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Trace formula on hyperbolic plane

Charles Li

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History of the slide

The slide is based on talks at Chinese University of Hong Kong on March 2006 and National Taiwan University on April 2006.

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Introduction

Trace formulas were introduced by Selberg in 1956. The original setting used differential geometry. Most of the modern settings use representation theory. In this slide, we will discuss Selberg’s original approach. We will derive Selberg trace formula on compact hyperbolic surface using modern setting.

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Hyperbolic plane

Let H be the upper half plane.

H = {z = x + iy : y > 0.}

A Riemannian metric is given by

\frac{dx^{2} + dy^{2}}{y^{2}} .

The area measure is given by

\frac{dxdy}{y^{2}} .

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[figure: SL_{2}(R)]

2 times 2 matrices with det 1. There is an group action on H by

(\begin{matrix}a & b \\ c & d \end{matrix}) z = \frac{az + b}{cz + d}.

The group action is an isometry.

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distance

The distance between two points is given by

ρ(z, w) = log \frac{|z − w̄| + |z − w|}{|z − w̄| − |z − w|}.

It is more convenient to use u function defined by

cosh ρ(z, w) = 1 + 2u(z, w).

u(z, w) = \frac{|z − w|^{2}}{4 Im z Im w} .

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The Laplace operator

Δ = −y^{2} (\frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} ) .

Define an action T_{g} on a function f of H by

(T_{g}f)(z) = f(gz).

Then

(ΔT_{g}f)(z) = (T_{g}Δf)(z).

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Eigenfunction

We are interesting in the eigenfunctions of Δ. In particular, we are interesting in those functions define on Γ\H, where Γ is a subgroup of SL_{2}(R). When Γ\H is compact, we can define an inner product

⟨f_{1}, f_{2}⟩ = ∫_{Γ\H} f_{1}(z)f_{2}(z)\frac{dxdy}{y^{2}} .

By Stokes’ theorem, Δ is self-adjoint and ⟨Δf, f⟩ > 0

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Integral operator

Suppose X is a measurable topological space, T : L^{2}(X) → L^{2}(X) is an integral operator with kernel function K(x, y). i.e.

T φ(x) = ∫_{X} K(x, y)φ(y)dy

Another way to express the kernel function is

K(x, y) = ∑_{i} T φ_{i}(x)\overline{φ_{i}(y)}

where {φ_{i}} is an orthonormal basis of L^{2}(X). This can be easily checked by

∫_{X} K(x, y)φ_{i}(y)dy = T φ_{i}(x).

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Trace of integral operators tr T = ∑ ⟨T φ_{i}, φ_{i}⟩

∑_{i} ∫_{X} T φ_{i}(x)\overline{φ_{i}(x)}dx = ∫_{X} ∑_{i} T φ_{i}(x)\overline{φ_{i}(x)}dx

= ∫_{X} K(x, x)dx.

(if all the calculations are well-defined.)

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Operators on functions of H

Let L(x, y) be a function on H × H. Suppose L is a function depends only on the distance of x and y. i.e L(x, y) = k(u(x, y)) for some function k. Define an operator on functions of H

T_{L}φ(x) = ∫_{H} L(x, y)φ(y)dy

Easily shown that T_{L} maps functions on Γ\H to functions on Γ\H.

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Spectral side

Assume Γ\H is compact. There exists an orthogonal basis {φ_{i}} of L^{2}(Γ\H) consists of eigenfunctions of Δ.

We can show that T_{L} and Δ commutes and thus we can further assume φ_{i} is eigenvector of T_{L}.

Therefore the trace is a summation related to the eigenvalues of the Laplacian.

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Geometric side

Recall that L(x, y) is a function of the distance. Therefore

L(γx, γy) = L(x, y).

T_{L}φ(x) = ∫_{H} L(x, y)φ(y)dy = ∫_{Γ\H} ∑_{γ∈Γ} L(x, γy)φ(y)dy

We get another expression of the kernel function

K(x, y) = ∑_{γ∈Γ} L(x, γy).

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Trace formula – Selberg’s original approach Divide Γ into conjugacy classes [γ]. Let Γ_{γ} be the elements commute with γ.

tr T_{L} = ∫_{Γ\H} K(x, x)dx = ∫_{Γ\H} ∑_{γ∈Γ} L(x, γx)dx

∑_{[γ]} ∫_{Γ\H} ∑_{δ∈Γ_{γ}\Γ} L(x, δ^{−1}γδx)dx = ∑_{[γ]} ∫_{Γ\H} ∑_{δ∈Γ_{γ}\Γ} L(δx, γδx)dx

= ∑_{[γ]} ∫_{Γ_{γ}\H} L(x, γx)dx.

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another model

Let G = SL_{2}(R). There is a map

G → H, g ↦ g(i) = \frac{ai + b}{ci + d}.

The stabilizer of i is given by

K = \{k_{θ} = (\begin{matrix}cos θ & sin θ \\ − sin θ & cos θ \end{matrix})\}.

Therefore

G/K ≅ H.

The left group multiplication is compatible with the group action on H.

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Iwasawa decomposition

g ∈ SL_{2}(R) can be written as

g = (\begin{matrix}1 & x \\ & 1 \end{matrix}) (\begin{matrix}y^{1/2} \\ & y^{−1/2} \end{matrix}) k_{θ}.

g(i) = x + iy.

Iwasawa decomposition ⇔ cartersian coordinate.

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Cartan decomposition

SL_{2}(R) = K\{(\begin{matrix}t \\ & t^{−1} \end{matrix})\}K

Suppose g = k_{θ_{1}} (\begin{matrix}e^{−r} \\ & e^{r} \end{matrix}) k_{θ}. Then (r, θ_{1}) is the geodesic coordinate of g(i).

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[figure: L^{2}(Γ\ SL_{2}(R))]

G = SL_{2}(R) is a group action on L^{2} = L^{2}(Γ\ SL_{2}(R)) by

R(g)φ(·) = φ(·g).

The action is called right regular representation. Because

Γ\H ≅ Γ\G/K.

Therefore

L^{2}(Γ\H) = L^{2}(Γ\G/K) = L^{2}(Γ\G)^{K}

the last notation denotes the set of right-K-invariant functions.

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Decomposition of representations

Suppose Γ\G is compact. Then

L^{2}(Γ\G) = ⨁ π

where each π = (π, V) is an irreducible subrepresentation of G. L^{2}(Γ\G)^{K} = ⨁ π^{K}.

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Casimir operator

Δ = −y^{2} (\frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} ) + y \frac{∂^{2}}{∂x∂θ} .

Define the Killing form

K(X, Y) = tr(ad X ad Y)

for X, Y ∈ 𝔰𝔩_{2}, the Lie algebra of SL_{2}. Then

Δ = ∑ X_{i}X_{i}^{∗}

where {X_{i}} is a basis of 𝔰𝔩_{2} and {X_{i}^{∗}} is its dual.

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More properties of Casimir operator

Δ is in the center of universal enveloping algebra. Δ commutes with π(g). Therefore Δ is an interwining operator. By Schur’s lemma, Δ is a scaler map on any irreducible representation, i.e. the whole irreducible component π is an eigenspace of Δ.

eigenfunction of Laplacian ⇔ π^{K}.

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Induced representations

An induced representation π_{s} consists of functions satisfies

φ((\begin{matrix}a & b \\ 0 & d \end{matrix}) g) = |\frac{a}{d}|^{\frac{s+1}{2}} φ(g).

The group action is defined by the usual right regular representation

π_{s}(g)φ(·) = φ(·g).

By Iwasawa decomposition, the function depends only on the value on K. There is a unique function that is right invariant on K:

φ_{s}^{0}((\begin{matrix}a & b \\ 0 & d \end{matrix}) k) = |\frac{a}{d}|^{\frac{s+1}{2}} .

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Subrepresentation theorem

Every unitary irreducible representation of SL_{2}(R) is a subrepresentation of π_{s}.

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Unitary representation with K-invariant function

There are two types of unitary representation with K-invariant functions.

1) principal series: s is purely imaginary. π_{s} is irreducible and has K-invariant element. The metric is defined as

⟨φ_{1}, φ_{2}⟩ = ∫_{K} φ_{1}(k)\overline{φ_{2}(k)}dk.

2) complementary series: s is a real number ∈ [−1, 1]. π_{s} is irreducible and has K-invariant element.

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Eigenvalue of Casimir operator

The eigenvalue of Casimir operator on π_{s} is

λ = \frac{1 − s^{2}}{4} .

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[figure: π(f)]

Let f be a function on G. Suppose (π, V) is a representation of G. Then π(f) is sort of the sum of all π(g) with coefficient f(g). i.e.

π(f) = ∫_{G} f(g)π(g)dg

or

π(f)v = ∫_{G} f(g)π(g)vdg.

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Test function f

The test function f we choose is a bi-K-invariant function. i.e. f(k_{θ_{1}}gk_{θ_{2}}) = f(g).

By Cartan deomposition k_{θ_{1}} (\begin{matrix}e^{−r/2} \\ & e^{r/2} \end{matrix}) k_{θ_{2}}, f is a function depends on r.

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Action of R(f)

Because f is left K-invariant, R(f) maps L^{2}(Γ\G) to L^{2}(Γ\G)^{K}. Because f is right K-invariant, R(f) annihilates (L^{2}(Γ\G)^{K})^{⊥}.

L^{2}(Γ\G) \overline{\begin{matrix}R(f) \\  \end{matrix}}L^{2}(Γ\G)

L^{2}(Γ\G)^{K}

orthog. proj.

\begin{matrix}R(f) \\ \end{matrix} L^{2}(Γ\G)^{K}

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L^{2}(Γ\H)  L^{2}(Γ\G)^{K}

Laplace operator Casimir operator eigenfunction π_{s}^{K}

eigenvalue λ \frac{1−s^{2}}{4}

L bi-K-invariant function

T_{L}  R(f)

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Spectral side of the trace formula

π_{s}(f) maps φ_{s}^{0} to a multiple of φ_{s}^{0}. Let λ = \frac{1−s^{2}}{4}

π_{s}(f)φ_{s}^{0} = h(λ)φ_{s}^{0},

π_{s}(f) annihilates (Cφ_{s}^{0})^{⊥}. Therefore

tr π_{s}(f) = h(λ).

Suppose R = ⨁ π_{s_{k}}, then

tr R(f) = ∑_{i} tr π_{s_{i}}(f) = ∑_{i} h(λ_{i}),

where λ_{i} = \frac{1−s_{i}^{2}}{4} .

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Evaluation of h(λ) π_{s}(f)φ_{s}^{0}(x) = h(λ)φ_{s}^{0}(x)

put x = e. h(λ) = ∫_{SL_{2}(R)} f(g)φ_{s}^{0}(g)dg = ∫_{B} f(b)φ_{s}^{0}(b)db

= ∫_{H} f((\begin{matrix}y^{1/2} & y^{−1/2}x \\ & y^{−1/2} \end{matrix}))y^{\frac{1+s}{2}} \frac{dxdy}{y^{2}} .

= ∫_{R^{∗}} (∫_{R} f((\begin{matrix}y^{1/2} & x \\ & y^{−1/2} \end{matrix}) dx) y^{s/2}d^{∗}y.

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The kernel function for R(f)

Write G = SL_{2}(R), X = Γ\G. R(f) is an operator on L^{2}(Γ\G). We want to show that it is an integral operator.

∫_{G} f(g)φ(xg)dg = ∫_{G} f(x^{−1}g)φ(g)dg

= ∫_{Γ\G} ∑_{γ∈Γ} f(x^{−1}γg)φ(γg)dg = ∫_{Γ\G} ∑_{γ∈Γ} f(x^{−1}γg)φ(g)dg

Therefore the kernel function is

K(x, y) = ∑_{γ∈Γ} f(x^{−1}γy).

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Geometric side – representation theoretical approach

tr R(f) = ∫_{Γ\G} K(x, x)dx = ∫_{Γ\G} ∑_{γ∈Γ} f(x^{−1}γx)dx

= ∑_{[γ]} ∫_{Γ\G} ∑_{δ∈Γ_{γ}\Γ} f(x^{−1}δ^{−1}γδx)dx

= ∑_{[γ]} ∫_{Γ_{γ}\G} f(x^{−1}γx)dx.

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Conjugacy classes

1) identity.

2) unipotent: conjugate to (\begin{matrix}1 & ∗ \\ & 1 \end{matrix}).

3) hyperbolic : conjugate to (\begin{matrix}a \\ & b \end{matrix}).

4) elliptic : none of the above. Conjugate to k_{θ}.

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[figure: G → Γ\G]

Assume the map is a covering map. Then Γ has no fixed point. Therefore Γ has no elliptic element(has one fixed point) and no unipotent element (has ∞ as fixed point). Thus Γ consists of identity and hyperbolic elements only.

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Identity term

When γ = e = identity:

Γ_{γ} = Γ. ∫_{Γ_{γ}\G} f(x^{−1}γx)dx = ∫_{Γ_{γ}\G} f(e)dx. = f(e) meas(Γ\G).

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Primitive element

An element γ in Γ is primitive if γ can’t be expressed as P^{ℓ} for some P ∈ Γ and ℓ ∈ Z. Every element in Γ can be expressed as P^{ℓ} for some primitive element P ∈ Γ and ℓ ∈ Z.

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orbital integral

If γ and γ′ are conjugate, say γ′ = δ^{−1}γδ. Then by change of variable g ↦ δgδ^{−1}. We can show that

δ^{−1}Γ_{γ}δ = Γ′_{γ′},

where Γ′ = δ^{−1}Γδ and ∫_{Γ_{γ}\G} f(g^{−1}γg)dg = ∫_{Γ′_{γ′} \G} f(g^{−1}γ′g)dg.

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When γ is diagonal

Suppose γ = P^{ℓ}. Then Γ_{γ} = diagonal element. Therefore Γ_{γ} is generated by P. Let P = (\begin{matrix}p^{1/2} \\ & p^{−1/2} \end{matrix}). Then ℓ log p is the geodesic length between z and γz. ∫_{Γ_{γ}\G} f(g^{−1}γg)dg =

∫_{R^{∗}/⟨P ⟩} ∫_{R} f((\begin{matrix}1 & −x \\ & 1 \end{matrix}) (\begin{matrix}p^{ℓ/2} \\ & p^{−ℓ/2} \end{matrix}) (\begin{matrix}1 & x \\ & 1 \end{matrix}))dx\frac{dy}{y}

= | log p| ∫_{R} f (\begin{matrix}p^{ℓ/2} & −(p^{ℓ/2} − p^{−ℓ/2})x \\ & p^{−ℓ/2} \end{matrix}) dx

= \frac{| log p|}{|p^{ℓ/2} − p^{−ℓ/2}|} ∫_{R} f (\begin{matrix}p^{ℓ/2} & −x \\ & p^{−ℓ/2} \end{matrix}) dx = \frac{| log p|}{|p^{ℓ/2} − p^{−ℓ/2}|}g(ℓ log p).

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Selberg trace formula (first form)

∑ h(λ_{i}) = meas(Γ\G)f(1) + ∑_{P} ∑_{ℓ=1}^{∞} \frac{| log p|}{|p^{ℓ/2} − p^{−ℓ/2}|}g(ℓ log p),

where P runs through all the primitive elements in Γ.

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relationship between f, h and g f

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h  \overline{}g

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relationship between h and g

h(λ) = ∫_{R} (∫_{R} f((\begin{matrix}e^{t/2} & x \\ & e^{−t/2} \end{matrix}) dx) e^{ts/2}dt.

g(t) = ∫_{R} f (\begin{matrix}e^{t/2} & x \\ & e^{−t/2} \end{matrix}) dx.

g is the Fourier transform of h !!!

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relationship between h and f

f(1) = \frac{1}{4π} ∫_{−∞}^{∞} h(r)r tanh(πr)dr.

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Selberg trace formula (final form)

∑ h(λ_{i}) = \frac{meas(Γ\G)}{4π} ∫_{−∞}^{∞} h(r)r tanh(πr)dr

+ ∑_{P} ∑_{ℓ=1}^{∞} \frac{| log p|}{|p^{ℓ/2} − p^{−ℓ/2}|}g(ℓ log p).

Geometric objects  Spectral objects
linear algebra sum of diagonal entries  sum of eigenvalue
of a square matrix  of the matriix
finite group  conjugacy classes  irreducible characters
topology  singular homology  deRham cohomlogy
differential  length of  eigenvalues of
geometry  closed geodesics  the Lapacian
physics  particles  waves
(classical mechanics)  (quantum mechanics)
number theory logarithms of powers of  zeros of ζ(s).

prime numbers

algebraic algebraic cycles motives

geometry

automorphic  rational conjugacy classes automorphic
form  representations

44-1

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Referece

Iwaniec, Spectral Methods of Automorphic forms. Selberg, Collected paper. Makean, H.P. Selberg’s Trace Formula as Applied to Compact Riemann Surface Comm on Pure and Applied Math, Vol XXV 1972. Lang, SL_{2}(R).