Γ
00 11 22 33
Γ Γ Γ
01 12 23
Γ Γ
02 13
Γ
03
with the smallest group Γ =Γ (p3) at the bottom, the largest groups (all conju-
03 0
gates of Γ = SL (Z)) along the top, the chain Γ = Γ (pj) along the left edge,
00 2 0j 0
andwithΓ ∩Γ =Γ for06i6j 6k 63. TheprojectorE maybeexpressed
ij jk ik ij
concretely as the averaging operator
1
E ϕ= ϕ|γ.
ij
|Γ \Γ |
03 ij
γ∈ΓX03\Γij
Because {ϕ| :ϕ∈B(pj−i)} extends to an orthonormal basis of A(p3), one has
pi
ϕ| (z )ϕ| (z )= E ϕ(z )ϕ(z ),
pi 1 pi 2 ij 1 2
ϕ∈BX(pj−i) ϕ∈XB(p3)
so our specialized goal (6) may be rewritten as
ϕ(z )ϕ(z )= E ϕ(z )ϕ(z )− E ϕ(z )ϕ(z )
1 2 03 1 2 02 1 2
ϕ∈XB∗(p3) ϕ∈XB(p3) ϕ∈XB(p3)
− E ϕ(z )ϕ(z )+ E ϕ(z )ϕ(z ),
13 1 2 12 1 2
ϕ∈B(p3) ϕ∈B(p3)
X X
or equivalently, with the definition E∗ :=E −E −E +E , as
03 03 02 13 12
ϕ(z )ϕ(z )= E∗ ϕ(z )ϕ(z ).
1 2 03 1 2
ϕ∈XB∗(p3) ϕ∈XB(p3)
In other words, we must show the following:
Lemma 2. E∗ defines the orthogonal projector onto the newspace A∗(p3).
03
ANALYTIC ISOLATION OF NEWFORMS OF GIVEN LEVEL 5
Remark 1. The conclusion of Lemma 2 is not altogether formal. For instance, it
fails if one replaces “3” by “2.”
The operators E ,E and E are self-adjoint idempotents with image in the
02 13 12
oldspaceandhencekernelcontainingthenewspace,whileE istheidentity,soE∗
03 03
restricts to the identity on newspace. To conclude that E∗ orthogonally projects
03
ontothe newspace,itremains onlyto verifythat itannihilates the oldspace;asthe
latterisspannedbytheimagesofE andE ,itsufficestoshowthatE∗ ◦E =0
02 13 03 02
andE∗ ◦E =0. Weverifyherethefirstoftheseidentities,theproofofthesecond
03 13
being similar. We claim that
E ◦E =E , E ◦E =E , E ◦E =E , (7)
03 02 02 02 02 02 12 02 12
and
E ◦E =E (8)
13 02 12
from which it follows that E∗ ◦E = E −E −E +E = 0, as required.
03 02 02 02 12 12
Theidentities(7)areconsequencesofthetransitivityoforthogonalprojectiononto
nested subspaces. The interesting identity is thus (8), which we may write thanks
to the third identity in (7) in the equivalent form E ◦E = E ◦E and then
13 02 12 02
in terms of averaging operators as the assertion that for all ϕ∈A ,
02
1 1
ϕ|γ = ϕ|γ.
|Γ \Γ | |Γ \Γ |
03 13 02 12
γ∈ΓX03\Γ13 γ∈ΓX02\Γ12
To that end, it suffices to verify that the natural map of coset spaces Γ \Γ →
03 13
Γ \Γ inducedbythe inclusionsΓ 6Γ ,Γ 6Γ is bijective. Theinjectivity
02 12 13 12 03 02
follows from the evident identity Γ ∩Γ = Γ , while the surjectivity, which is
13 02 03
the crux of the whole matter, is given as follows:
Lemma 3. The map Γ ×Γ ∋(γ ,γ )7→γ γ ∈Γ is surjective.
02 13 1 2 1 2 12
Proof. We must show that every γ ∈SL (Q) satisfying
2
Z p−1Z
γ ∈
p2Z Z
(cid:20) (cid:21)
arises as the product γ =γ γ of two γ ,γ ∈SL (Q) satisfying
1 2 1 2 2
Z Z Z p−1Z
γ ∈ , γ ∈ .
1 p2Z Z 2 p3Z Z
(cid:20) (cid:21) (cid:20) (cid:21)
To simplify calculations, we conjugate by (p ) and reduce modulo p3. Set o :=
1
Z/p3, p := pZ/p3 < o. By the surjectivity of the natural map SL (Z) → SL (o),
2 2
we reduce to verifying that every
a b o o
γ = ∈ ∩SL (o)
c d p o 2
(cid:18) (cid:19) (cid:20) (cid:21)
arises as the product γ =γ γ of some
1 2
o p o o
γ ∈ ∩SL (o), γ ∈ ∩SL (o).
1 p o 2 2 p2 o 2
(cid:20) (cid:21) (cid:20) (cid:21)
To that end, we note that det(γ)=1,c∈p implies a∈o× and take
1 0 a b
γ := , γ := .
1 c/a 1 2 0 d−bc/a
(cid:18) (cid:19) (cid:18) (cid:19)
ANALYTIC ISOLATION OF NEWFORMS OF GIVEN LEVEL 6
This completes the proof of Lemma 3, hence of Lemma 2, hence of the q =p3 case
of Theorem 1. (cid:3)
Remark 2. The “standard” approach to proving something like Theorem 1 (see
e.g. [14] or §9) would be to check it one Hecke-irreducible subspace at a time
usingexplicitformulasderivedfromAtkin–Lehnertheory. The presentobservation
is that it is in some cases more efficient to work with the congruence subgroups
themselves.
3. Statement of the main local result
We now formulate the main result of this article, whichas indicated earliermay
be understood as a local, flexibly applicable form of Theorem 1 (see [12, §7.1] for
an application).
Let k be a non-archimedean local field with ring of integers o, maximal ideal p,
uniformizer̟,andq :=#o/p. LetGbeaclosedsubgroupofGL (k)thatcontains
2
SL (k). Equip G with some Haar measure dg. By a segment, we shall mean a
2
nonempty finite consecutive set of integers, denoted m..n :={m,m+1,...,n} for
someintegersm,nwithm6n. The cardinalityofa segmentℓis denoted#ℓ,thus
#m..n:=|m−n|+1. For eachsegmentℓ=m..n, define the congruence subgroup
o p−m ×
K = ∩G<G.
ℓ pn o
(cid:20) (cid:21)
ItistheintersectionwithGoftheunitgroupoftheEichlerorderoflevel|m−n|=
#ℓ−1 indexed by ℓ, regarded as a geodesic segment on the Bruhat–Tits tree (see
e.g. [18] or [17] or [6, §1.2]). Denote by 1 the characteristic function.
Kℓ
Let π be a smooth representation of G. For f ∈ C∞(G), denote by π(f) ∈
c
End(π) the operator π(f)v := f(g)π(g)vdg. For each segment ℓ, denote by
g∈G
π[ℓ] := πKℓ the subspace of vecRtors fixed by Kℓ. The standard projector onto π[ℓ]
is the averaging operator e := π(vol(K )−11 ) ∈ End(π); it is an idempotent
ℓ ℓ Kℓ
projectorwithimageπ[ℓ]. Thestandard complement ofπ[ℓ]isthekernelofe . One
ℓ
has
π =π[ℓ]⊕ker(e ). (9)
ℓ
Note that ℓ ⊇ ℓ′ implies π[ℓ] ⊇ π[ℓ′]. Set π[ℓ]♭ := π[ℓ′], with the sum taken
ℓ′(ℓ
over all proper subsegments ℓ′ of ℓ. The standard complement of π[ℓ]♭ in π[ℓ] is
P
π[ℓ]♯ :={v∈π[ℓ]:eℓ′v =0 for all ℓ′ (ℓ}.
One has
π[ℓ]=π[ℓ]♯⊕π[ℓ]♭. (10)
Thestandard projector ontoπ[ℓ]♯ isthe projectorπ →π[ℓ]♯ affordedbythedecom-
positions (9) and (10).
If π is unitary, then “standard projector” and “standard complement” have the
same meanings as “orthogonalprojector” and “orthogonalcomplement.”
Theorem 4. Let ℓ = m..n be a segment with #ℓ − 1 = |m − n| > 3. Then
e∗ :=e −e −e +e is the standard projector onto π[ℓ]♯.
ℓ m..n m+1..n m..n−1 m+1..n−1
Remark 3. The general case of Theorem 4 reduces to the case G = SL (k) by
2
det(G)
convolving against the characteristic function of .
1
(cid:20) (cid:21)
ANALYTIC ISOLATION OF NEWFORMS OF GIVEN LEVEL 7
Remark 4. WehaveassumedinTheorem4neitherthatπisirreduciblenorgeneric,
hence this result lies somewhat shallower than the fundamental results of local
newvector theory [5, 15]. For instance, in the case G = GL , it does not depend
2
upon the existence of the Kirillov model.
4. Interpretation for generic representations of GL
2
We record here what Theorem 4 says when G = GL (k) and π is a generic
2
irreducible representation with unramified central character. In that case, local
newvector theory [5, 15] says that one may attach to π a nonnegative integer c(π),
its log-conductor, with the property that π[0..n] 6= 0 if and only if n > c(π); one
knows then moreover that dimπ[0..n]=max(0,n+1−c(π)) and that π[0..n]♯ =0
unless n = c(π), in which case dimπ[0..n]♯ = 1. Since the subgroups Kℓ,Kℓ′ are
conjugate whenever #ℓ = #ℓ′, it follows more generally for any segment ℓ that
dimπ[ℓ]=max(0,#ℓ−c(π)) and
1 if #ℓ−1=c(π),
dimπ[ℓ]♯ = (11)
(0 otherwise.
Theorem 4 thus implies the following:
Corollary 5. For G = GL (k), π as above, and ℓ satisfying the assumptions of
2
Theorem 4, one has e∗ =0 unless c(π)=#ℓ−1, in which case e∗ is the standard
ℓ ℓ
projector onto the one-dimensional space π[ℓ]=π[ℓ]♯.
Remark 5. Ifπisirreducibleandnon-generic,thenitisone-dimensional,ande∗ =0
ℓ
under the assumptions of Theorem 4.
Remark 6. The formulation and proof of Corollary 5 extend with the usual mod-
ifications to representations having ramified central character: Let π be a generic
irreducible representationof G=GL (k) with centralcharacterω :k× →C×. De-
2
note by χ : o× → C× the restriction of ω. Let k := c(χ) denote the log-conductor
ofχ,i.e.,thesmallestk ∈Z forwhichχrestrictstriviallytoo×∩1+pk. Assume
>0
thatk >1,i.e.,thatχisnontrivial,orequivalentlythatω isramified. Letℓ=m..n
be a segment. If |m−n| > k, then χ induces a character K ∋ a b 7→ χ(d); by
ℓ c d
abuse of notation, we denote this character also by χ: K →C×. Define e :=0
ℓ (cid:0) (cid:1) ℓ,χ
unless|m−n|>k,inwhichcasesete :=π(vol(K )−1χ−11 ). Denotebyπ[ℓ,χ]
ℓ,χ ℓ Kℓ
the image ofe ,thus π[ℓ,χ]={v ∈π :gv =χ(g)v for all g ∈K }. Define π[ℓ,χ]♯
ℓ,χ ℓ
intermsofπ[ℓ,χ]asin§3ande∗ intermsofe asinTheorem4. Localnewvec-
ℓ,χ ℓ,χ
tor theory gives a formula analogous to (11) for π[ℓ,χ]♯. The proof of Corollary 5
shows in this context that e∗ = 0 unless c(π) = #ℓ−1, in which case e∗ is the
ℓ,χ ℓ,χ
standard projector onto the one-dimensional space π[ℓ,χ]=π[ℓ,χ]♯.
5. Deduction of Theorem 1 from Theorem 4
Theorem1followsfromTheorem4viaastandard“adelic-to-classical”argument
(seee.g. [7]); forthe sakeofcompleteness andvarietyofexposition,werecordhere
a direct “local-to-classical” proof of this implication. Thus, let q be cubefull. For
squarefree integers d,e dividing q, denote by E the orthogonalprojection from
d..de
A(q) to the subspace {ϕ| : ϕ ∈ A( q )}. As in §2, Theorem 1 amounts to the
d de
assertion that the operator E∗ := µ(d)µ(e)E defines the orthogonal
1..q d,e|q d..de
projectionontoA∗(q). Forthe samereasonsas inthe proofofLemma2, E∗ acts
P 1..q
ANALYTIC ISOLATION OF NEWFORMS OF GIVEN LEVEL 8
by the identity onthe newspace,so it remains only to verify that it annihilates the
oldspace. The oldspace is the sum over all p|q of the subspaces
A(q/p) and {ϕ| :ϕ∈A(q/p)} (12)
p
so we reduce to verifying for each such p that E∗ annihilates the spaces (12). To
1..q
thatend,denotebyπthespanofthefunctionsϕ|γ takenoverallϕ∈A(q)andallγ
inthegroupΓ:=SL (Q)∩R (q)[1/p],whereR (q):= Z Z 6M (Z)istheorder
2 0 0 qZZ 2
for which SL (Q)∩R (q) = Γ (q). Regard Γ as a subgroup of G := SL (Q ). It
2 0 0 2 p
(cid:2) (cid:3)
is dense. Each ϕ∈π is invariant under some congruence subgroup of Γ (q), hence
0
underΓ∩U forsomeopensubgroupU ofG. Consequently,theleftactionofΓonπ
givenby γ·ϕ:=ϕ|γ−1 extends continuouslyto a smoothunitary representationof
G,whichwecontinuetodenotebyπ. Factorq =q pnwhere(q ,p)=1,anddenote
0 0
by ℓ the segment ℓ:=0..n. The subspace A(q) is recovered from π as A(q)=π[ℓ],
while π[ℓ]♭ is the span of (12). By the Chinese remainder theorem, we may factor
E∗ =E∗ ◦e∗| (13)
1..q 1..q0 ℓ A(q)
where E∗ is defined analogously to E∗ and e∗ is the operator on π defined in
1..q0 1..q ℓ
§3. By Theorem 4, the subspaces (12) are annihilated by e∗, hence (by (13)) by
ℓ
E∗ , as required.
1..q
6. Reduction to an operator calculus for idempotents
We reduce here the proof of Theorem 4 to that of the following:
Lemma 6. Let ℓ,ℓ′ be segments. Suppose that ℓ ⊆ ℓ′ or ℓ ⊇ ℓ′ or #ℓ∩ℓ′ > 2.
Then eℓ◦eℓ′ =eℓ∩ℓ′.
Remark 7. By analogy to composition formulas arising in microlocal analysis, it
maybeinstructivetothinkofthecharacteristicfunction1 ofℓasasymbol,e asits
ℓ ℓ
quantization,andthe conclusionofLemma 6 asthe assertionthatthe composition
of the quantization of two such symbols 1ℓ,1ℓ′ is the quantization of their product
1ℓ1ℓ′ =1ℓ∩ℓ′ in nice enough cases.
Remark 8. Suppose G = GL2(k). The identity eℓ ◦eℓ′ = eℓ∩ℓ′ fails in general if
#ℓ∩ℓ′ < 2 and neither segment contains the other, but continues to hold if π is
irreduciblewithunramifiedcentralcharacterandlog-conductorc(π)>2,thepoint
being that in such cases, one can simultaneously diagonalize the operators e by a
ℓ
basis of characteristic functions of o×-cosets in the Kirillov model, corresponding
classicallytothe Fouriercoefficientsofnewformsofconductordivisiblebyp2 being
supported on integers coprime to p; see §9.
Assuming Lemma6forthe moment, wenowdeduce Theorem4byanargument
similar to that in the proof of Lemma 2. Take ℓ=m..n with |m−n|>3; we must
show that e∗ is the standard projector onto π[ℓ]♯. By the identity e∗ =e∗◦e and
ℓ ℓ ℓ ℓ
the definition of π[ℓ]♯, we see that e∗ annihilates the standard complement of π[ℓ]
ℓ
andrestrictstothe identityonπ[ℓ]♯,soourmaintaskistoshowthatitannihilates
π[ℓ]♭, or equivalently, that
e∗ℓ ◦eℓ′ =0 (14)
for all ℓ′ (ℓ. Since eℓ′ =eℓ′′ ◦eℓ′ for any ℓ′′ ⊇ℓ′, it suffices to establish (14) when
ℓ′ is a maximal proper subsegment ℓ′ ( ℓ. We verify this when ℓ′ = m+1..n; the
case ℓ′ = m..n−1 is similar. Our assumption on |m−n| implies that for each
ANALYTIC ISOLATION OF NEWFORMS OF GIVEN LEVEL 9
segmentℓ′′ ∈{m..n,m+1..n,m..n−1,m+1..n−1}arisingin the definition of e∗,
ℓ
one has #ℓ′∩ℓ′′ >2, so by Lemma 6, we have
e∗ℓ ◦eℓ′ =em..n∩ℓ′−em+1..n∩ℓ′ −em+1..n−1∩ℓ′ −em+1..n−1∩ℓ′
which simplifies to e −e −e −e =0, as required.
m+1..n m+1..n m+1..n−1 m+1..n−1
7. Group-theoretic proof
WerecordhereaproofofLemma6similartothatofLemma2. Thecaseinwhich
one of ℓ,ℓ′ contains the other follows from the transitivity of standard projectors
onto nested subspaces, so we need only consider the case that #ℓ∩ℓ′ > 2 and
neithercontainstheother. Byasymmetryargument,wereducefurthertoshowing
that
em..n◦em′..n′ =em′..n whenever m<m′ <n<n′. (15)
Note especially that (15) implies |m′−n| > 1. Since em′..n = em′..n◦em′..n′, we
reduce to verifying that em..nv =em′..nv for all v ∈π[m′..n′]. For such v, one has
1
e v = π(γ)v,
m..n
|Km..n/Km..n′|
γ∈Km.X.n/Km..n′
1
em′..nv = π(γ)v.
|Km′..n/Km′..n′|
γ∈Km′.X.n/Km′..n′
soourtaskreducestoverifyingthatthenaturalmapofcosetspacesKm..n/Km..n′ →
Km′..n/Km′..n′ induced by the inclusions Km..n 6 Km′..n, Km..n′ 6 Km′..n′ is bi-
jective. TheinjectivityfollowsfromKm..n∩Km′..n′ =Km..n′,thesurjectivityfrom
the claim Km..n·Km′..n′ =Km′..n for whose proof we must verify that any
a b o p−m′ ×
γ = ∈ ∩G
c d pn o
(cid:18) (cid:19) (cid:20) (cid:21)
arises as γ =γ γ for some
1 2
o p−m × o p−m′ ×
γ1 ∈ pn o ∩G, γ2 ∈ pn′ o ∩G.
(cid:20) (cid:21) (cid:20) (cid:21)
From det(γ)∈o×, b∈p−m′,c∈pn, n>m′ we obtain bc∈p and hence a,d∈o×,
justifying the choice
1 a b
γ := , γ :=
1 c/a 1 2 0 d−bc/a
(cid:18) (cid:19) (cid:18) (cid:19)
for which γ ∈ SL (k) < G and γ = γ−1γ ∈ G. The required congruences are
1 2 2 1
clear. This completes the proof.
8. Probabilistic proof
We record here an alternative proof of the key identity (15). For notational
simplicity we suppose that G = SL (k) (cf. Remark 3). Fix an arbitrary linear
2
functional v∗ :π →C. It suffices to show that
v∗(em..nem′..n′v)=v∗(em′..nv) for v ∈π[m..n′] (16)
under the assumptions (15) on the indices. Denote by X the (q+1)-regular tree,
where q := #o/p, and by Xm..n′ the set of non-backtracking paths x = (x →
m
ANALYTIC ISOLATION OF NEWFORMS OF GIVEN LEVEL 10
xm+1 →···→xn′) on X. There is a well-knowninjection G/Km..n′ →Xm..n′ ob-
tainedbyidentifyingX withthesetofhomothetyclasses[L]oflatticesL⊆k2 and
mappingthecosetgKm..n′ tothetupleoflatticeclasses([gLm],[gLm+1],...,[gLn′])
where Lj :=p−j ×o has stabilizer Kj..j. Denote by Xm..n′ the image of G/Km..n′
in Xm..n′ and by G ∋ g 7→ [g] ∈ Xm..n′ the induced surjection. Consider the map
φ : π[m..n′] → CXm..n′ from π to the space of complex-valued functions on Xm..n′
that sends a vector v ∈π[m..n′] to the function φ(v):Xm..n′ →C given by
φ(v)([g]):=v∗(π(g)v),
which is well-defined. Given x,y ∈ Xm..n′ and a subsegment ℓ = p..p′ ⊆ m..n′,
write x|ℓ = y|ℓ to denote that the subpaths (xp → ··· → xp′), (yp → ··· → yp′)
coincide. The subset Xm..n′ of Xm..n′ has the property that if x ∈ Xm..n′ and
y ∈ Xm..n′ satisfy x|ℓ = y|ℓ for some subsegment ℓ ⊆ m..n′, then y ∈ Xm..n′. For
f : Xm..n′ → C and a subsegment ℓ ⊆ m..n′, denote by ρℓf : Xm..n′ → C the
function given by the expectation ρ f(x) = Ef(y) taken over y = (y → y →
ℓ m m+1
···yn′) chosen uniformly at random from the finite set of non-backtracking paths
for which x| =y| . One verifies directly from the definitions that
ℓ ℓ
φ(e v)=ρ φ(v), (17)
ℓ ℓ
so to establish (16), our task reduces to showing for all f :Xm..n′ →C that
ρm..nρm′..n′f =ρm′..nf. (18)
Itsufficestotestthisequalityonthecharacteristicfunctionf :=1 ofanarbitrary
x
non-backtracking path x = (xm → ··· → xn′) ∈ Xm..n′. The RHS of (18) is then
the uniform distribution on the finite set of non-backtracking paths y = (y →
m
···→yn′) for which y|m′..n =x|m′..n, and so (18) follows from:
Lemma 7. Suppose m6m′ <n6n′. Then the following probability distributions
on Xm..n′ coincide:
• Theuniformdistributionρm′..n1x onthefinitesetofnon-backtrackingpaths
y =(ym →···→yn′) for which y|m′..n =x|m′..n.
• The distribution ρm..nρm′..n1x generated iteratively as follows:
(1) Start with the deterministic subpath (ym′ → ··· → yn) := (xm′ →
···x ).
n
(2) Choose uniformly at random a forward extension (yn → ··· → yn′)
satisfying the non-backtracking condition that (y →y ) not be the
n n+1
inverse of (x →x ).
n n−1
(3) Independentlychooseuniformlyatrandomabackwardextension(y →
m
··· → ym′) for which (ym′ → ym′−1) is not the inverse of (xm′ →
xm′+1).
Proof. The assumption n > m′ implies that the subpath (xm′ → ··· → xn) con-
tains the (possibly identical) edges (xm′ → xm′+1) and (xn−1 → xn), so the non-
backtracking condition on the forward path (yn → ··· → yn′) is independent of
that on the backwardpath (ym →···→ym′). (cid:3)
9. A third proof
We sketch here a proof of Corollary5 (thus G=GL (k)) which is more compli-
2
cated and less self-contained than the above proofs, but which some readers may
findillustrative;itisalsosimilarinspirittowhatistypicallydoneclassicallyalong
