Colin de Verdi`ere's meromorphic continuation of Eisenstein series

28 Pages·2012·0.45 MB·English

(March 12, 2012)
Colin de Verdi`ere’s meromorphic continuation of Eisenstein series
Paul Garrett [email protected] http://www.math.umn.edu/ garrett/
(cid:101)
1. Harmonic analysis on H
2. Meromorphic continuation up to the critical line
3. Sobolev inequality/imbedding
4. Eventually-vanishing constant terms
5. Compactness of Sob(Γ\H) →L2(Γ\H)
a
6. Discreteness of cuspforms
7. Meromorphic continuation beyond the critical line
8. Discrete decomposition of truncated Eisenstein series
9. Appendix: Friedrichs extensions
10. Appendix: simplest Maass-Selberg relation
We elaborate the brief note [Colin de Verdi`ere 1981] on meromorphic continuation of Eisenstein series, and
related harmonic analysis of automorphic forms. See also [Colin de Verdi`ere 1982,83].
The context of [Colin de Verdi`ere 1981] is not elementary: it uses technical aspects of [Friedrichs 1934,35]’s
canonical self-adjoint extensions of symmetric unbounded operators on Hilbert spaces, and uses Sobolev
spaces and Schwartz’ distributions. The compactness of the inclusion map of Friedrichs-Sobolev spaces of
automorphic forms with constant terms vanishing above y = a, into L2(Γ\H), proves the compactness of
the resolvent of the Friedrichs self-adjoint extension ∆˜ of the restriction of the invariant Laplacian to that
a
subspace, giving its meromorphy. Eisenstein series differ from Eisenstein-series-like functions in the domain
of ∆˜ by elementary functions, giving the meromorphic continuation of the Eisenstein series.
a
A noteworthy preliminary result, reminiscent of [Avakumovi´c 1956],[Roelcke 1956], [Selberg 1956],
immediately extends Eisenstein series E to Re(s) > 1. Analytic continuation of the zeta function ζ(s)
s 2
to Re(s) > 0 is a corollary, the simplest example of [Langlands 1967/76] and [Langlands 1971] arguments
about meromorphic continuation of automorphic L-functions.
The compactness of the imbedding of Friedrichs’ L2 Sobolev-like spaces of automorphic forms into L2 also
provesthatthespaceofL2cuspformsdecomposesdiscretelywithrespecttotheinvariantLaplacian,although
this is not a trivial corollary, for reasons we explain.
The precise import of the compactness argument is widely misunderstood. Often, the description of the
compactness argument (with corollaries about discrete decomposition of cuspforms) does not distinguish
these (correct) arguments from similar (incorrect) arguments purportedly proving that truncated Eisenstein
are eigenfunctions for the Laplacian. Yet, Colin-de-Verdi`ere’s argument does discretely decompose spaces
containing truncated Eisenstein series, by self-adjoint extensions ∆˜ of restrictions of the Laplacian
a
∆ to subspaces. These operators ∆˜ are not differential operators, as becomes clear below. in
a
[Colin de Verdi`ere 1982,83] these and other variants are usefully called pseudo-Laplacians.
As will be clarified later: for fixed cut-off height y =a, the pseudo-Laplacian constructed as the self-adjoint
Friedrichs’ extension ∆˜ of the restriction of ∆, does have compact resolvent on the subspace L2(Γ\H)
a a
of L2(Γ\H) consisting of automorphic forms with constant term vanishing above y = a. Thus, ∆˜ has a
a
basisofeigenvectors. Inparticular,theorthogonalcomplementtocuspformsinL2(Γ\H) hasanorthogonal
a
basis of ∆˜ -eigenvectors, consisting of truncated Eisenstein ∧aE whose constant term vanishes on y = a.
a s
There is no paradox, because ∆˜ is designed to ignore order-zero distributions supported on the line y =a.
a
Computed distributionally, (∆−s(s−1))∧a E is a distribution supported on (images of) y = a. When
s
the constant term does not vanish on that line, the resulting distribution is of order one, and ∧aE is not
s
in the domain of ∆˜ . When the constant term vanishes, the resulting distribution is of order zero, and the
a
truncation ∧aE is in the domain of ∆˜ , since the zero-order distribution is ignored.
s a
The simplest example Γ\H with Γ=SL (Z) and H=SL (R)/SO(2) illustrates the mechanism.
2 2
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Paul Garrett: Colin de Verdi`ere’s meromorphic continuation of Eisenstein series (March 12, 2012)
1. Harmonic analysis on H
[1.1] Invariant Laplacian
The usual SL (R)-invariant Laplacian on the upper half-plane H≈G/K is
2
(cid:16) ∂2 ∂2 (cid:17)
∆ = y2 +
∂x2 ∂y2
Parametrize ∆-eigenvalues as usual by
λ = λ = s(s−1)
s
Let
(cid:18) (cid:19) (cid:18) (cid:19)
1 x t 0
N = { : x∈R} A+ = { : t>0}
0 1 0 t−1
[1.2] Density of automorphic test functions
IntegrationbypartsonC∞(Γ\G)showsthat∆isasymmetric(unbounded)operatoronL2(Γ\H). Toshow
c
that it is densely defined, show that C∞(Γ\H), defined to be right K-invariant functions in C∞(Γ\G), is
c c
dense in L2(Γ\H), as follows.
Fix 0<1≤b<b(cid:48) <∞, and take a smooth cut-off function 0≤τ ≤1 on (0,∞) with

1 (for b(cid:48) ≤y)

τ(y) =
0 (for 0≤y ≤b)
For t>0, define a smooth cut-off by
ϕ (y) = τ(y/t) (for t>0)
t
Let Φ (z)=ϕ (Im(z)). With Γ the upper-triangular elements of Γ=SL (Z), the corresponding pseudo-
t t ∞ 2
Eisenstein series is
(cid:88)
Ψ (z) = Φ (Im(γ·z))
t t
γ∈Γ∞\Γ
We claim that (1−Ψ )·f →f in L2(Γ\H) as t→+∞, for all f ∈L2(Γ\H). Indeed,
t
(cid:90) (cid:12) (cid:12)2 (cid:90) (cid:12) (cid:12)2 (cid:90) (cid:90)
(cid:12)(1−Ψ )f −f(cid:12) = (cid:12)Ψ ·f(cid:12) − |Φ ·f|2 ≤ |f|2 −→ 0
(cid:12) t (cid:12) (cid:12) t (cid:12) t
Γ\H Γ\H Γ∞\H Γ∞\{y≥t}
because the tails of the integral of |f|2 go to 0, by convergence of the integral of the L2 norm of f.
[1.3] Friedrichs extension of ∆ on C∞(Γ\H)
c
Precisediscussionofanunboundedoperatoranditsresolventrequireaspecifieddomain. Take[1] C∞(Γ\H)
c
as the domain of ∆.
[1] Generally, taking a domain to be test functions requires some sort of generalized vanishing on the boundary in
the self-adjoint extension, if there is a boundary. In boundary-less situations such as Γ\H, this is often appropriate.
For the operators ∆a later, the interaction with boundary properties is visible. For example, see [Grubb 2009], for
extensive examples and a modern discussion of boundary conditions versus extensions of operators.
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Paul Garrett: Colin de Verdi`ere’s meromorphic continuation of Eisenstein series (March 12, 2012)
Let ∆˜ be the Friedrichs extension of ∆ to a self-adjoint (unbounded) operator on L2(Γ\H). The Friedrichs
construction shows that the domain of ∆˜ is contained in a Sobolev-like space:
(cid:16) (cid:17)
domain∆˜ ⊂ Sob(+1) = completion of C∞(Γ\H) under (cid:104)v,w(cid:105) =(cid:104)v,w(cid:105)+(cid:104)−∆v,w(cid:105)
c Fr
The domain of ∆˜ contains[2] the smaller Sobolev space
(cid:16) (cid:17)
Sob(+2) = completion of C∞(Γ\H) under (cid:104)v,w(cid:105) =(cid:104)v,w(cid:105)+(cid:104)∆v,∆w(cid:105)
c Sob(+2)
[1.3.1] Remark: The Sobolev spaces above are defined as completions of test functions, and there is no
immediate need to make comparisons to other characterizations.
2. Meromorphic continuation up to the critical line
ThequotientΓ\Histheunionofacompactpart,whose(conceivablycomplicated)geometrydoesnotmatter,
and a geometrically trivial non-compact part:
Γ\H = X ∪X ( compact X , cusp neighborhood X )
cpt ∞ cpt ∞
where
X = image of {x+iy :y ≥y } = Γ {x+iy :y ≥y } ≈ circle×ray
∞ o ∞ o
Define a smooth cut-off function τ as usual: fix b<b(cid:48) large enough so that the image of {z ∈H:y >b} in
the quotient is in X , let
∞

1 (for y >b(cid:48))

τ(y) =
0 (for y <b)
Form a pseudo-Eisenstein series h by automorphizing the smoothly cut-off function τ(Im(z))·ys:
s
(cid:88)
h (z) = τ(Im(γz))·Im(γz)s
s
γ∈Γ∞\Γ
Since τ is supported on y ≥ b for large b, for any z ∈ H there is at most one non-vanishing summand in
the expression for h , and convergence is not an issue. Thus, the pseudo-Eisenstein series h is entire as a
s s
function-valued function of s. Let
E˜ = h −(∆˜ −λ)−1(∆−λ)h (where λ=s(s−1))
s s s
[2.0.1] Remark: From Friedrichs, the resolvent (∆˜ −λ)−1 exists as a bounded operator for s ∈ C for λ
s
not a non-positive real number, because of the non-positive-ness of ∆. Further, for λ not a non-positive
s
real, this resolvent is a holomorphic operator-valued function. Thus, E˜ is holomorphic for Re(s) > 1 and
s 2
Im(s)(cid:54)=0.
[2] Infact,anyself-adjointextensionT of∆willhavedomaincontainingSob(+2),withT definedtherebyextending
bycontinuityintheSob(+2)topology. Thisisseenasfollows. ForL2(Γ\H)-Cauchyv inthedomainofT,iflimTv
i i
exists in the topology of L2(Γ\H), then v ⊕Tv is Cauchy in L2(Γ\H)⊕L2(Γ\H). Graphs of self-adjoint operators,
i i
whetherunboundedorbounded,areclosed. Thus,butonlybecauseweassumedthelimitexists,limTv =T(limv ).
i i
ThisargumentdoesnottouchuponL2(Γ\H)-continuityofT,but,rather,provesthatT iscontinuousintheSob(+2)
topology.
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Paul Garrett: Colin de Verdi`ere’s meromorphic continuation of Eisenstein series (March 12, 2012)
[2.0.2] Remark: The smooth function (∆−λ)h is supported on the image of b≤y ≤b(cid:48) in Γ\H, which is
s
compact. Thus, it is in L2(Γ\H). It might seem E˜ vanishes, if it is forgotten that the indicated resolvent
s
maps to the domain of ∆˜ inside L2(Γ\H), and that h is not in L2(Γ\H) for Re(s)> 1. Indeed, since h is
s 2 s
not in L2(Γ\H) and (∆˜ −λ)−1(∆−λ)h is in L2(Γ\H), the difference cannot vanish.
s
[2.0.3] Theorem: Withλ=s(s−1)notnon-positivereal,u=E˜ −h istheuniqueelementofthedomain
s s
of ∆˜ such that
(∆˜ −λ)u = −(∆−λ)h
s
Thus, E˜ is the usual Eisenstein series E for Re(s) > 1, and gives an analytic continuation of E to
s s s
Re(s)> 1 with s(cid:54)∈(1,1].
2 2
Proof: Uniqueness follows from Friedrichs’ construction and construction of resolvents, because ∆˜ −λ is a
bijection of its domain to L2(Γ\H).
On the other hand, for Re(s)> 1 and s(cid:54)∈(01,1], E˜ −h is in L2(Γ\H), and is smooth, so is in the domain
2 2 s s
of ∆˜. Abbreviate
H = (∆−λ)h
s s
Then it is legitimate to compute
(cid:16) (cid:17) (cid:16) (cid:17)
(∆˜ −λ)(E˜ −h ) = (∆˜ −λ) (h −(∆˜ −λ)−1H )−h = (∆˜ −λ) −(∆˜ −λ)−1H = −H
s s s s s s s
Thus, E˜ −hs is a solution. Certainly E −h is a solution. ///
s s s
[2.0.4] Remark: Thus, the Eisenstein series E has an analytic continuation to Re(s) > 1 and s (cid:54)∈ (1,1]
s 2 2
as an h +L2(Γ\H)-valued function. Further, Friedrichs gives a bound for the L2-norm of E −h via an
s s s
estimate on the operator norm of (∆˜ −λ)−1. The L2-norm of (∆−λ)h is not difficult to estimate, since its
s
support is b≤y ≤b(cid:48):
(cid:90) 1(cid:90) b(cid:48) dxdy
|(∆−λ)h |2 ≤ (|∆h |+|λh |)2 (cid:28) |λ|2
s L2 s s y2 b,b(cid:48)
0 b
Since ∆˜ is negative-definite, Friedrichs gives
1 1
||(∆˜ −λ)−1|| ≤ = (for σ > 1, t(cid:54)=0)
Im(λ) 2(σ− 1)t 2
2
Thus,
1
|Es−hs|L2 = ||(∆˜ −λ)−1||·|(∆−λ)hs|L2 (cid:28) (σ− 1)t ·|s(s−1)|21
2
[2.0.5] Remark: Granting that the Eisenstein series E has constant term ys + c y1−s, the analytic
s s
continuation of E to Re(s) > 1 analytically continues c to Re(s) > 1. Since c = ξ(2s − 1)/ξ(2s)
s 2 s 2 s
with ξ(s) the completed zeta-function
ξ(s) = π−s/2Γ(s/2)ζ(s)
this yields the analytic continuation of ζ(s) to Re(s)>0, off the interval [0,1].
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Paul Garrett: Colin de Verdi`ere’s meromorphic continuation of Eisenstein series (March 12, 2012)
3. Sobolev inequality/imbedding
The self-adjoint extensions of differential operators typically have domains including not-necessarily-smooth
functions, requiring a finer description of the spaces Sob(+1) occuring in Friedrichs’ construction for the
case of second-order operators.
In particular, as needed later, computations relevant to Sobolev-norm behavior of pseudo-Eisenstein series
is clarified.
[3.1] Another description of Sob(+1)
This description applies to general Γ,G,K.
Consider functions on Γ\H≈Γ\G/K as right K-invariant functions on Γ\G. We use the G-invariant trace
pairing[3]
(cid:104)x,y(cid:105) = trace(xy) (with x,y ∈g)
Thispairingisnegative-definiteontheLiealgebrakofK,andpositive-definiteontheorthogonalcomplement
p of k in g. Thus, we can choose a negative-orthonormal basis {θ } of k, that is, with (cid:104)θ ,θ (cid:105) = −δ with
i i j ij
Kronecker delta. We can choose an orthonormal basis {x } for p.
j
For any such choice, the Casimir element Ω in the universal enveloping algebra Ug is expressible as
(cid:88) (cid:88)
Ω = x2 − θ2
j i
j i
The Lie algebra g of G acts on the right on Γ\G. The restriction of Ω to right K-invariant functions on G
is the invariant Laplacian ∆ on G/K, up to a constant. On test functions f on Γ\G, integration by parts
gives
(cid:90) (cid:90) (cid:90) (cid:90) (cid:90)
(cid:88) (cid:88) (cid:88) (cid:88)
Ωf ·f = x2f ·f − θ2f ·f = − x f ·x f + θ f ·θ f
j i j j i i
Γ\G j Γ\G i Γ\G j Γ\G i Γ\G
For right K-invariant f, this computes
(cid:90) (cid:90)
(cid:88)
−∆f ·f = |x f|2
j
Γ\G/K j Γ\G
Of course, typically the derivatives x f are not right K-invariant, but this is harmless.
j
Thus, on one hand, a Sob(+1) norm (cid:104),(cid:105) attached to ∆ is expressible as
1
(cid:90) (cid:90) (cid:90)
(cid:88)
(cid:104)f,f(cid:105) = (1−∆)f ·f = |f|2 + |x f|2
1 j
Γ\G/K Γ\G/K j Γ\G
On the other hand, the computation shows that L2(Γ\G) norms of first derivatives (given by g) of
f ∈C∞(Γ\G/K) are dominated by the Sob(+1) norm of f.
c
[3] For simple linear Lie algebras g, this R-bilinear pairing is a multiple of the Killing form.
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Paul Garrett: Colin de Verdi`ere’s meromorphic continuation of Eisenstein series (March 12, 2012)
[3.2] Constant terms and local Sobolev spaces
Although Γ \N is compact, the constant term maps
∞
(cid:90)
f −→ f(ng)dn
Γ∞\N
do not map C∞(Γ\G/K) → C∞(N\G/K). This prevents comparison of (global) Sobolev spaces.
c c
Nevertheless, local Sobolev spaces are readily compared: for compact C ⊂G, let
(cid:90)
ν (f) = (1−Ω)f ·f (for f ∈C∞(G/K))
C
C
Let
Sobloc (+1) = local +1-index Sobolev space on N\G/K
N\G/K
be the quasi-completion of C∞(N\G/K) with respect to the collection of these semi-norms. The constant-
term map respects these semi-norms, since Γ \N is compact. Thus, we have a continuous map
∞
c : Sob(+1) −→ Sobloc (+1)
P N\G/K
ThedimensionofN\G/K ismuchlowerthanthatofΓ\G/K. ForG=SL (R)oranyreal-rank1group,the
2
dimension of N\G/K is 1. The (local) Sobolev imbedding/inequality shows that constant terms of Sob(+1)
functions are continuous, since
Sobloc (+1) ⊂ Co(N\G/K)
N\G/K
In fact, the local Sobolev theory shows that functions in Sobloc (+1) satisfy a non-trivial Lipschitz
N\G/K
condition.
[3.3] Pseudo-Eisenstein series in Sob(+1)
Weneedasimplesufficientconditionforpseudo-EisensteinseriestobeinSob(+1). WereverttoG=SL (R)
2
and Γ=SL (Z), for simplicity.
2
With large b>0, let ϕ∈Co[b,∞) be smooth, except possibly at y =a with fixed a>b, but continuous at
y = a and possessing left and right derivatives at y = a. We claim that the pseudo-Eisenstein series Ψ is
ϕ
in Sob(+1) if
(cid:90) ∞ (cid:12) ∂ϕ(cid:12)2 dy
|ϕ|2+(cid:12)y (cid:12) < ∞
(cid:12) ∂y(cid:12) y2
0
Proof: To discuss right derivatives, we must look at automorphic forms on the group G, rather than on the
domain H. Let the Iwasawa decomposition of an element of G be g = na(g)k with n ∈ N, a(g) ∈ A+, and
k ∈K. Let √
(cid:18) y 0 (cid:19)
a =
y 0 √1
y
and let Φ(g) = ϕ(y), where a(g) = a . The Sob(+1) hypothesis on ϕ implies that ϕ is locally in the +1
y
Sobolevspace. Thus, locally, anyfirst-derivativeisinthe0th Sobolevspace, thatis, locallyL2. Thisimplies
local integrability of ϕ and Φ.
The right action of α∈g on a smooth function f on G is
∂ (cid:12)
(αf)(g) = (cid:12) f(g·etα)
∂t(cid:12)t=0
The right action of g commutes with the left action of G, so we can unwind:
(cid:90) (cid:90) (cid:90)
(cid:88)
|αΨ |2 = αΨ ·αΦ = αΦ(γg)·αΦ(g)dg
ϕ ϕ
Γ\G Γ∞\G Γ∞\Gγ∈Γ∞\Γ
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Paul Garrett: Colin de Verdi`ere’s meromorphic continuation of Eisenstein series (March 12, 2012)
Since ϕ is supported on y ≥b, the same is true of αϕ, and by reduction theory αΦ(γg)αΦ(a )(cid:54)=0 only for
g
γ ∈Γ . Thus,
∞
(cid:90) (cid:90) (cid:90)
|αΨ |2 = αΦ·αΦ = |αΦ|2
ϕ
Γ\G Γ∞\G N\G
Let (,) be the Killing form (or trace form) on g. It is negative-definite on the Lie algebra k of K, and
positive-definite on the orthogonal complement p of k in g. Modify B(,) by reversing its sign on k, giving a
positive-definite K-invariant form B+(,) on g, and corresponding K-invariant length.
Typically, the derivative αf of a right K-invariant function is no longer right K-invariant, but we still have
∂ (cid:12) ∂ (cid:12)
(αf)(g·k) = (cid:12) f(gk·etα) = (cid:12) f(g·et·kαk−1 ·k)
∂t(cid:12)t=0 ∂t(cid:12)t=0
∂ (cid:12)
= (cid:12) f(g·et·kαk−1) (for α∈g, k ∈K, g ∈G)
∂t(cid:12)t=0
Let K have total measure 1. For α ∈ g with B+(α,α) ≤ 1, using an Iwasawa decomposition G = NA+K,
we have
(cid:90) (cid:90) ∞(cid:90) dy (cid:90) ∞ dy
|αΦ|2 ≤ |αΦ(a k)|2 dk ≤ sup |βΦ(a )|2
y y2 y y2
N\G 0 K 0 β∈g:B+(β,β)≤1
Let
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
1 0 0 1 0 1
h = X = θ = (in g)
0 −1 0 0 −1 0
√ √
The elements h and 2X−θ are in p, of length 2. The element θ is in k, of length 2. Any β ∈g is a linear
combination,
b b
β = ah+bX+cθ = ah+ (2X−θ)+(c+ )θ
2 2
Thus, for B+(β,β)≤1, there is a uniform bound on the coefficients a,b,c. Thus, to uniformly bound βΦ it
suffices to show XΦ(a )=0, θΦ(a )=0, and to bound hΦ(a ).
y y y
Since Φ is right K-invariant, θΦ=0. Since Φ is left N-invariant,
∂ (cid:12) ∂ (cid:12) ∂ (cid:12)
XΦ(a ) = (cid:12) Φ(a etX) = (cid:12) Φ(et·yXy−1a ) = (cid:12) Φ(a ) = 0
y ∂t(cid:12)t=0 y ∂t(cid:12)t=0 y ∂t(cid:12)t=0 y
Finally,
∂ (cid:12) ∂ (cid:12) ∂ϕ
hΦ(a ) = (cid:12) Φ(a eth) = (cid:12) ϕ(y·e2t) = 2y
y ∂t(cid:12)t=0 y ∂t(cid:12)t=0 ∂y
Thus, in summary,
(cid:90) (cid:90) ∞(cid:12) ∂ϕ(cid:12)2 dy
|αΨ |2 (cid:28) (cid:12)y (cid:12) (uniform implied constant)
ϕ (cid:12) ∂y(cid:12) y2
Γ\G 0
We should prove that Ψ is a Sob(+1)-limit of elements of C∞(Γ\H). In fact, as should be anticipated, it
ϕ c
is a limit of elements Ψ with η ∈C∞(0,∞). However, given the above comparison and prior development,
η c
the argument is straightforward. ///
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Paul Garrett: Colin de Verdi`ere’s meromorphic continuation of Eisenstein series (March 12, 2012)
4. Eventually-vanishing constant terms
Suitable restrictions ∆ of ∆ to subspaces of L2(Γ\H), where constant terms vanishing above a fixed height
a
y =a, have Friedrichs extensions with compact resolvents.
[4.1] Constant terms vanishing for y > a
For ϕ∈C∞(0,∞), the corresponding pseudo-Eisenstein series is
c
(cid:88)
Ψ (z) = ϕ(Im(γz)) ∈ C∞(Γ\H)
ϕ c
γ∈Γ∞\Γ
Fix a>b(cid:48). Denote the collection of all pseudo-Eisenstein series with test function ϕ supported on [a,∞) by
Ψ = {Ψ :ϕsmooth on (0,+∞), compact support inside [a,+∞)}
≥a ϕ
The collection of L2(Γ\H) functions with constant terms vanishing[4] in y >a is best defined as
L2(Γ\H) = Ψ⊥ = orthogonal complement to Ψ in L2(Γ\H)
a ≥a ≥a
Equivalently, since Ψ ⊂C∞(Γ\H), we can also characterize L2(Γ\H) as the collection of distributions on
≥a c a
Γ\H coming from elements of L2(Γ\H) and annihilating all pseudo-Eisenstein series in Ψ .
≥a
[4.1.1] Proposition: Corresponding test functions are dense in L2(Γ\H) , that is,
a
(cid:16) (cid:17)
L2(Γ\H) = L2(Γ\H)-closure of L2(Γ\H) ∩ C∞(Γ\H)
a a c
Proof: As earlier, fix 0<b<b(cid:48) <∞, and take a smooth cut-off function 0≤τ ≤1 on (0,∞) with

1 (for b(cid:48) ≤y)

τ(y) =
0 (for 0≤y ≤b)
let ϕ (y)=τ(y/t) and Φ (z)=ϕ (Im(z)). Form the corresponding pseudo-Eisenstein series
t t t
(cid:88)
Ψ (z) = ϕ (Im(γ·z))
t t
γ∈Γ∞\Γ
We already proved that (1−Ψ )·f →f in L2(Γ\H). We claim that, for f ∈L2(Γ\H), the constant term of
t
(1−Ψ )·f vanishes for y ≥ a for large t. Indeed, elementary reduction theory assures us that, for large t
t
and y ≥a, Ψ (γ·z)(cid:54)=0 only for γ ∈Γ . Then
t ∞
[4] The constant term c f of a function f on Γ\H is usually defined (somewhat imprecisely) by
P
(cid:90) (cid:18)1 ∗(cid:19)
c f(z) = f(nz)dn (with N = )
P 0 1
N∩Γ\N
Forfixeda,theusualcharacterizationofL2(Γ\H)functionsf withconstanttermsvanishinginy≥awouldbethat
c f(z) = 0 for y ≥ a. The intention is clear, but L2 functions do not have pointwise values. The definition via
P
pseudo-Eisenstein series avoids certain specious arguments.
8
Paul Garrett: Colin de Verdi`ere’s meromorphic continuation of Eisenstein series (March 12, 2012)
(cid:90) 1
(cid:0) (cid:1)
c (1−Ψ )·f (iy) = (1−Ψ )f(x+iy)dx
P t t
0
(cid:90) 1
= (1−ϕ )(y) f(x+iy)dx = (1−ϕ )(y)·c f(iy)
t t P
0
Thus,fory ≥aandlarget,whenc f vanishessodoestheconstanttermof(1−Ψ )·f. Thus,testfunctions
P t
in L2(Γ\H) are dense in L2(Γ\H) . ///
a a
˜
[4.2] The operators ∆ , ∆
a a
Let
C∞(Γ\H) = L2(Γ\H) ∩ C∞(Γ\H)
c a a c
Let ∆ be the unbounded operator on L2(Γ\H) defined by taking the operator ∆, but with domain
a a
C∞(Γ\H) . The density of test functions in L2(Γ\H) proves the symmetry of ∆ , extending integration
c a a a
by parts on test functions. Let ∆˜ be the Friedrichs extension of ∆ to a self-adjoint unbounded operator
a a
on L2(Γ\H) . Let Sob(+1) be the completion of C∞(Γ\H)∩L2(Γ\H) with the Sob(+1)-topology, and
a a c a
similarly for Sob(+2) . By definition, the subspaces of test functions are dense in Sob(+1) and Sob(+2)
a a a
with their finer topologies. Friedrichs’ construction has the property
Sob(+2) ⊂ domain∆˜ ⊂ Sob(+1)
a a a
˜
[4.3] Distributional explication of ∆
a
Let T be the order-zero distribution on Γ\H given by
a
T (f) = (c f)(a) (for f ∈C∞(Γ\H) )
a P c a
Asobservedearlier, theconstant-termmapssendsSob(+1)toSobloc (+1),andthelatteriscontainedin
N\G/K
continuous functions on N\G/K, so T is a continuous functional on Sob(+1). Let A be the distributions
a
on (0,∞) supported at {a}, and understand by A ◦c the composition of the constant-term map with
P
distributions on N\G/K ≈(0,∞) supported on {a}.
[4.3.1] Lemma: The domain in L2(Γ\H) of Friedrichs’ extension ∆˜ is
a a
domain∆˜ = {f ∈L2(Γ\H) : ∆f ∈L2(Γ\H) +A ◦c } (distributional derivative ∆f)
a a a P
The extension ∆˜ is
a
∆˜ f = g (for ∆f ∈g+A ◦c withg ∈L2(Γ\H) )
a P a
In fact, the same assertions hold with A ◦c replaced by C·T .
P a
Proof: The proof consists of a review of Friedrichs’ construction, computing the adjoint of a differential
operator on test functions distributionally. Friedrichs characterizes the resolvent (1−∆˜ )−1 by requiring
a
that it map to Sob(+1) , and requiring
a
(cid:104)(1−∆˜ )−1v,(1−∆)f(cid:105) = (cid:104)v,f(cid:105) (for v ∈L2(Γ\H) , for f ∈C∞(Γ\H) )
a a c a
The existence of (1 − ∆˜ )−1v follows from Riesz-Fischer. Since f is a test function, we can compute
a
distributionally:
(cid:104)v,f(cid:105) = (cid:104)(1−∆˜ )−1v,(1−∆)f(cid:105) = (cid:104)(1−∆)(1−∆˜ )−1v,f(cid:105)
a a
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Paul Garrett: Colin de Verdi`ere’s meromorphic continuation of Eisenstein series (March 12, 2012)
where the pairing is extended from L2(Γ\H) ×C∞(Γ\H) to
a c a
(cid:0)distributions on Γ\H vanishing on Ψ (cid:1) × C∞(Γ\H)
≥a c a
The distribution u=(1−∆)(1−∆˜ )−1v is not completely determined by the conditions
a

(cid:104)u,f(cid:105) = (cid:104)v,f(cid:105) (for all f ∈C∞(Γ\H) )
 c a
(cid:104)u,f(cid:105) = 0 (for all f ∈Ψ )
≥a
Distributionsu−v annihilatingC∞(Γ\H) arenecessarilysupportedon(theimageof)thetailabovey =a,
c a
namely, on
(cid:0) (cid:1)
Y = Γ\ Γ·{x+iy ∈H:y ≥a}
∞
TestfunctionsanddistributionsonthetailY canbedecomposedintoFouriercomponents,becauseΓ \N
∞ ∞
iscompact. Thus, foradistributionu−v toannihilateC∞(Γ\H) requiresnotonlythatu−v besupported
c a
on Y , but, also, that all but the 0th Fourier component of u−v vanish. Thus, u−v is equal to its 0th
∞
Fourier component c (u−v). Annihilation of Ψ implies that u−v =c (u−v) is supported only on the
P ≥a P
boundary ∂Y . Thus, the collection of possible distributions is contained in A ◦c .
∞ P
Conversely, if w ∈ Sob(+1) and (1−∆)w−v = η◦c with η ∈ A, then (cid:104)(1−∆)w−v,f(cid:105) = 0 for both
a P
f ∈C∞(Γ\H) and f ∈Ψ , so w =(∆˜ −λ)−1v.
c a ≥a a
Identifying N\G/K ≈ (0,+∞) by taking the y-coordinate, the distributions A supported on the single
point{a}arefinitelinearcombinationsofDiracdelta(ata)anditsderivatives. However,thespecificsofthe
situationsharplylimittheorderofpossibledistributions, vialocalSobolevtheory, asfollows. Applicationof
the second-order differential operator 1−∆ maps Sob(+1) to the local Sobolev space Sobloc (−1) on Γ\H.
a Γ\H
Application of the constant-term integral produces an element of the local Sobolev space Sobloc (−1),
N\G/K
which we identify with Sobloc(−1) on (0,+∞). Standard Fourier series computations show that Dirac delta
δ at y =a isinSobloc(−1−ε)forall ε>0, butnot inSobloc(−1). Thus, δ(cid:48) ∈Sobloc(−3−ε)forall ε>0,
a 2 2 a 2
but δ(cid:48) (cid:54)∈Sobloc(−3), and so on. That is, only δ itself can arise in this fashion. Thus,
a 2 a
(1−∆)(1−∆˜ )−1v − v ∈ C·T (for all v ∈L2(Γ\H) )
a a a
as claimed. ///
[4.3.2] Remark: In particular, it is conceivable that ∆˜ has eigenvectors whose distributional derivatives
a
include multiples of the distribution T . Indeed, below we will discuss in some detail the fact that truncated
a
Eisenstein series ∧aE whose constant terms ys+c y1−s vanish on the cut-off line y =a are eigenfunctions
s s
for ∆˜ . Such truncated Eisenstein series are not eigenfunctions for ∆, nor for the Friedrichs self-adjoint
a
extension ∆˜ of ∆ on L2(Γ\H).
5. Compactness of Sob(+1) → L2(Γ\H)
a a
We claim that the inclusion Sob(+1) →L2(Γ\H) , from Sob(+1) with its finer topology, is compact.
a a a
For proof, [Colin de Verdi`ere 1981] cites [Lax-Phillips 1976] p. 206, to which we add some details. The total
boundedness criterion for relative compactness requires that, given ε > 0, the image of the unit ball B in
Sob(+1) in L2(Γ\H) can be covered by finitely-many balls of radius (cid:15).
a a
The idea is that the usual Rellich lemma reduces the issue to an estimate on the tail, which follows from the
Sob(+1) condition.
a
10