Weyl’s Law for Heisenberg Manifolds Chung, Khosravi, Petridis, Toth Outline Weyl’s Law for Heisenberg Manifolds Derrick Chung 1 Mahta Khosravi 2 Yiannis Petridis 1 John Toth 3 1 The Graduate Center and Lehman College, City University of New York 2 Institute for Advanced Study 3 McGill University November 10, 2006
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Weyl’s Law for Heisenberg Manifolds Chung, Khosravi, Petridis
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Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
Outline
Weyl’s Law for Heisenberg Manifolds
Derrick Chung1 Mahta Khosravi2 Yiannis Petridis1
John Toth3
1The Graduate Center and Lehman College, City University of New York2Institute for Advanced Study 3McGill University
November 10, 2006
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
The setting
The Heisenberg group
Heisenberg manifolds
Why do we care?
The spectrum ofHeisenberg manifolds
Classicallattice-countingproblems
Results andConjectures
Weyl’s Law
Exponent pairs
Evidence
Average over metrics
Higher dimensions
Hint of proof of Th. 1
Open problems
Outline
The settingThe Heisenberg groupHeisenberg manifoldsWhy do we care?The spectrum of Heisenberg manifoldsClassical lattice-counting problems
Results and ConjecturesWeyl’s LawExponent pairsEvidenceAverage over metricsNumericsHigher dimensionsHint of proof of Th. 1
Open problems
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
The setting
The Heisenberg group
Heisenberg manifolds
Why do we care?
The spectrum ofHeisenberg manifolds
Classicallattice-countingproblems
Results andConjectures
Weyl’s Law
Exponent pairs
Evidence
Average over metrics
Higher dimensions
Hint of proof of Th. 1
Open problems
The Heisenberg group
Heisenberg algebra:
hn = 〈X1, . . . ,Xn,Y1, . . . ,Yn,Z 〉
[Xi ,Xj ] = [Yi ,Yj ] = [Xi ,Z ] = [Yi ,Z ] = 0
[Xi ,Yj ] = δijZ
Heisenberg Group:
Hn =
1 x z
0 Inty
0 0 1
, x, y ∈ Rn, z ∈ R
X (x, y, z) =
0 x z0 0 ty0 0 0
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
The setting
The Heisenberg group
Heisenberg manifolds
Why do we care?
The spectrum ofHeisenberg manifolds
Classicallattice-countingproblems
Results andConjectures
Weyl’s Law
Exponent pairs
Evidence
Average over metrics
Higher dimensions
Hint of proof of Th. 1
Open problems
hn = {X (x, y, z)} ⊂ gl(n + 2,R)
{X (x, y, 0), x, y ∈ Rn} ≡ R2n
Center, derived subalgebra: zn = {X (0, 0, z)}
hn = R2n + zn
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
The setting
The Heisenberg group
Heisenberg manifolds
Why do we care?
The spectrum ofHeisenberg manifolds
Classicallattice-countingproblems
Results andConjectures
Weyl’s Law
Exponent pairs
Evidence
Average over metrics
Higher dimensions
Hint of proof of Th. 1
Open problems
Heisenberg Manifolds
(Γ \ Hn, g)
Γ uniform discrete, g left-invariant metric
S1 ↪→ Γ \ Hn
↓T 2n
Circle bundle over a torusTake r ∈ Zn
+, rj |rj+1. Define
Γr =
1 x z
0 Inty
0 0 1
, xi ∈ riZ, y ∈ Zn, z ∈ Z
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
The setting
The Heisenberg group
Heisenberg manifolds
Why do we care?
The spectrum ofHeisenberg manifolds
Classicallattice-countingproblems
Results andConjectures
Weyl’s Law
Exponent pairs
Evidence
Average over metrics
Higher dimensions
Hint of proof of Th. 1
Open problems
Theorem (Gordon-Wilson)
(a) ∃r:(Γ \ Hn, g) ∼ (Γr \ Hn, g)
(b) ∃φ ∈ Inn(Hn) : hn = R2n ⊕ zn, rel. φ∗(g)
φ∗(g) =
[h2n×2n 0
0 g2n+1
]
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
The setting
The Heisenberg group
Heisenberg manifolds
Why do we care?
The spectrum ofHeisenberg manifolds
Classicallattice-countingproblems
Results andConjectures
Weyl’s Law
Exponent pairs
Evidence
Average over metrics
Higher dimensions
Hint of proof of Th. 1
Open problems
Why do we care?
1. Hn model for CR manifolds (Folland, Stein, Beals,Greiner)
2. Integrable: Butler, J. Geom. Phys. 36(2000)One integral is NOT analytic functionTaimanov: Analytic integrals constrain π1(M)
3. Isospectral manifolds: Gordon, Wilson, Gornet, Pescen = 1, (Γ \ H1, g) determined by its spectrum amongHeisenbergn > 1, if r1r2 · · · rn = r ′1r
′2 · · · r ′n, continuous families
Spec(Γ \ Hn, gt) = Spec(Γ′ \ Hn, g′t)
Almost Inner Automorphisms exist in abundance.
4. (H1, g) is a model geometry in classification of3-manifolds
5. Fourier coefficients of automorphic forms are∫Γ∩N\N φ(ng) dn where Γ ∩ N \ N is Heisenberg
manifold.
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
The setting
The Heisenberg group
Heisenberg manifolds
Why do we care?
The spectrum ofHeisenberg manifolds
Classicallattice-countingproblems
Results andConjectures
Weyl’s Law
Exponent pairs
Evidence
Average over metrics
Higher dimensions
Hint of proof of Th. 1
Open problems
Why do we care?
1. Hn model for CR manifolds (Folland, Stein, Beals,Greiner)
2. Integrable: Butler, J. Geom. Phys. 36(2000)One integral is NOT analytic functionTaimanov: Analytic integrals constrain π1(M)
3. Isospectral manifolds: Gordon, Wilson, Gornet, Pescen = 1, (Γ \ H1, g) determined by its spectrum amongHeisenbergn > 1, if r1r2 · · · rn = r ′1r
′2 · · · r ′n, continuous families
Spec(Γ \ Hn, gt) = Spec(Γ′ \ Hn, g′t)
Almost Inner Automorphisms exist in abundance.
4. (H1, g) is a model geometry in classification of3-manifolds
5. Fourier coefficients of automorphic forms are∫Γ∩N\N φ(ng) dn where Γ ∩ N \ N is Heisenberg
manifold.
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
The setting
The Heisenberg group
Heisenberg manifolds
Why do we care?
The spectrum ofHeisenberg manifolds
Classicallattice-countingproblems
Results andConjectures
Weyl’s Law
Exponent pairs
Evidence
Average over metrics
Higher dimensions
Hint of proof of Th. 1
Open problems
Why do we care?
1. Hn model for CR manifolds (Folland, Stein, Beals,Greiner)
2. Integrable: Butler, J. Geom. Phys. 36(2000)One integral is NOT analytic functionTaimanov: Analytic integrals constrain π1(M)
3. Isospectral manifolds: Gordon, Wilson, Gornet, Pescen = 1, (Γ \ H1, g) determined by its spectrum amongHeisenbergn > 1, if r1r2 · · · rn = r ′1r
′2 · · · r ′n, continuous families
Spec(Γ \ Hn, gt) = Spec(Γ′ \ Hn, g′t)
Almost Inner Automorphisms exist in abundance.
4. (H1, g) is a model geometry in classification of3-manifolds
5. Fourier coefficients of automorphic forms are∫Γ∩N\N φ(ng) dn where Γ ∩ N \ N is Heisenberg
manifold.
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
The setting
The Heisenberg group
Heisenberg manifolds
Why do we care?
The spectrum ofHeisenberg manifolds
Classicallattice-countingproblems
Results andConjectures
Weyl’s Law
Exponent pairs
Evidence
Average over metrics
Higher dimensions
Hint of proof of Th. 1
Open problems
Why do we care?
1. Hn model for CR manifolds (Folland, Stein, Beals,Greiner)
2. Integrable: Butler, J. Geom. Phys. 36(2000)One integral is NOT analytic functionTaimanov: Analytic integrals constrain π1(M)
3. Isospectral manifolds: Gordon, Wilson, Gornet, Pescen = 1, (Γ \ H1, g) determined by its spectrum amongHeisenbergn > 1, if r1r2 · · · rn = r ′1r
′2 · · · r ′n, continuous families
Spec(Γ \ Hn, gt) = Spec(Γ′ \ Hn, g′t)
Almost Inner Automorphisms exist in abundance.
4. (H1, g) is a model geometry in classification of3-manifolds
5. Fourier coefficients of automorphic forms are∫Γ∩N\N φ(ng) dn where Γ ∩ N \ N is Heisenberg
manifold.
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
The setting
The Heisenberg group
Heisenberg manifolds
Why do we care?
The spectrum ofHeisenberg manifolds
Classicallattice-countingproblems
Results andConjectures
Weyl’s Law
Exponent pairs
Evidence
Average over metrics
Higher dimensions
Hint of proof of Th. 1
Open problems
Why do we care?
1. Hn model for CR manifolds (Folland, Stein, Beals,Greiner)
2. Integrable: Butler, J. Geom. Phys. 36(2000)One integral is NOT analytic functionTaimanov: Analytic integrals constrain π1(M)
3. Isospectral manifolds: Gordon, Wilson, Gornet, Pescen = 1, (Γ \ H1, g) determined by its spectrum amongHeisenbergn > 1, if r1r2 · · · rn = r ′1r
′2 · · · r ′n, continuous families
Spec(Γ \ Hn, gt) = Spec(Γ′ \ Hn, g′t)
Almost Inner Automorphisms exist in abundance.
4. (H1, g) is a model geometry in classification of3-manifolds
5. Fourier coefficients of automorphic forms are∫Γ∩N\N φ(ng) dn where Γ ∩ N \ N is Heisenberg
manifold.
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
The setting
The Heisenberg group
Heisenberg manifolds
Why do we care?
The spectrum ofHeisenberg manifolds
Classicallattice-countingproblems
Results andConjectures
Weyl’s Law
Exponent pairs
Evidence
Average over metrics
Higher dimensions
Hint of proof of Th. 1
Open problems
Why do we care?
1. Hn model for CR manifolds (Folland, Stein, Beals,Greiner)
2. Integrable: Butler, J. Geom. Phys. 36(2000)One integral is NOT analytic functionTaimanov: Analytic integrals constrain π1(M)
3. Isospectral manifolds: Gordon, Wilson, Gornet, Pescen = 1, (Γ \ H1, g) determined by its spectrum amongHeisenbergn > 1, if r1r2 · · · rn = r ′1r
′2 · · · r ′n, continuous families
Spec(Γ \ Hn, gt) = Spec(Γ′ \ Hn, g′t)
Almost Inner Automorphisms exist in abundance.
4. (H1, g) is a model geometry in classification of3-manifolds
5. Fourier coefficients of automorphic forms are∫Γ∩N\N φ(ng) dn where Γ ∩ N \ N is Heisenberg
manifold.
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
The setting
The Heisenberg group
Heisenberg manifolds
Why do we care?
The spectrum ofHeisenberg manifolds
Classicallattice-countingproblems
Results andConjectures
Weyl’s Law
Exponent pairs
Evidence
Average over metrics
Higher dimensions
Hint of proof of Th. 1
Open problems
The spectrum of Heisenberg manifolds
Deninger-Singhof, Colin de Verdiere, Gordon-WilsonLet Lr = {X (x, y, 0), xi ∈ riZ, y ∈ Zn}
J =
[0 In−In 0
]±id2
j be the eigenvalues of h−1J
The Spectrum
1. Type I: Spec(Lr \ R2n, h)
2. Type II: µ(y , t1, t2, . . . , tn) =4π2y2
g2n+1+
n∑j=1
2πyd2j (2tj +
1), y ∈ N, ti ∈ Z+,with multiplicity 2ynr1r2 · · · rn.
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
The setting
The Heisenberg group
Heisenberg manifolds
Why do we care?
The spectrum ofHeisenberg manifolds
Classicallattice-countingproblems
Results andConjectures
Weyl’s Law
Exponent pairs
Evidence
Average over metrics
Higher dimensions
Hint of proof of Th. 1
Open problems
The spectrum of Heisenberg manifolds
Deninger-Singhof, Colin de Verdiere, Gordon-WilsonLet Lr = {X (x, y, 0), xi ∈ riZ, y ∈ Zn}
J =
[0 In−In 0
]±id2
j be the eigenvalues of h−1J
The Spectrum
1. Type I: Spec(Lr \ R2n, h)
2. Type II: µ(y , t1, t2, . . . , tn) =4π2y2
g2n+1+
n∑j=1
2πyd2j (2tj +
1), y ∈ N, ti ∈ Z+,with multiplicity 2ynr1r2 · · · rn.
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
The setting
The Heisenberg group
Heisenberg manifolds
Why do we care?
The spectrum ofHeisenberg manifolds
Classicallattice-countingproblems
Results andConjectures
Weyl’s Law
Exponent pairs
Evidence
Average over metrics
Higher dimensions
Hint of proof of Th. 1
Open problems
The spectrum of Heisenberg manifolds
Deninger-Singhof, Colin de Verdiere, Gordon-WilsonLet Lr = {X (x, y, 0), xi ∈ riZ, y ∈ Zn}
J =
[0 In−In 0
]±id2
j be the eigenvalues of h−1J
The Spectrum
1. Type I: Spec(Lr \ R2n, h)
2. Type II: µ(y , t1, t2, . . . , tn) =4π2y2
g2n+1+
n∑j=1
2πyd2j (2tj +
1), y ∈ N, ti ∈ Z+,with multiplicity 2ynr1r2 · · · rn.
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
The setting
The Heisenberg group
Heisenberg manifolds
Why do we care?
The spectrum ofHeisenberg manifolds
Classicallattice-countingproblems
Results andConjectures
Weyl’s Law
Exponent pairs
Evidence
Average over metrics
Higher dimensions
Hint of proof of Th. 1
Open problems
The spectrum of Heisenberg manifolds
Deninger-Singhof, Colin de Verdiere, Gordon-WilsonLet Lr = {X (x, y, 0), xi ∈ riZ, y ∈ Zn}
J =
[0 In−In 0
]±id2
j be the eigenvalues of h−1J
The Spectrum
1. Type I: Spec(Lr \ R2n, h)
2. Type II: µ(y , t1, t2, . . . , tn) =4π2y2
g2n+1+
n∑j=1
2πyd2j (2tj +
1), y ∈ N, ti ∈ Z+,with multiplicity 2ynr1r2 · · · rn.
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
The setting
The Heisenberg group
Heisenberg manifolds
Why do we care?
The spectrum ofHeisenberg manifolds
Classicallattice-countingproblems
Results andConjectures
Weyl’s Law
Exponent pairs
Evidence
Average over metrics
Higher dimensions
Hint of proof of Th. 1
Open problems
Heat kernel, determinant of Laplace operator:Furutani, de Gosson J. Geom. Phys. 48(2003)Question: Are (Hn, g) Quantum Integrable?
Example
I On H1
g0 =
[I2 00 2π
]I 2π factors:
1
2πµ(y , t) =
(y2 + y(2t + 1)
)= y(y + 2t + 1) = yx
with x > y , x 6≡ y(mod 2)
I Leads to lattice-point counting with weight y belowhyperbola xy = λ and line x = yUse Z2 and L = {(x , y) ∈ Z2, x ≡ y(mod 2)}
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
The setting
The Heisenberg group
Heisenberg manifolds
Why do we care?
The spectrum ofHeisenberg manifolds
Classicallattice-countingproblems
Results andConjectures
Weyl’s Law
Exponent pairs
Evidence
Average over metrics
Higher dimensions
Hint of proof of Th. 1
Open problems
Heat kernel, determinant of Laplace operator:Furutani, de Gosson J. Geom. Phys. 48(2003)Question: Are (Hn, g) Quantum Integrable?
Example
I On H1
g0 =
[I2 00 2π
]I 2π factors:
1
2πµ(y , t) =
(y2 + y(2t + 1)
)= y(y + 2t + 1) = yx
with x > y , x 6≡ y(mod 2)
I Leads to lattice-point counting with weight y belowhyperbola xy = λ and line x = yUse Z2 and L = {(x , y) ∈ Z2, x ≡ y(mod 2)}
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
The setting
The Heisenberg group
Heisenberg manifolds
Why do we care?
The spectrum ofHeisenberg manifolds
Classicallattice-countingproblems
Results andConjectures
Weyl’s Law
Exponent pairs
Evidence
Average over metrics
Higher dimensions
Hint of proof of Th. 1
Open problems
Heat kernel, determinant of Laplace operator:Furutani, de Gosson J. Geom. Phys. 48(2003)Question: Are (Hn, g) Quantum Integrable?
Example
I On H1
g0 =
[I2 00 2π
]
I 2π factors:
1
2πµ(y , t) =
(y2 + y(2t + 1)
)= y(y + 2t + 1) = yx
with x > y , x 6≡ y(mod 2)
I Leads to lattice-point counting with weight y belowhyperbola xy = λ and line x = yUse Z2 and L = {(x , y) ∈ Z2, x ≡ y(mod 2)}
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
The setting
The Heisenberg group
Heisenberg manifolds
Why do we care?
The spectrum ofHeisenberg manifolds
Classicallattice-countingproblems
Results andConjectures
Weyl’s Law
Exponent pairs
Evidence
Average over metrics
Higher dimensions
Hint of proof of Th. 1
Open problems
Heat kernel, determinant of Laplace operator:Furutani, de Gosson J. Geom. Phys. 48(2003)Question: Are (Hn, g) Quantum Integrable?
Example
I On H1
g0 =
[I2 00 2π
]I 2π factors:
1
2πµ(y , t) =
(y2 + y(2t + 1)
)= y(y + 2t + 1) = yx
with x > y , x 6≡ y(mod 2)
I Leads to lattice-point counting with weight y belowhyperbola xy = λ and line x = yUse Z2 and L = {(x , y) ∈ Z2, x ≡ y(mod 2)}
Weyl’s Law forHeisenbergManifolds
Chung, Khosravi,Petridis, Toth
The setting
The Heisenberg group
Heisenberg manifolds
Why do we care?
The spectrum ofHeisenberg manifolds
Classicallattice-countingproblems
Results andConjectures
Weyl’s Law
Exponent pairs
Evidence
Average over metrics
Higher dimensions
Hint of proof of Th. 1
Open problems
Heat kernel, determinant of Laplace operator:Furutani, de Gosson J. Geom. Phys. 48(2003)Question: Are (Hn, g) Quantum Integrable?
Example
I On H1
g0 =
[I2 00 2π
]I 2π factors:
1
2πµ(y , t) =
(y2 + y(2t + 1)
)= y(y + 2t + 1) = yx
with x > y , x 6≡ y(mod 2)
I Leads to lattice-point counting with weight y belowhyperbola xy = λ and line x = yUse Z2 and L = {(x , y) ∈ Z2, x ≡ y(mod 2)}