Introduction

Singular reduction and quantization

1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions

CIRM, Luminy – October 8, 2018

Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

Pablo Ramacher
Fachbereich Mathematik und Informatik
Philipps-Universität Marburg
1Introduction
Introduction

Let M be a closed connected Riemannian manifold and K a
compact Lie group acting isometrically on M. Further, let

1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law∞

2

P0 : C (M) −→ L (M)be an elliptic, positive, symmetric K -invariant PDO withprincipal symbol p and self-adjoint extension P.

Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions2

Introduction
Introduction

Let M be a closed connected Riemannian manifold and K a
compact Lie group acting isometrically on M. Further, let

1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law∞

2

P0 : C (M) −→ L (M)be an elliptic, positive, symmetric K -invariant PDO withprincipal symbol p and self-adjoint extension P.
,→ (L2 (M), P) represents a quantization of (T ∗ M, p).

Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions2

Introduction
Introduction

Let M be a closed connected Riemannian manifold and K a
compact Lie group acting isometrically on M. Further, let

1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law∞

2

P0 : C (M) −→ L (M)be an elliptic, positive, symmetric K -invariant PDO withprincipal symbol p and self-adjoint extension P.
,→ (L2 (M), P) represents a quantization of (T ∗ M, p).
Let J : T ∗ M −→ k∗ be the moment map and Ωζ := J−1 (ζ).
Classical mechanics
Ωζ /K

←→

Quantum mechanics
M
L2 (M) 'L2 (M)σ

Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actionsb
σ∈K

symplectic reductionPeter-Weyl decomposition2

Introduction
Introduction

Let M be a closed connected Riemannian manifold and K a
compact Lie group acting isometrically on M. Further, let

1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law∞

2

P0 : C (M) −→ L (M)be an elliptic, positive, symmetric K -invariant PDO withprincipal symbol p and self-adjoint extension P.
,→ (L2 (M), P) represents a quantization of (T ∗ M, p).
Let J : T ∗ M −→ k∗ be the moment map and Ωζ := J−1 (ζ).
Classical mechanics
Ωζ /K

←→

Quantum mechanics
M
L2 (M) 'L2 (M)σ

Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actionsb
σ∈K

symplectic reductionPeter-Weyl decomposition

Creed: • Correspondence principle (Bohr)• [Q, R] = 0 (Guillemin-Sternberg)
2

Questions:• spectrum of P on L2 (M)σ ↔ flow of p on Ω0 /K• shape of eigenfunctions of P ↔ symmetries•

equiv. cohomology HK∗ (T ∗ M) ↔ cohomology H ∗ (Ω0 /K )• ergodicity, heat asymptotics, index theory, ...

Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

3

Questions:• spectrum of P on L2 (M)σ ↔ flow of p on Ω0 /K• shape of eigenfunctions of P ↔ symmetries•

equiv. cohomology HK∗ (T ∗ M) ↔ cohomology H ∗ (Ω0 /K )• ergodicity, heat asymptotics, index theory, ...

Problem: In the presence of singular orbits, serious difficulties
arise.

Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

3

Questions:Introduction• spectrum of P on L2 (M)σ ↔ flow of p on Ω0 /K• shape of eigenfunctions of P ↔ symmetries•

equiv. cohomology HK∗ (T ∗ M) ↔ cohomology H ∗ (Ω0 /K )• ergodicity, heat asymptotics, index theory, ...

Problem: In the presence of singular orbits, serious difficulties
arise.
Ultimately, one has to understand the asymptotic behavior of
the Witten integral
ZZ
I(µ) =
eiµ J(x,ξ)(X ) a(x, ξ, X ) dX dx dξ, µ → +∞,T ∗M

1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actionsk3

Questions:Introduction• spectrum of P on L2 (M)σ ↔ flow of p on Ω0 /K• shape of eigenfunctions of P ↔ symmetries•

equiv. cohomology HK∗ (T ∗ M) ↔ cohomology H ∗ (Ω0 /K )• ergodicity, heat asymptotics, index theory, ...

Problem: In the presence of singular orbits, serious difficulties
arise.
Ultimately, one has to understand the asymptotic behavior of
the Witten integral
ZZ
I(µ) =
eiµ J(x,ξ)(X ) a(x, ξ, X ) dX dx dξ, µ → +∞,T ∗M

1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actionsk

via the stationary phase theorem, where a ∈ C∞c and
Φ(x, ξ, X ) = J(x, ξ)(X )has, in general, a singular critical set.

3

Questions:Introduction• spectrum of P on L2 (M)σ ↔ flow of p on Ω0 /K• shape of eigenfunctions of P ↔ symmetries•

equiv. cohomology HK∗ (T ∗ M) ↔ cohomology H ∗ (Ω0 /K )• ergodicity, heat asymptotics, index theory, ...

Problem: In the presence of singular orbits, serious difficulties
arise.
Ultimately, one has to understand the asymptotic behavior of
the Witten integral
ZZ
I(µ) =
eiµ J(x,ξ)(X ) a(x, ξ, X ) dX dx dξ, µ → +∞,T ∗M

1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actionsk

via the stationary phase theorem, where a ∈ C∞c and
Φ(x, ξ, X ) = J(x, ξ)(X )has, in general, a singular critical set.
,→overcome this problem by using resolution of singularities.
3

Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

1.
Equivariant spectral geometry

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

4

The spectral function of an elliptic operator
Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

5

The spectral function of an elliptic operator
Introduction

Consider• a compact Riemannian manifold M of dimension n,• a positive, symmetric, elliptic PDO P0 : C∞ (M) −→ L2 (M)of order m on M with principal symbol p(x, ξ),• its self-adjoint extension P with spectral resolution Ej ,ONB of eigenfunctions ϕj , and eigenvalues λj .

1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

5

The spectral function of an elliptic operator
Introduction
1. Equivariant spectral
geometry

ConsiderThe spectral function of anelliptic operator• a compact Riemannian manifold M of dimension n,

Equivariant Weyl law• a positive, symmetric, elliptic PDO P0 : C∞ (M) −→ L2 (M)of order m on M with principal symbol p(x, ξ),Classical and quantum
ergodicity

ONB of eigenfunctions ϕj , and eigenvalues λj .
pDefine for µ > 0, µj := m λj , and x, y ∈ MXµj ≤µϕj (x)ϕj (y )spectral function of Q :=Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity• its self-adjoint extension P with spectral resolution Ej ,e(x, y , µ) :=Convex bounds foreigenfunctions

Symmetries and ergodicity

3. Equivariant
cohomology

√
m

P.

Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

5

The spectral function of an elliptic operator
Introduction
1. Equivariant spectral
geometry

ConsiderThe spectral function of anelliptic operator• a compact Riemannian manifold M of dimension n,

Equivariant Weyl law• a positive, symmetric, elliptic PDO P0 : C∞ (M) −→ L2 (M)of order m on M with principal symbol p(x, ξ),Classical and quantum
ergodicity

ONB of eigenfunctions ϕj , and eigenvalues λj .
pDefine for µ > 0, µj := m λj , and x, y ∈ MXϕj (x)ϕj (y )spectral function of Q :=Symmetries and ergodicity

3. Equivariant
cohomology

√
m

P.

Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actionsµj ≤µLocal Weyl law. By Hörmander one hasZµne(x, x, µ) =dξ + O(µn−1 ),(2π)n p(x,ξ)<1

Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity• its self-adjoint extension P with spectral resolution Ej ,e(x, y , µ) :=Convex bounds foreigenfunctionsµ → +∞.

5

The spectral function of an elliptic operator
Introduction
1. Equivariant spectral
geometry

ConsiderThe spectral function of anelliptic operator• a compact Riemannian manifold M of dimension n,

Equivariant Weyl law• a positive, symmetric, elliptic PDO P0 : C∞ (M) −→ L2 (M)of order m on M with principal symbol p(x, ξ),Classical and quantum
ergodicity

ONB of eigenfunctions ϕj , and eigenvalues λj .
pDefine for µ > 0, µj := m λj , and x, y ∈ MXϕj (x)ϕj (y )spectral function of Q :=Symmetries and ergodicity

3. Equivariant
cohomology

√
m

P.

Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actionsµj ≤µLocal Weyl law. By Hörmander one hasZµne(x, x, µ) =dξ + O(µn−1 ),(2π)n p(x,ξ)<1

Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity• its self-adjoint extension P with spectral resolution Ej ,e(x, y , µ) :=Convex bounds foreigenfunctionsµ → +∞.

Proof: Via Fourier integral operators:
5Approximate sµ (x, y ) = e(x, y , µ + 1) − e(x, y , µ)Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

6

Approximate sµ (x, y ) = e(x, y , µ + 1) − e(x, y , µ) by the
Schwartz kernel of
s̃µ =

∞
X

Introduction
1. Equivariant spectral
geometry%(µ − µj )Ej

j=0

where % ∈ S(R, R+ ), %(0) = 1, supp %b ⊂ (−δ, δ) for some δ > 0.

The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

6

Approximate sµ (x, y ) = e(x, y , µ + 1) − e(x, y , µ) by the
Schwartz kernel of
s̃µ =

∞
X

1. Equivariant spectral
geometry%(µ − µj )Ej

j=0

where % ∈ S(R, R+ ), %(0) = 1, supp %b ⊂ (−δ, δ) for some δ > 0.
Now,
Z
d
s̃µ =
eitµ %(t)U(t)dt,
U(t) = e−itQ .
R

Introduction

The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

6

Approximate sµ (x, y ) = e(x, y , µ + 1) − e(x, y , µ) by the
Schwartz kernel of
s̃µ =

∞
X

Introduction
1. Equivariant spectral
geometry%(µ − µj )Ej

j=0

where % ∈ S(R, R+ ), %(0) = 1, supp %b ⊂ (−δ, δ) for some δ > 0.
Now,
Z
d
s̃µ =
eitµ %(t)U(t)dt,
U(t) = e−itQ .
R

The kernel of U(t) is locally given by an oscillatory integralZŨ(t, x̃, ỹ ) =ei(ψ(t,x̃,η)−hỹ ,ηi) a(t, x̃, η)dη, x̃, ỹ ∈ Rn ,The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actionsRn

6

Approximate sµ (x, y ) = e(x, y , µ + 1) − e(x, y , µ) by the
Schwartz kernel of
s̃µ =

∞
X

Introduction
1. Equivariant spectral
geometry%(µ − µj )Ej

j=0

The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions

where % ∈ S(R, R+ ), %(0) = 1, supp %b ⊂ (−δ, δ) for some δ > 0.
Now,
Z
d
s̃µ =
eitµ %(t)U(t)dt,
U(t) = e−itQ .
R

The kernel of U(t) is locally given by an oscillatory integralZŨ(t, x̃, ỹ ) =ei(ψ(t,x̃,η)−hỹ ,ηi) a(t, x̃, η)dη, x̃, ỹ ∈ Rn ,Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actionsRn0
where a ∈ Sphg
, and ψ solves the Hamilton-Jacobi problem

r 
∂ψ 
∂ψ
m
+ p x,
= 0,
∂t
∂ x̃

ψ(0, x̃, η) = hx̃, ηi.

Asymptotics for Ks̃µ (x, y ) then yield the assertion.
6

Now, let K be a compact Lie group acting isometrically on Mband P0 be K -invariant. Define for σ ∈ KXeσ (x, y , µ) :=ϕj (x)ϕj (y )reduced spectral function.
µj ≤µ,

ϕj ∈L2 (M)σIntroduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

7

Now, let K be a compact Lie group acting isometrically on Mband P0 be K -invariant. Define for σ ∈ KXeσ (x, y , µ) :=ϕj (x)ϕj (y )reduced spectral function.

Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl lawµj ≤µ,

Convex bounds foreigenfunctions

ϕj ∈L2 (M)σSubconvex bounds for
Hecke–Maass forms

Approximate eσ (x, y , µ + 1) − eσ (x, y , µ) byX
Ks̃µ ◦Πσ (x, y ) =%(µ − µj )ϕj (x)ϕj (y )µj ≤µ,

ϕj ∈L2 (M)σ

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

7

Now, let K be a compact Lie group acting isometrically on Mband P0 be K -invariant. Define for σ ∈ KXeσ (x, y , µ) :=ϕj (x)ϕj (y )reduced spectral function.

Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl lawµj ≤µ,

Convex bounds foreigenfunctions

ϕj ∈L2 (M)σSubconvex bounds for
Hecke–Maass forms

Approximate eσ (x, y , µ + 1) − eσ (x, y , µ) byX
Ks̃µ ◦Πσ (x, y ) =%(µ − µj )ϕj (x)ϕj (y )µj ≤µ,

ϕj ∈L2 (M)σ

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae

where

Residue formulae for circle
actionsZχσ (k )π(k ) dk

Πσ = d σK

is the projector onto L2 (M)σ ,• dσ = dim σ,• χσ character of σ,• (π, L2 (M)) regular representation.
7

Introduction

Ks̃µ ◦Πσ (x, y ) is a superposition of Witten-type integrals of theform
Z Z
Ix,y (µ) =eiµΦx,y (ω,k ) a(x, y , ω, k ) dk dω,ΣxKnwhere a ∈ C∞
c , Σx ≡ {ω ∈ R : q(x, ω) = 1}, (Y , κ) arecoordinates, andΦx,y (ω, k ) := hκ(x) − κ(k · y ), ωiRn

1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions8

Introduction

Ks̃µ ◦Πσ (x, y ) is a superposition of Witten-type integrals of theform
Z Z
Ix,y (µ) =eiµΦx,y (ω,k ) a(x, y , ω, k ) dk dω,ΣxKnwhere a ∈ C∞
c , Σx ≡ {ω ∈ R : q(x, ω) = 1}, (Y , κ) arecoordinates, andΦx,y (ω, k ) := hκ(x) − κ(k · y ), ωiRn

1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology

has critical set

Atiyah-Bott-Berline-Vergne
localization
Residue formulae

Crit Φx,y = {(ω, k ) : κ(x) − κ(k · y ) ∈ Nω Σx , (k · y , ω) ∈ Ω0 }.
Ω0 is a stratified space=⇒

Residue formulae for circle
actions

Caustic behaviour

8Introduction

Ks̃µ ◦Πσ (x, y ) is a superposition of Witten-type integrals of theform
Z Z
Ix,y (µ) =eiµΦx,y (ω,k ) a(x, y , ω, k ) dk dω,ΣxKnwhere a ∈ C∞
c , Σx ≡ {ω ∈ R : q(x, ω) = 1}, (Y , κ) arecoordinates, andΦx,y (ω, k ) := hκ(x) − κ(k · y ), ωiRn

1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology

has critical set

Atiyah-Bott-Berline-Vergne
localization
Residue formulae

Crit Φx,y = {(ω, k ) : κ(x) − κ(k · y ) ∈ Nω Σx , (k · y , ω) ∈ Ω0 }.
Ω0 is a stratified space=⇒

Residue formulae for circle
actions

Caustic behaviour,→ Study asymptotic behavior via stationary phase methodand desingularization.

8

Equivariant Weyl law
Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions9

Equivariant Weyl law
Introduction
1. Equivariant spectral
geometry

Theorem (Reduced local Weyl law, 2016)

The spectral function of anelliptic operatoreσ (x, x, µ) = µn−κxdσ [πσ|Kx : 1](2π)n−κx

Equivariant Weyl law

Zdξ +Ox,σ (µ),{(x,ξ)∈Ω0 , p(x,ξ)<1}

where Ox,σ (µ) = Ox,σ (µn−κx −1 ), κx = dim K · x.

Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions9

Equivariant Weyl law
Introduction
1. Equivariant spectral
geometry

Theorem (Reduced local Weyl law, 2016)

The spectral function of anelliptic operatoreσ (x, x, µ) = µn−κxdσ [πσ|Kx : 1](2π)n−κx

Equivariant Weyl law

Zdξ +Ox,σ (µ),{(x,ξ)∈Ω0 , p(x,ξ)<1}

where Ox,σ (µ) = Ox,σ (µn−κx −1 ), κx = dim K · x.
Coefficients and exponents exhibit caustic behaviour in x.

Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions9

Equivariant Weyl law
Introduction
1. Equivariant spectral
geometry

Theorem (Reduced local Weyl law, 2016)

The spectral function of anelliptic operatoreσ (x, x, µ) = µn−κxdσ [πσ|Kx : 1](2π)n−κx

Equivariant Weyl law

Zdξ +Ox,σ (µ),{(x,ξ)∈Ω0 , p(x,ξ)<1}

where Ox,σ (µ) = Ox,σ (µn−κx −1 ), κx = dim K · x.
Coefficients and exponents exhibit caustic behaviour in x.
Example. Let M = S 2 , G = SO(2), P0 = −∆ the Laplace
operator, Yk ,m the spherical harmonic functions witheigenvalues k (k + 1), Pk ,m the Legendre polynomials. Then(0 ≤ ϕ < 2π, 0 ≤ θ < π)(k
θ = 0, π
2k+
1|Yk ,0 (ϕ, θ)|2 =|Pk ,0 (cos θ)|2 ≈−14π(sin θ)θ ∈ (0, π).

Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions9

Equivariant Weyl law
Introduction
1. Equivariant spectral
geometry

Theorem (Reduced local Weyl law, 2016)

The spectral function of anelliptic operatoreσ (x, x, µ) = µn−κxdσ [πσ|Kx : 1](2π)n−κx

Equivariant Weyl law

Zdξ +Ox,σ (µ),{(x,ξ)∈Ω0 , p(x,ξ)<1}

where Ox,σ (µ) = Ox,σ (µn−κx −1 ), κx = dim K · x.
Coefficients and exponents exhibit caustic behaviour in x.
Example. Let M = S 2 , G = SO(2), P0 = −∆ the Laplace
operator, Yk ,m the spherical harmonic functions witheigenvalues k (k + 1), Pk ,m the Legendre polynomials. Then(0 ≤ ϕ < 2π, 0 ≤ θ < π)(k
θ = 0, π
2k+
1|Yk ,0 (ϕ, θ)|2 =|Pk ,0 (cos θ)|2 ≈−14π(sin θ)θ ∈ (0, π).

Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions,→ resolution of singularities yields pointwise description ofeσ (x, x, µ) and eigenfunctions.
9Consider the orbit type decomposition of M
Introduction

M = M1 ∪ · · · ∪ Mℵ .

1. Equivariant spectral
geometry
The spectral function of anelliptic operator

Theorem (Singular reduced local Weyl law, 2016)n−κ

eσ (x, x, µ) =µdσ
Lx,σ (τ1 , τ2 , . . . ) + Ox,σ (µn−κ−1) ,(2π)n−κwhere Lx,σ , Ox,σ are rational functions in τℵ = dist (x, Mℵ )bounded in x and κ is the maximal orbit dimension.

Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

10

Consider the orbit type decomposition of M
Introduction

M = M1 ∪ · · · ∪ Mℵ .

1. Equivariant spectral
geometry
The spectral function of anelliptic operator

Theorem (Singular reduced local Weyl law, 2016)n−κ

eσ (x, x, µ) =µdσ
Lx,σ (τ1 , τ2 , . . . ) + Ox,σ (µn−κ−1) ,(2π)n−κwhere Lx,σ , Ox,σ are rational functions in τℵ = dist (x, Mℵ )bounded in x and κ is the maximal orbit dimension.

Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology

Corollary (Reduced Weyl’s law, 2010)

Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actionsdσ [πσ|H : 1]eσ (x, x, µ) dx =vol[(Ω ∩ S ∗ M)/K ]µn−κ + Oµ ,(n − κ)(2π)n−κMZwhere Oµ = O(µn−κ−1 (log µ)Λ ), Λ ∈ N, andS ∗ M = {(x, ξ) ∈ T ∗ M : p(x, ξ) = 1}.

10

Convex bounds for eigenfunctions
Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions11

Convex bounds for eigenfunctions
Non-equivariant bounds. As a consequence of Hörmander’slocal Weyl law one obtains

Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator

n−1
2m

kϕj k∞  λjSeeger-Sogge: Lp -bounds.

convex bound.

Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions11

Convex bounds for eigenfunctions
Non-equivariant bounds. As a consequence of Hörmander’slocal Weyl law one obtains

Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator

n−1
2m

kϕj k∞  λjconvex bound.

Seeger-Sogge: Lp -bounds.
Guiding idea. If the ϕj are simultaneous eigenfunctions of afamily of commuting differential operators, better bounds
should hold.

Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions11

Convex bounds for eigenfunctions
Non-equivariant bounds. As a consequence of Hörmander’slocal Weyl law one obtains

Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator

n−1
2m

kϕj k∞  λjconvex bound.

Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

Seeger-Sogge: Lp -bounds.
Guiding idea. If the ϕj are simultaneous eigenfunctions of afamily of commuting differential operators, better bounds
should hold.
Equivariant bounds. Assume that K acts on M with orbits ofthe same dimension κ ≤ n − 1.

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

Corollary (2018)

For ϕj ∈ L2 (M)σ one hasn−κ−1

kϕj k∞ σ λj 2mequivariant convex bound.

11

Convex bounds for eigenfunctions
Non-equivariant bounds. As a consequence of Hörmander’slocal Weyl law one obtains

Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator

n−1
2m

kϕj k∞  λjconvex bound.

Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

Seeger-Sogge: Lp -bounds.
Guiding idea. If the ϕj are simultaneous eigenfunctions of afamily of commuting differential operators, better bounds
should hold.
Equivariant bounds. Assume that K acts on M with orbits ofthe same dimension κ ≤ n − 1.

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

Corollary (2018)

For ϕj ∈ L2 (M)σ one hasn−κ−1

kϕj k∞ σ λj 2mequivariant convex bound.

Similarly, equivariant Lp -bounds can be derived.

11

Subconvex bounds for Hecke–Maas forms
Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

12

Subconvex bounds for Hecke–Maas forms
Spherical bounds. Let• G = SL(2, R), K = SO(2),• Γ ⊂ G arithmetic co-compact discrete subgroup,• ∆ Laplace operator on Γ\G/K ,• ϕj ONB of eigenfunctions of ∆ and the Hecke operators.

Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

12

Subconvex bounds for Hecke–Maas forms
Spherical bounds. Let• G = SL(2, R), K = SO(2),• Γ ⊂ G arithmetic co-compact discrete subgroup,• ∆ Laplace operator on Γ\G/K ,• ϕj ONB of eigenfunctions of ∆ and the Hecke operators.
Iwaniec-Sarnak proved for ϕj ∈ L2 (Γ\G/K ) and any ε > 0
5/24+εkϕj k∞ ε λjspherical subconvex bound.

Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

12

Subconvex bounds for Hecke–Maas forms
Spherical bounds. Let• G = SL(2, R), K = SO(2),• Γ ⊂ G arithmetic co-compact discrete subgroup,• ∆ Laplace operator on Γ\G/K ,• ϕj ONB of eigenfunctions of ∆ and the Hecke operators.
Iwaniec-Sarnak proved for ϕj ∈ L2 (Γ\G/K ) and any ε > 0
5/24+εkϕj k∞ ε λjspherical subconvex bound.

Non-spherical bounds. Let ϕj be an ONB of eigenfunctions of
the Laplace operator ∆G on Γ\G and the Hecke operators.

Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

Theorem (–,Wakatsuki, 2018)b
For ϕj ∈ L2 (Γ\G) one has uniformly in σ ∈ K
5/24+εkϕj k∞ ε λj

non-spherical subconvex bound.

12

Subconvex bounds for Hecke–Maas forms
Spherical bounds. Let• G = SL(2, R), K = SO(2),• Γ ⊂ G arithmetic co-compact discrete subgroup,• ∆ Laplace operator on Γ\G/K ,• ϕj ONB of eigenfunctions of ∆ and the Hecke operators.
Iwaniec-Sarnak proved for ϕj ∈ L2 (Γ\G/K ) and any ε > 0
5/24+εkϕj k∞ ε λjspherical subconvex bound.

Non-spherical bounds. Let ϕj be an ONB of eigenfunctions of
the Laplace operator ∆G on Γ\G and the Hecke operators.

Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

Theorem (–,Wakatsuki, 2018)b
For ϕj ∈ L2 (Γ\G) one has uniformly in σ ∈ K
5/24+εkϕj k∞ ε λj

non-spherical subconvex bound.

Proof: Via asymptotics for Ks̃µ ◦Πσ (x, y ) in a neighborhood of
the diagonal and arithmetic amplification.

12

Application. Let C be the Casimir operator on G, so thatb 3 σl ≡ l ∈ Z. For fixed Casimir∆ = −C + l 2 /4 on L2 (Γ\G)σl , K
eigenvalue one obtains

Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law

kϕj k∞ ε (1 + |l|)5/12+ε ,ϕj ∈ L2 (Γ\G)σl .

Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions13

Application. Let C be the Casimir operator on G, so thatb 3 σl ≡ l ∈ Z. For fixed Casimir∆ = −C + l 2 /4 on L2 (Γ\G)σl , K
eigenvalue one obtains

Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law

kϕj k∞ ε (1 + |l|)5/12+ε ,ϕj ∈ L2 (Γ\G)σl .

,→ Previously shown by Venkatesh using equidistribution
methods. Best known bound by Reznikov reads (1 + |l|)1/3+ε .

Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions13

Application. Let C be the Casimir operator on G, so thatb 3 σl ≡ l ∈ Z. For fixed Casimir∆ = −C + l 2 /4 on L2 (Γ\G)σl , K
eigenvalue one obtains

Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law

kϕj k∞ ε (1 + |l|)5/12+ε ,ϕj ∈ L2 (Γ\G)σl .

,→ Previously shown by Venkatesh using equidistribution
methods. Best known bound by Reznikov reads (1 + |l|)1/3+ε .
Corollary (–, Wakatsuki, 2018)

If f : H → C is a classical automorphic form of weight l ∈ N,then
kf k∞ ε l 5/12+ε .

Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions13

Application. Let C be the Casimir operator on G, so thatb 3 σl ≡ l ∈ Z. For fixed Casimir∆ = −C + l 2 /4 on L2 (Γ\G)σl , K
eigenvalue one obtains

Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law

kϕj k∞ ε (1 + |l|)5/12+ε ,ϕj ∈ L2 (Γ\G)σl .

,→ Previously shown by Venkatesh using equidistribution
methods. Best known bound by Reznikov reads (1 + |l|)1/3+ε .
Corollary (–, Wakatsuki, 2018)

If f : H → C is a classical automorphic form of weight l ∈ N,then
kf k∞ ε l 5/12+ε .

Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

1
31
Remark. The best previously known exponent was 12 − 33
= 66
and shown by Das-Sengupta.

13

Application. Let C be the Casimir operator on G, so thatb 3 σl ≡ l ∈ Z. For fixed Casimir∆ = −C + l 2 /4 on L2 (Γ\G)σl , K
eigenvalue one obtains

Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law

kϕj k∞ ε (1 + |l|)5/12+ε ,ϕj ∈ L2 (Γ\G)σl .

,→ Previously shown by Venkatesh using equidistribution
methods. Best known bound by Reznikov reads (1 + |l|)1/3+ε .
Corollary (–, Wakatsuki, 2018)

If f : H → C is a classical automorphic form of weight l ∈ N,then
kf k∞ ε l 5/12+ε .

Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

1
31
Remark. The best previously known exponent was 12 − 33
= 66
and shown by Das-Sengupta.

Equivariant subconvex bounds can be shown also for rather
general algebraic groups, based on previous work of Marshall.
13

Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2.
Equivariant quantum ergodicity

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

14

Classical and quantum ergodicity
Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

15

Classical and quantum ergodicity
Introduction

As before, let M be a compact Riemannian manifold M ofdimension n,∆ : C∞ (M) −→ L2 (M)the Laplacian, and ϕj an ONB of eigenfunctions ϕj witheigenvalues λj .

1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

15

Classical and quantum ergodicity
Introduction

As before, let M be a compact Riemannian manifold M ofdimension n,∆ : C∞ (M) −→ L2 (M)the Laplacian, and ϕj an ONB of eigenfunctions ϕj witheigenvalues λj .
Classical mechanics:

1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity• S ∗ M co-shpere bundle ≡ phase space of free particle• a ∈ C ∞ (S ∗ M) function ≡ observable∗∗• Φt : S M → S M geodesic flow ≡ motion

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

15

Classical and quantum ergodicity
Introduction

As before, let M be a compact Riemannian manifold M ofdimension n,∆ : C∞ (M) −→ L2 (M)the Laplacian, and ϕj an ONB of eigenfunctions ϕj witheigenvalues λj .
Classical mechanics:

1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity• S ∗ M co-shpere bundle ≡ phase space of free particle• a ∈ C ∞ (S ∗ M) function ≡ observable∗∗• Φt : S M → S M geodesic flow ≡ motion

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

Quantum mechanics:• L2 (M) ≡ space of states• A self-adjoint operator ≡ observable• hAϕ, ϕiL2 ≡ expectation values

15Classical and quantum ergodicity
Introduction

As before, let M be a compact Riemannian manifold M ofdimension n,∆ : C∞ (M) −→ L2 (M)the Laplacian, and ϕj an ONB of eigenfunctions ϕj witheigenvalues λj .
Classical mechanics:

1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity• S ∗ M co-shpere bundle ≡ phase space of free particle• a ∈ C ∞ (S ∗ M) function ≡ observable∗∗• Φt : S M → S M geodesic flow ≡ motion

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

Quantum mechanics:• L2 (M) ≡ space of states• A self-adjoint operator ≡ observable• hAϕ, ϕiL2 ≡ expectation values

Quantization map. S l (M) 3 a 7−→ Op~ (a), ~ Planck’s constant.
15

By the correspondence principle,
distributionof trajectories←→

distributionof eigenfunctions

Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions16

By the correspondence principle,
distributionof trajectories←→

Introduction

distributionof eigenfunctions
2,→ Study weak convergence of the measures |ϕj | dM.

1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions16

By the correspondence principle,
distributionof trajectories←→

Introduction

distributionof eigenfunctions
2,→ Study weak convergence of the measures |ϕj | dM.
Quantum ergodicity theorem. Consider the distributionsµj : C∞ (S ∗ M) −→ C,

a 7→ hOp~ (a)ϕj , ϕj iL2

1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

If Φt is ergodic, Shnirelman, CdV, and Zelditch showed that ∃subsequence {ϕjk } of density 1 such thatµjk → d(S ∗ M),|ϕjk |2 dM → dM.

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions16

By the correspondence principle,
distributionof trajectories←→

Introduction

distributionof eigenfunctions
2,→ Study weak convergence of the measures |ϕj | dM.
Quantum ergodicity theorem. Consider the distributionsµj : C∞ (S ∗ M) −→ C,

a 7→ hOp~ (a)ϕj , ϕj iL2

1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

If Φt is ergodic, Shnirelman, CdV, and Zelditch showed that ∃subsequence {ϕjk } of density 1 such thatµjk → d(S ∗ M),|ϕjk |2 dM → dM.

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

Problem. Study ergodicity in the presence of symmetries.
,→ Both the classical and quantum system behave lesschaotically. By dividing out symmetries (i.e. order), ergodic
properties should emerge.

16

By the correspondence principle,
distributionof trajectories←→

Introduction

distributionof eigenfunctions
2,→ Study weak convergence of the measures |ϕj | dM.
Quantum ergodicity theorem. Consider the distributionsµj : C∞ (S ∗ M) −→ C,

a 7→ hOp~ (a)ϕj , ϕj iL2

1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

If Φt is ergodic, Shnirelman, CdV, and Zelditch showed that ∃subsequence {ϕjk } of density 1 such thatµjk → d(S ∗ M),|ϕjk |2 dM → dM.

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

Problem. Study ergodicity in the presence of symmetries.
,→ Both the classical and quantum system behave lesschaotically. By dividing out symmetries (i.e. order), ergodic
properties should emerge.
,→ Singular symplectic reduction (Sjamaar-Lerman-Bates)and desingularization yield
16Symmetries and ergodicity
Introductionb.
e t be the reduced geodesic flow on (Ω0 ∩ Mreg )/K , σ ∈ K
Let Φ

1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions17

Symmetries and ergodicity
Introductionb.
e t be the reduced geodesic flow on (Ω0 ∩ Mreg )/K , σ ∈ K
Let ΦTheorem (Equivariant quantum ergodicity, Küster, –, 2016)

1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law

e t is ergodic, and let ϕσ be an ONB of L2 (M)σ of
Assume that Φjeigenfunctions of −∆. Then ∃ subsequence {ϕσjk } of density 1such that ∀a ∈ C∞ (M)Zdµa
Op(a)ϕσjk , ϕσjk L2 → c ·.
S ∗ M ∩ Ωreg vol O
Furthermore, |ϕσjk |2 dM → c · dM/vol O.

Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions17

Symmetries and ergodicity
Introductionb.
e t be the reduced geodesic flow on (Ω0 ∩ Mreg )/K , σ ∈ K
Let ΦTheorem (Equivariant quantum ergodicity, Küster, –, 2016)

1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law

e t is ergodic, and let ϕσ be an ONB of L2 (M)σ of
Assume that Φjeigenfunctions of −∆. Then ∃ subsequence {ϕσjk } of density 1such that ∀a ∈ C∞ (M)Zdµa
Op(a)ϕσjk , ϕσjk L2 → c ·.
S ∗ M ∩ Ωreg vol O
Furthermore, |ϕσjk |2 dM → c · dM/vol O.

Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions,→ already shown by Kordyukov if M/K is an orbifold.

17Symmetries and ergodicity
Introductionb.
e t be the reduced geodesic flow on (Ω0 ∩ Mreg )/K , σ ∈ K
Let ΦTheorem (Equivariant quantum ergodicity, Küster, –, 2016)

1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law

e t is ergodic, and let ϕσ be an ONB of L2 (M)σ of
Assume that Φjeigenfunctions of −∆. Then ∃ subsequence {ϕσjk } of density 1such that ∀a ∈ C∞ (M)Zdµa
Op(a)ϕσjk , ϕσjk L2 → c ·.
S ∗ M ∩ Ωreg vol O
Furthermore, |ϕσjk |2 dM → c · dM/vol O.

Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions,→ already shown by Kordyukov if M/K is an orbifold.
Example. Let M ⊂ R3 be a surface of revolution,el,m (ϕ, θ) = fl,m (θ)eilϕ a basis of L2 (M)σl . Then|el,m |2 dM → c · dM/R,
R distance to symmetry axis.
17

Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

3.
Equivariant cohomology

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions18

Atiyah-Bott-Berline-Vergne localization
Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions19

Atiyah-Bott-Berline-Vergne localization
Consider• a manifold M with C∞ -action of a compact Lie group K ,• A := (S(k∗ ) ⊗ Λ(M))K ,• D%(X ) := d(%(X )) − ιXe (%(X )), % ∈ A, X ∈ k,∗
• HG(M) := Ker D/Im D equivariant cohomology of M.

Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions19

Atiyah-Bott-Berline-Vergne localization
Consider• a manifold M with C∞ -action of a compact Lie group K ,• A := (S(k∗ ) ⊗ Λ(M))K ,• D%(X ) := d(%(X )) − ιXe (%(X )), % ∈ A, X ∈ k,∗
• HG(M) := Ker D/Im D equivariant cohomology of M.
em = 0}
Let % ∈ A have compact support, M0 = {m ∈ M : Y
where Y ∈ k, χNM0 equivariant Euler form. If D% = 0,
Z
Z
%(Y )
%(Y ) =
Atiyah-Berline-Bott-Vergne localization.
M
M0 χNM0 (Y )Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions19

Atiyah-Bott-Berline-Vergne localization
Consider• a manifold M with C∞ -action of a compact Lie group K ,• A := (S(k∗ ) ⊗ Λ(M))K ,• D%(X ) := d(%(X )) − ιXe (%(X )), % ∈ A, X ∈ k,∗
• HG(M) := Ker D/Im D equivariant cohomology of M.
em = 0}
Let % ∈ A have compact support, M0 = {m ∈ M : Y
where Y ∈ k, χNM0 equivariant Euler form. If D% = 0,
Z
Z
%(Y )
%(Y ) =
Atiyah-Berline-Bott-Vergne localization.
M
M0 χNM0 (Y )Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions∗

Let (M, ω) be a compact Hamiltonian K -space, J : M → k ,JX (p) = J(p)(X ) the moment map, ω̄ := J − ω. As a specialcase, if K = T is a torus, localization implies thatZ
X 7−→
ei ω̄(X )Duistermaat-Heckman integral
Mis given by its exact stationary phase approximation. Its t-FTJ∗ (ω n /n!) is a piecewise polynomial measure on t∗ .

19

Residue formulae
Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

20

Residue formulae
IntroductionMore generally, consider the tempered distribution
Z
k 3 X 7−→ L% (X ) :=ei ω̄(X ) %(X ),% ∈ Λ∗K (M).
M

1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

20

Residue formulae
IntroductionMore generally, consider the tempered distribution
Z
k 3 X 7−→ L% (X ) :=ei ω̄(X ) %(X ),% ∈ Λ∗K (M).
M

∗ −1

∗

−1

H (J

The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

If 0 ∈ k∗ is a regular value of J, Kirwan showed that(π )
ι∗K : HK∗ (M) −→ HK (J−1 (0)) −→1. Equivariant spectral
geometry

2. Equivariant
quantum ergodicity(0)/K )Classical and quantum
ergodicity
Symmetries and ergodicity−1

is surjective, where ι : J−1(0) ,→ M, π : J(0) → J

−1(0)/K .

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

20

Residue formulae
IntroductionMore generally, consider the tempered distribution
Z
k 3 X 7−→ L% (X ) :=ei ω̄(X ) %(X ),% ∈ Λ∗K (M).
M

∗ −1

∗

−1

H (J

The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

If 0 ∈ k∗ is a regular value of J, Kirwan showed that(π )
ι∗K : HK∗ (M) −→ HK (J−1 (0)) −→1. Equivariant spectral
geometry

2. Equivariant
quantum ergodicity(0)/K )Classical and quantum
ergodicity
Symmetries and ergodicity−1

is surjective, where ι : J∗−1,→ Express H (Jk-FT of L% .

−1(0) ,→ M, π : J(0) → J

−1(0)/K .

(0)/K ) in terms of HK (M) by studying the

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

20

Residue formulae
IntroductionMore generally, consider the tempered distribution
Z
k 3 X 7−→ L% (X ) :=ei ω̄(X ) %(X ),% ∈ Λ∗K (M).
M

∗ −1

∗

−1

H (J

The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

If 0 ∈ k∗ is a regular value of J, Kirwan showed that(π )
ι∗K : HK∗ (M) −→ HK (J−1 (0)) −→1. Equivariant spectral
geometry

2. Equivariant
quantum ergodicity(0)/K )Classical and quantum
ergodicity
Symmetries and ergodicity−1

is surjective, where ι : J∗−1,→ Express H (Jk-FT of L% .

−1(0) ,→ M, π : J(0) → J

−1(0)/K .

(0)/K ) in terms of HK (M) by studying the

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

Let T ⊂ G be a maximal torus. Using localization and
Paley-Wiener theorems, Kirwan-Jeffrey showed for % ∈ HK∗ (M)ZX K(%ei ω̄ ) = Res
uF
Residue formulae,
J−1 (0)/K

F

where F are the components of M T and uF (X ) ≡R%(X )eiω.
F χNF (X )20

Proof. Via stationary phase asymptotics for the Witten integral
Z Z
Iζ (µ) :=kMeiµ (J−ζ) a

n

ω
dX ∼
n!

∞
X

Introduction
1. Equivariant spectral
geometryµ−κ−j Lj (ζ),

j=0where ζ ∈ k∗ , µ → +∞, a ∈ C∞
c , κ := dim K /Hprin , andCrit(J − ζ) = {(p, X ) : p ∈ Ωζ , X ∈ kp }The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

is clean, that is, (J(p) − ζ)(X ) is a Morse-Bott function.

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions21

Proof. Via stationary phase asymptotics for the Witten integral
Z Z
Iζ (µ) :=kMeiµ (J−ζ) a

n

ω
dX ∼
n!

∞
X

Introduction
1. Equivariant spectral
geometryµ−κ−j Lj (ζ),

j=0where ζ ∈ k∗ , µ → +∞, a ∈ C∞
c , κ := dim K /Hprin , andCrit(J − ζ) = {(p, X ) : p ∈ Ωζ , X ∈ kp }The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

is clean, that is, (J(p) − ζ)(X ) is a Morse-Bott function.
Remarks.
• Full asymptotic expansion required.

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions• ζ is a classical parameter, µ a quantum parameter.
• Continuity of Lj (ζ) in ζ needed =⇒ Limits can beinterchanged, [Q, R] = 0 is fulfilled.

21

Proof. Via stationary phase asymptotics for the Witten integral
Z Z
Iζ (µ) :=kMeiµ (J−ζ) a

n

ω
dX ∼
n!

∞
X

Introduction
1. Equivariant spectral
geometryµ−κ−j Lj (ζ),

j=0where ζ ∈ k∗ , µ → +∞, a ∈ C∞
c , κ := dim K /Hprin , andCrit(J − ζ) = {(p, X ) : p ∈ Ωζ , X ∈ kp }The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

is clean, that is, (J(p) − ζ)(X ) is a Morse-Bott function.
Remarks.
• Full asymptotic expansion required.

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions• ζ is a classical parameter, µ a quantum parameter.
• Continuity of Lj (ζ) in ζ needed =⇒ Limits can beinterchanged, [Q, R] = 0 is fulfilled.
Problem. If 0 ∈ k∗ is not a regular value of J, the critical set issingular =⇒ stationary phase fails.
21

Residue formulae for circle actions
Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

22

Residue formulae for circle actions
In what follows, assume that
• (M, ω) is a (not necessarily compact) 2n-dimensional
Hamiltonian K -space with K = SO(2) ' S 1 .
• 0 ∈ k∗ ' R is not a regular value.

Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

22

Residue formulae for circle actions
In what follows, assume that
• (M, ω) is a (not necessarily compact) 2n-dimensional
Hamiltonian K -space with K = SO(2) ' S 1 .
• 0 ∈ k∗ ' R is not a regular value.
By the Marle-Guillemin-Sternberg normal form of the
momentum map, the Witten integral reduces to integrals
Z Z
Jζ (µ) :=eixµ(hQ w,wi−ζ) b(x, w) dw dx,
RR2n

where Q ∈ M(2n, R) is symmetric and non-degenerate.

Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

22

Residue formulae for circle actions
In what follows, assume that
• (M, ω) is a (not necessarily compact) 2n-dimensional
Hamiltonian K -space with K = SO(2) ' S 1 .
• 0 ∈ k∗ ' R is not a regular value.
By the Marle-Guillemin-Sternberg normal form of the
momentum map, the Witten integral reduces to integrals
Z Z
Jζ (µ) :=eixµ(hQ w,wi−ζ) b(x, w) dw dx,
R

Proposition (Kuester,Konstantis,–, 2018)
∞
X
j=0

µ−1−j

∞
X

Θl+j,l (ζµ)ζ l

1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology

R2nwhere Q ∈ M(2n, R) is symmetric and non-degenerate.

Jζ (µ) ∼Introduction

Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

Singular stationary phase,l=0

where Θj,l are known coefficients with |Θj,l (ζµ)| j,l 1 given by
distributions supported on the strata of {hQ w, wi = 0}, andpossibly not continuous in ζ.
22Consider• S 2 ⊂ R3 ' so(3) co-adjoint orbit of SO(3) with symplectic
form ωS 2 (X̃ , Ỹ )ξ =: − hξ, [X , Y ]i
• Hamiltonian action of K = SO(2) on S 2 with moment map
JS 2 (ξ1 , ξ2 , ξ3 ) = −ξ3 .
•[M = S2 × S2 =Mℵ = Mreg ∪ Msingℵ

with product symplectic form and K -action.

Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions23

Consider• S 2 ⊂ R3 ' so(3) co-adjoint orbit of SO(3) with symplectic
form ωS 2 (X̃ , Ỹ )ξ =: − hξ, [X , Y ]i
• Hamiltonian action of K = SO(2) on S 2 with moment map
JS 2 (ξ1 , ξ2 , ξ3 ) = −ξ3 .
•[M = S2 × S2 =Mℵ = Mreg ∪ Msingℵ

with product symplectic form and K -action.

Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicityΩ0 = ξ ∈ M ⊂ R6 : ξ3 + ξ6 = 0 ' ΣT 2 ,are stratified spacesΩ0 /S 1 ' ΣS 1 ' S 2

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions23

Consider• S 2 ⊂ R3 ' so(3) co-adjoint orbit of SO(3) with symplectic
form ωS 2 (X̃ , Ỹ )ξ =: − hξ, [X , Y ]i
• Hamiltonian action of K = SO(2) on S 2 with moment map
JS 2 (ξ1 , ξ2 , ξ3 ) = −ξ3 .
•[M = S2 × S2 =Mℵ = Mreg ∪ Msingℵ

with product symplectic form and K -action.

Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity
Classical and quantum
ergodicity
Symmetries and ergodicityΩ0 = ξ ∈ M ⊂ R6 : ξ3 + ξ6 = 0 ' ΣT 2 ,Ω0 /S 1 ' ΣS 1 ' S 2

are stratified spaces and we define the stratified Kirwan map
∗

∗ −1

(π )
ι
e ℵ : H ∗1 (Mℵ ) −→K
HS 1 (Ω0 ∩ Mℵ ) −→ H ∗ ((Ω0 ∩ Mℵ )/S 1 ).
S

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions23

Consider• S 2 ⊂ R3 ' so(3) co-adjoint orbit of SO(3) with symplectic
form ωS 2 (X̃ , Ỹ )ξ =: − hξ, [X , Y ]i
• Hamiltonian action of K = SO(2) on S 2 with moment map
JS 2 (ξ1 , ξ2 , ξ3 ) = −ξ3 .
•[M = S2 × S2 =Mℵ = Mreg ∪ Msing

Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicityℵwith product symplectic form and K -action.

Classical and quantum
ergodicity
Symmetries and ergodicityΩ0 = ξ ∈ M ⊂ R6 : ξ3 + ξ6 = 0 ' ΣT 2 ,Ω0 /S 1 ' ΣS 1 ' S 2

are stratified spaces and we define the stratified Kirwan map
∗

∗ −1

(π )
ι
e ℵ : H ∗1 (Mℵ ) −→K
HS 1 (Ω0 ∩ Mℵ ) −→ H ∗ ((Ω0 ∩ Mℵ )/S 1 ).
S

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions

Theorem (Kuester, Konstantis, –, 2018)

Res

X
F

Z
uF =
(Ω0 ∩Mreg )/S 1

e reg (%ei ω̄ )+K

Ze sing (%ei ω̄ ).
K(Ω0 ∩Msing )/S 1

23

Introduction
1. Equivariant spectral
geometry
The spectral function of anelliptic operator
Equivariant Weyl law
Convex bounds foreigenfunctions
Subconvex bounds for
Hecke–Maass forms

2. Equivariant
quantum ergodicity

Thank you!

Classical and quantum
ergodicity
Symmetries and ergodicity

3. Equivariant
cohomology
Atiyah-Bott-Berline-Vergne
localization
Residue formulae
Residue formulae for circle
actions24