Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Selberg’s twisted trace formula and asymptotics
of equivariant analytic torsion
Bingxiao LIU
Max Planck Institute for Mathematics

CIRM-Luminy, 9th October 2018

Bingxiao LIU

Twisted trace formula

1/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Contents

1

Overview

2

Definition of twisted orbital integrals

3

An explicit geometric formula4

Asymptotics of the equivariant Ray-Singer torsion

Bingxiao LIU

Twisted trace formula

2/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Overview

In this talkTwo parts:

1. Evaluate twisted orbital integrals on symmetric spaces by an
explicit geometric formula.

2. Apply computations to equivariant analytic torsion of locallysymmetric spaces.

Bingxiao LIU

Twisted trace formula

3/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Overview

In this talkTwo parts:

1. Evaluate twisted orbital integrals on symmetric spaces by an
explicit geometric formula.

2. Apply computations to equivariant analytic torsion of locallysymmetric spaces.

Bingxiao LIU

Twisted trace formula

3/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Overview

In this talkTwo parts:

1. Evaluate twisted orbital integrals on symmetric spaces by an
explicit geometric formula.

2. Apply computations to equivariant analytic torsion of locallysymmetric spaces.

Bingxiao LIU

Twisted trace formula

3/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Overview

In this talkTwo parts:

1. Evaluate twisted orbital integrals on symmetric spaces by an
explicit geometric formula.

2. Apply computations to equivariant analytic torsion of locallysymmetric spaces.

Bingxiao LIU

Twisted trace formula

3/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Overview

General setting

X = G/K symmetric space, σ automorphism of (G, K, B).
∆X Laplace-Beltrami on X.
Twisted orbital integral Tr[γσ] [exp(t∆X /2)]: integration on orbitof γ in G under σ-twisted conjugation.
Such integrals appear when evaluating σ-equivariant traces on
Z = Γ\X, compact locally symmetric space.

Bingxiao LIU

Twisted trace formula

4/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Overview

General setting

X = G/K symmetric space, σ automorphism of (G, K, B).
∆X Laplace-Beltrami on X.
Twisted orbital integral Tr[γσ] [exp(t∆X /2)]: integration on orbitof γ in G under σ-twisted conjugation.
Such integrals appear when evaluating σ-equivariant traces on
Z = Γ\X, compact locally symmetric space.

Bingxiao LIU

Twisted trace formula

4/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Overview

General setting

X = G/K symmetric space, σ automorphism of (G, K, B).
∆X Laplace-Beltrami on X.
Twisted orbital integral Tr[γσ] [exp(t∆X /2)]: integration on orbitof γ in G under σ-twisted conjugation.
Such integrals appear when evaluating σ-equivariant traces on
Z = Γ\X, compact locally symmetric space.

Bingxiao LIU

Twisted trace formula

4/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Overview

General setting

X = G/K symmetric space, σ automorphism of (G, K, B).
∆X Laplace-Beltrami on X.
Twisted orbital integral Tr[γσ] [exp(t∆X /2)]: integration on orbitof γ in G under σ-twisted conjugation.
Such integrals appear when evaluating σ-equivariant traces on
Z = Γ\X, compact locally symmetric space.

Bingxiao LIU

Twisted trace formula

4/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Overview

General setting

X = G/K symmetric space, σ automorphism of (G, K, B).
∆X Laplace-Beltrami on X.
Twisted orbital integral Tr[γσ] [exp(t∆X /2)]: integration on orbitof γ in G under σ-twisted conjugation.
Such integrals appear when evaluating σ-equivariant traces on
Z = Γ\X, compact locally symmetric space.

Bingxiao LIU

Twisted trace formula

4/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Overview

Twisted orbital integrals
When γσ is semisimple, we have the explicit local formula:

Theorem A (Bismut 2011 for σ = 1, L. 2018)
For t > 0, we haveTr[γσ] [exp(t(∆X /2 − c/2)]Zkexp(−|a|2 /2t)k −|Y0k |2 /2t dY0
=
J
(Y
)e
.
γσ
0
(2πt)p/2(2πt)q/2
kσ (γ)

Jγσ explicitly defined analytic function on kσ (γ).

Bingxiao LIU

Twisted trace formula

5/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Overview

Twisted orbital integrals
When γσ is semisimple, we have the explicit local formula:

Theorem A (Bismut 2011 for σ = 1, L. 2018)
For t > 0, we haveTr[γσ] [exp(t(∆X /2 − c/2)]Zkexp(−|a|2 /2t)k −|Y0k |2 /2t dY0
=
J
(Y
)e
.
γσ
0
(2πt)p/2(2πt)q/2
kσ (γ)

Jγσ explicitly defined analytic function on kσ (γ).

Bingxiao LIU

Twisted trace formula

5/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Overview

Twisted orbital integrals
When γσ is semisimple, we have the explicit local formula:

Theorem A (Bismut 2011 for σ = 1, L. 2018)
For t > 0, we haveTr[γσ] [exp(t(∆X /2 − c/2)]Zkexp(−|a|2 /2t)k −|Y0k |2 /2t dY0
=
J
(Y
)e
.
γσ
0
(2πt)p/2(2πt)q/2
kσ (γ)

Jγσ explicitly defined analytic function on kσ (γ).

Bingxiao LIU

Twisted trace formula

5/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Overview

Twisted orbital integrals
When γσ is semisimple, we have the explicit local formula:

Theorem A (Bismut 2011 for σ = 1, L. 2018)
For t > 0, we haveTr[γσ] [exp(t(∆X /2 − c/2)]Zkexp(−|a|2 /2t)k −|Y0k |2 /2t dY0
=
J
(Y
)e
.
γσ
0
(2πt)p/2(2πt)q/2
kσ (γ)

Jγσ explicitly defined analytic function on kσ (γ).

Bingxiao LIU

Twisted trace formula

5/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Overview

A theorem of Bismut-Ma-Zhang
λ highest weight of compact form U of G, Ed rep. of G with dλ.
GP −→ M with flat connection, Fd = P ×G Ed flat vector bundles
on M .
Td real analytic torsions for Fd .
Theorem(Bismut-Ma-Zhang, 2017)
There is a locally computable diff. form W on M associated with λsuch that as d → +∞, the leading term of Td is given by
Z
n+1
d
W.
M

Also true for asymptotic analytic torsion forms.
Bismut-Vasserot(1989), Bergeron-Venkatesh(2013), Müller(2012),
Müller-Pfaff(2013), Ksenia Fedosova(2015)....
Bingxiao LIU

Twisted trace formula

6/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Overview

A theorem of Bismut-Ma-Zhang
λ highest weight of compact form U of G, Ed rep. of G with dλ.
GP −→ M with flat connection, Fd = P ×G Ed flat vector bundles
on M .
Td real analytic torsions for Fd .
Theorem(Bismut-Ma-Zhang, 2017)
There is a locally computable diff. form W on M associated with λsuch that as d → +∞, the leading term of Td is given by
Z
n+1
d
W.
M

Also true for asymptotic analytic torsion forms.
Bismut-Vasserot(1989), Bergeron-Venkatesh(2013), Müller(2012),
Müller-Pfaff(2013), Ksenia Fedosova(2015)....
Bingxiao LIU

Twisted trace formula

6/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Overview

A theorem of Bismut-Ma-Zhang
λ highest weight of compact form U of G, Ed rep. of G with dλ.
GP −→ M with flat connection, Fd = P ×G Ed flat vector bundles
on M .
Td real analytic torsions for Fd .
Theorem(Bismut-Ma-Zhang, 2017)
There is a locally computable diff. form W on M associated with λsuch that as d → +∞, the leading term of Td is given by
Z
n+1
d
W.
M

Also true for asymptotic analytic torsion forms.
Bismut-Vasserot(1989), Bergeron-Venkatesh(2013), Müller(2012),
Müller-Pfaff(2013), Ksenia Fedosova(2015)....
Bingxiao LIU

Twisted trace formula

6/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Overview

A theorem of Bismut-Ma-Zhang
λ highest weight of compact form U of G, Ed rep. of G with dλ.
GP −→ M with flat connection, Fd = P ×G Ed flat vector bundles
on M .
Td real analytic torsions for Fd .
Theorem(Bismut-Ma-Zhang, 2017)
There is a locally computable diff. form W on M associated with λsuch that as d → +∞, the leading term of Td is given by
Z
n+1
d
W.
M

Also true for asymptotic analytic torsion forms.
Bismut-Vasserot(1989), Bergeron-Venkatesh(2013), Müller(2012),
Müller-Pfaff(2013), Ksenia Fedosova(2015)....
Bingxiao LIU

Twisted trace formula

6/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Overview

A theorem of Bismut-Ma-Zhang
λ highest weight of compact form U of G, Ed rep. of G with dλ.
GP −→ M with flat connection, Fd = P ×G Ed flat vector bundles
on M .
Td real analytic torsions for Fd .
Theorem(Bismut-Ma-Zhang, 2017)
There is a locally computable diff. form W on M associated with λsuch that as d → +∞, the leading term of Td is given by
Z
n+1
d
W.
M

Also true for asymptotic analytic torsion forms.
Bismut-Vasserot(1989), Bergeron-Venkatesh(2013), Müller(2012),
Müller-Pfaff(2013), Ksenia Fedosova(2015)....
Bingxiao LIU

Twisted trace formula

6/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Overview

A theorem of Bismut-Ma-Zhang
λ highest weight of compact form U of G, Ed rep. of G with dλ.
GP −→ M with flat connection, Fd = P ×G Ed flat vector bundles
on M .
Td real analytic torsions for Fd .
Theorem(Bismut-Ma-Zhang, 2017)
There is a locally computable diff. form W on M associated with λsuch that as d → +∞, the leading term of Td is given by
Z
n+1
d
W.
M

Also true for asymptotic analytic torsion forms.
Bismut-Vasserot(1989), Bergeron-Venkatesh(2013), Müller(2012),
Müller-Pfaff(2013), Ksenia Fedosova(2015)....
Bingxiao LIU

Twisted trace formula

6/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Overview

A theorem of Bismut-Ma-Zhang
λ highest weight of compact form U of G, Ed rep. of G with dλ.
GP −→ M with flat connection, Fd = P ×G Ed flat vector bundles
on M .
Td real analytic torsions for Fd .
Theorem(Bismut-Ma-Zhang, 2017)
There is a locally computable diff. form W on M associated with λsuch that as d → +∞, the leading term of Td is given by
Z
n+1
d
W.
M

Also true for asymptotic analytic torsion forms.
Bismut-Vasserot(1989), Bergeron-Venkatesh(2013), Müller(2012),
Müller-Pfaff(2013), Ksenia Fedosova(2015)....
Bingxiao LIU

Twisted trace formula

6/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Overview

A fixed point formula : following BMZ
Equivariant case: Fd |d∈N sequence of flat vector bundles on Z = Γ\Xequipped with action of σ.
Bismut-Zhang(1994): equivariant analytic torsion can beevaluated on the fixed point set.
jWγσlocally computable forms on fixed point set of σ in Z:
W -invariants by BMZ.

Theorem B (L. 2018)
As d → +∞,d−m(σ)−1 Tσ (Fd )
X
X

j max
=
Vol(Γ ∩ Zσ (γ)\X(γσ))
sj (γσ)d [Wγσ]+ o(1).
|{z}| {z }
certain elliptic
jclasses [γ]σfixed pts of σ in Z

Bingxiao LIU

oscillating

Twisted trace formula

7/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Overview

A fixed point formula : following BMZ
Equivariant case: Fd |d∈N sequence of flat vector bundles on Z = Γ\Xequipped with action of σ.
Bismut-Zhang(1994): equivariant analytic torsion can beevaluated on the fixed point set.
jWγσlocally computable forms on fixed point set of σ in Z:
W -invariants by BMZ.

Theorem B (L. 2018)
As d → +∞,d−m(σ)−1 Tσ (Fd )
X
X

j max
=
Vol(Γ ∩ Zσ (γ)\X(γσ))
sj (γσ)d [Wγσ]+ o(1).
|{z}| {z }
certain elliptic
jclasses [γ]σfixed pts of σ in Z

Bingxiao LIU

oscillating

Twisted trace formula

7/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Overview

A fixed point formula : following BMZ
Equivariant case: Fd |d∈N sequence of flat vector bundles on Z = Γ\Xequipped with action of σ.
Bismut-Zhang(1994): equivariant analytic torsion can beevaluated on the fixed point set.
jWγσlocally computable forms on fixed point set of σ in Z:
W -invariants by BMZ.

Theorem B (L. 2018)
As d → +∞,d−m(σ)−1 Tσ (Fd )
X
X

j max
=
Vol(Γ ∩ Zσ (γ)\X(γσ))
sj (γσ)d [Wγσ]+ o(1).
|{z}| {z }
certain elliptic
jclasses [γ]σfixed pts of σ in Z

Bingxiao LIU

oscillating

Twisted trace formula

7/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Overview

A fixed point formula : following BMZ
Equivariant case: Fd |d∈N sequence of flat vector bundles on Z = Γ\Xequipped with action of σ.
Bismut-Zhang(1994): equivariant analytic torsion can beevaluated on the fixed point set.
jWγσlocally computable forms on fixed point set of σ in Z:
W -invariants by BMZ.

Theorem B (L. 2018)
As d → +∞,d−m(σ)−1 Tσ (Fd )
X
X

j max
=
Vol(Γ ∩ Zσ (γ)\X(γσ))
sj (γσ)d [Wγσ]+ o(1).
|{z}| {z }
certain elliptic
jclasses [γ]σfixed pts of σ in Z

Bingxiao LIU

oscillating

Twisted trace formula

7/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Overview

A fixed point formula : following BMZ
Equivariant case: Fd |d∈N sequence of flat vector bundles on Z = Γ\Xequipped with action of σ.
Bismut-Zhang(1994): equivariant analytic torsion can beevaluated on the fixed point set.
jWγσlocally computable forms on fixed point set of σ in Z:
W -invariants by BMZ.

Theorem B (L. 2018)
As d → +∞,d−m(σ)−1 Tσ (Fd )
X
X

j max
=
Vol(Γ ∩ Zσ (γ)\X(γσ))
sj (γσ)d [Wγσ]+ o(1).
|{z}| {z }
certain elliptic
jclasses [γ]σfixed pts of σ in Z

Bingxiao LIU

oscillating

Twisted trace formula

7/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Overview

Our tools

1. Hypoelliptic Laplacian of Bismut interpolates between
the two sides of our formula for twisted orbital integrals.

2. Using Selberg’s twisted trace formula and Theorem A,
we will obtain Theorem B.

Bingxiao LIU

Twisted trace formula

8/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Overview

Our tools

1. Hypoelliptic Laplacian of Bismut interpolates between
the two sides of our formula for twisted orbital integrals.

2. Using Selberg’s twisted trace formula and Theorem A,
we will obtain Theorem B.

Bingxiao LIU

Twisted trace formula

8/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Overview

Our tools

1. Hypoelliptic Laplacian of Bismut interpolates between
the two sides of our formula for twisted orbital integrals.

2. Using Selberg’s twisted trace formula and Theorem A,
we will obtain Theorem B.

Bingxiao LIU

Twisted trace formula

8/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

Geometric setting(G, K, θ, B) connected real reductive Lie group.
g, k Lie algebras of G, K.
Cartan decomposition: g = p ⊕ k.
X = G/K symmetric space, p : G → X.
T X = G ×K p, B induces a Riemannian metric on X.
X ' p is of nonpositive sectional curvature.

Bingxiao LIU

Twisted trace formula

9/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

Geometric setting(G, K, θ, B) connected real reductive Lie group.
g, k Lie algebras of G, K.
Cartan decomposition: g = p ⊕ k.
X = G/K symmetric space, p : G → X.
T X = G ×K p, B induces a Riemannian metric on X.
X ' p is of nonpositive sectional curvature.

Bingxiao LIU

Twisted trace formula

9/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

Geometric setting(G, K, θ, B) connected real reductive Lie group.
g, k Lie algebras of G, K.
Cartan decomposition: g = p ⊕ k.
X = G/K symmetric space, p : G → X.
T X = G ×K p, B induces a Riemannian metric on X.
X ' p is of nonpositive sectional curvature.

Bingxiao LIU

Twisted trace formula

9/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

Geometric setting(G, K, θ, B) connected real reductive Lie group.
g, k Lie algebras of G, K.
Cartan decomposition: g = p ⊕ k.
X = G/K symmetric space, p : G → X.
T X = G ×K p, B induces a Riemannian metric on X.
X ' p is of nonpositive sectional curvature.

Bingxiao LIU

Twisted trace formula

9/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

Geometric setting(G, K, θ, B) connected real reductive Lie group.
g, k Lie algebras of G, K.
Cartan decomposition: g = p ⊕ k.
X = G/K symmetric space, p : G → X.
T X = G ×K p, B induces a Riemannian metric on X.
X ' p is of nonpositive sectional curvature.

Bingxiao LIU

Twisted trace formula

9/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

Geometric setting(G, K, θ, B) connected real reductive Lie group.
g, k Lie algebras of G, K.
Cartan decomposition: g = p ⊕ k.
X = G/K symmetric space, p : G → X.
T X = G ×K p, B induces a Riemannian metric on X.
X ' p is of nonpositive sectional curvature.

Bingxiao LIU

Twisted trace formula

9/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

Geometric setting(G, K, θ, B) connected real reductive Lie group.
g, k Lie algebras of G, K.
Cartan decomposition: g = p ⊕ k.
X = G/K symmetric space, p : G → X.
T X = G ×K p, B induces a Riemannian metric on X.
X ' p is of nonpositive sectional curvature.

Bingxiao LIU

Twisted trace formula

9/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

The twist σσ ∈ Aut(G, θ, B), acts on X isometrically.
Σσ ⊂ Aut(G) compact subgroup generated by σ:Gσ = G o Σσ , K σ = K o Σσ , X = Gσ /K σ .
If h ∈ G, twisted conjugation,
C σ (h)γ = hγσ(h−1 ) ∈ G.
Twisted centralizer of γ ∈ G:Zσ (γ) = {h ∈ G : C σ (h)γ = γ}.

Bingxiao LIU

Twisted trace formula

10/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

The twist σσ ∈ Aut(G, θ, B), acts on X isometrically.
Σσ ⊂ Aut(G) compact subgroup generated by σ:Gσ = G o Σσ , K σ = K o Σσ , X = Gσ /K σ .
If h ∈ G, twisted conjugation,
C σ (h)γ = hγσ(h−1 ) ∈ G.
Twisted centralizer of γ ∈ G:Zσ (γ) = {h ∈ G : C σ (h)γ = γ}.

Bingxiao LIU

Twisted trace formula

10/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

The twist σσ ∈ Aut(G, θ, B), acts on X isometrically.
Σσ ⊂ Aut(G) compact subgroup generated by σ:Gσ = G o Σσ , K σ = K o Σσ , X = Gσ /K σ .
If h ∈ G, twisted conjugation,
C σ (h)γ = hγσ(h−1 ) ∈ G.
Twisted centralizer of γ ∈ G:Zσ (γ) = {h ∈ G : C σ (h)γ = γ}.

Bingxiao LIU

Twisted trace formula

10/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

The twist σσ ∈ Aut(G, θ, B), acts on X isometrically.
Σσ ⊂ Aut(G) compact subgroup generated by σ:Gσ = G o Σσ , K σ = K o Σσ , X = Gσ /K σ .
If h ∈ G, twisted conjugation,
C σ (h)γ = hγσ(h−1 ) ∈ G.
Twisted centralizer of γ ∈ G:Zσ (γ) = {h ∈ G : C σ (h)γ = γ}.

Bingxiao LIU

Twisted trace formula

10/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

The twist σσ ∈ Aut(G, θ, B), acts on X isometrically.
Σσ ⊂ Aut(G) compact subgroup generated by σ:Gσ = G o Σσ , K σ = K o Σσ , X = Gσ /K σ .
If h ∈ G, twisted conjugation,
C σ (h)γ = hγσ(h−1 ) ∈ G.
Twisted centralizer of γ ∈ G:Zσ (γ) = {h ∈ G : C σ (h)γ = γ}.

Bingxiao LIU

Twisted trace formula

10/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

Twisted orbital integrals
ρE : K σ → Aut(E) unitary representation.
F = G ×K E = Gσ ×K σ E Hermitian vector bundle on X.
Q a Gσ -invariant integral operator acting on C b (X, F ) withkernel q(g) ∈ End(E),|q(g −1 g 0 )| ≤ C exp(−C 0 d2 (pg, pg 0 )).
If γσ semisimple, twisted orbital integralZTr[γσ] [Q] =TrE [ρE (σ)q(g −1 γσ(g))]dg.
Zσ (γ)\G

We need a geometric interpretation for Tr[γσ] [Q].

Bingxiao LIU

Twisted trace formula

11/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

Twisted orbital integrals
ρE : K σ → Aut(E) unitary representation.
F = G ×K E = Gσ ×K σ E Hermitian vector bundle on X.
Q a Gσ -invariant integral operator acting on C b (X, F ) withkernel q(g) ∈ End(E),|q(g −1 g 0 )| ≤ C exp(−C 0 d2 (pg, pg 0 )).
If γσ semisimple, twisted orbital integralZTr[γσ] [Q] =TrE [ρE (σ)q(g −1 γσ(g))]dg.
Zσ (γ)\G

We need a geometric interpretation for Tr[γσ] [Q].

Bingxiao LIU

Twisted trace formula

11/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

Twisted orbital integrals
ρE : K σ → Aut(E) unitary representation.
F = G ×K E = Gσ ×K σ E Hermitian vector bundle on X.
Q a Gσ -invariant integral operator acting on C b (X, F ) withkernel q(g) ∈ End(E),|q(g −1 g 0 )| ≤ C exp(−C 0 d2 (pg, pg 0 )).
If γσ semisimple, twisted orbital integralZTr[γσ] [Q] =TrE [ρE (σ)q(g −1 γσ(g))]dg.
Zσ (γ)\G

We need a geometric interpretation for Tr[γσ] [Q].

Bingxiao LIU

Twisted trace formula

11/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

Twisted orbital integrals
ρE : K σ → Aut(E) unitary representation.
F = G ×K E = Gσ ×K σ E Hermitian vector bundle on X.
Q a Gσ -invariant integral operator acting on C b (X, F ) withkernel q(g) ∈ End(E),|q(g −1 g 0 )| ≤ C exp(−C 0 d2 (pg, pg 0 )).
If γσ semisimple, twisted orbital integralZTr[γσ] [Q] =TrE [ρE (σ)q(g −1 γσ(g))]dg.
Zσ (γ)\G

We need a geometric interpretation for Tr[γσ] [Q].

Bingxiao LIU

Twisted trace formula

11/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

Twisted orbital integrals
ρE : K σ → Aut(E) unitary representation.
F = G ×K E = Gσ ×K σ E Hermitian vector bundle on X.
Q a Gσ -invariant integral operator acting on C b (X, F ) withkernel q(g) ∈ End(E),|q(g −1 g 0 )| ≤ C exp(−C 0 d2 (pg, pg 0 )).
If γσ semisimple, twisted orbital integralZTr[γσ] [Q] =TrE [ρE (σ)q(g −1 γσ(g))]dg.
Zσ (γ)\G

We need a geometric interpretation for Tr[γσ] [Q].

Bingxiao LIU

Twisted trace formula

11/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

Twisted orbital integrals
ρE : K σ → Aut(E) unitary representation.
F = G ×K E = Gσ ×K σ E Hermitian vector bundle on X.
Q a Gσ -invariant integral operator acting on C b (X, F ) withkernel q(g) ∈ End(E),|q(g −1 g 0 )| ≤ C exp(−C 0 d2 (pg, pg 0 )).
If γσ semisimple, twisted orbital integralZTr[γσ] [Q] =TrE [ρE (σ)q(g −1 γσ(g))]dg.
Zσ (γ)\G

We need a geometric interpretation for Tr[γσ] [Q].

Bingxiao LIU

Twisted trace formula

11/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

Semisimple isometries

If φ ∈ Isom(X), displacement functiondφ (x) = d(x, φ(x)).
dφ convex function on X.
φ semisimple ⇔ dφ reaches its infimum mφ in X.
X(φ) = d−1
φ (mφ ) closed convex subset of X.

Bingxiao LIU

Twisted trace formula

12/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

Semisimple isometries

If φ ∈ Isom(X), displacement functiondφ (x) = d(x, φ(x)).
dφ convex function on X.
φ semisimple ⇔ dφ reaches its infimum mφ in X.
X(φ) = d−1
φ (mφ ) closed convex subset of X.

Bingxiao LIU

Twisted trace formula

12/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

Semisimple isometries

If φ ∈ Isom(X), displacement functiondφ (x) = d(x, φ(x)).
dφ convex function on X.
φ semisimple ⇔ dφ reaches its infimum mφ in X.
X(φ) = d−1
φ (mφ ) closed convex subset of X.

Bingxiao LIU

Twisted trace formula

12/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

Semisimple isometries

If φ ∈ Isom(X), displacement functiondφ (x) = d(x, φ(x)).
dφ convex function on X.
φ semisimple ⇔ dφ reaches its infimum mφ in X.
X(φ) = d−1
φ (mφ ) closed convex subset of X.

Bingxiao LIU

Twisted trace formula

12/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

Semisimple isometries

If φ ∈ Isom(X), displacement functiondφ (x) = d(x, φ(x)).
dφ convex function on X.
φ semisimple ⇔ dφ reaches its infimum mφ in X.
X(φ) = d−1
φ (mφ ) closed convex subset of X.

Bingxiao LIU

Twisted trace formula

12/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

The minimizing set for γσ

Take γ ∈ G such that γσ ∈ Isom(X) semisimple.
After conjugation, one has
γ = ea k −1 , a ∈ p, k ∈ K, Ad(k −1 )σ(a) = a.
mγσ = |a|.
X(γσ) symmetric space associated with Zσ (γ):X(γσ) = Zσ (γ)/Kσ (γ), Kσ (γ) = K ∩ Zσ (γ).
X ' p, X(γσ) ' pσ (γ).

Bingxiao LIU

Twisted trace formula

13/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

The minimizing set for γσ

Take γ ∈ G such that γσ ∈ Isom(X) semisimple.
After conjugation, one has
γ = ea k −1 , a ∈ p, k ∈ K, Ad(k −1 )σ(a) = a.
mγσ = |a|.
X(γσ) symmetric space associated with Zσ (γ):X(γσ) = Zσ (γ)/Kσ (γ), Kσ (γ) = K ∩ Zσ (γ).
X ' p, X(γσ) ' pσ (γ).

Bingxiao LIU

Twisted trace formula

13/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

The minimizing set for γσ

Take γ ∈ G such that γσ ∈ Isom(X) semisimple.
After conjugation, one has
γ = ea k −1 , a ∈ p, k ∈ K, Ad(k −1 )σ(a) = a.
mγσ = |a|.
X(γσ) symmetric space associated with Zσ (γ):X(γσ) = Zσ (γ)/Kσ (γ), Kσ (γ) = K ∩ Zσ (γ).
X ' p, X(γσ) ' pσ (γ).

Bingxiao LIU

Twisted trace formula

13/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

The minimizing set for γσ

Take γ ∈ G such that γσ ∈ Isom(X) semisimple.
After conjugation, one has
γ = ea k −1 , a ∈ p, k ∈ K, Ad(k −1 )σ(a) = a.
mγσ = |a|.
X(γσ) symmetric space associated with Zσ (γ):X(γσ) = Zσ (γ)/Kσ (γ), Kσ (γ) = K ∩ Zσ (γ).
X ' p, X(γσ) ' pσ (γ).

Bingxiao LIU

Twisted trace formula

13/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

The minimizing set for γσ

Take γ ∈ G such that γσ ∈ Isom(X) semisimple.
After conjugation, one has
γ = ea k −1 , a ∈ p, k ∈ K, Ad(k −1 )σ(a) = a.
mγσ = |a|.
X(γσ) symmetric space associated with Zσ (γ):X(γσ) = Zσ (γ)/Kσ (γ), Kσ (γ) = K ∩ Zσ (γ).
X ' p, X(γσ) ' pσ (γ).

Bingxiao LIU

Twisted trace formula

13/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

The minimizing set for γσ

Take γ ∈ G such that γσ ∈ Isom(X) semisimple.
After conjugation, one has
γ = ea k −1 , a ∈ p, k ∈ K, Ad(k −1 )σ(a) = a.
mγσ = |a|.
X(γσ) symmetric space associated with Zσ (γ):X(γσ) = Zσ (γ)/Kσ (γ), Kσ (γ) = K ∩ Zσ (γ).
X ' p, X(γσ) ' pσ (γ).

Bingxiao LIU

Twisted trace formula

13/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

Normal coordinates to X(γσ)
Normal bundle to X(γσ):
geodesic coordinates

NX(γσ)/X = Zσ (γ) ×Kσ (γ) p⊥
σ (γ) ============== X.

Bingxiao LIU

Twisted trace formula

14/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

Geometric twisted orbital integrals
20q(x, x0 ) kernel of Q, |q(x, x0 )| ≤ Ce−Cd (x,x ) .
Geometric formula for twisted orbital integral:Z
Tr[γσ] [Q] =TrF [γσq(f, γσf )] r(f ) df.
|{z}
p⊥
(γ)
σ
Jacobian

Bingxiao LIU

Twisted trace formula

15/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

Geometric twisted orbital integrals
20q(x, x0 ) kernel of Q, |q(x, x0 )| ≤ Ce−Cd (x,x ) .
Geometric formula for twisted orbital integral:Z
Tr[γσ] [Q] =TrF [γσq(f, γσf )] r(f ) df.
|{z}
p⊥
(γ)
σ
Jacobian

Bingxiao LIU

Twisted trace formula

15/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

Geometric twisted orbital integrals
20q(x, x0 ) kernel of Q, |q(x, x0 )| ≤ Ce−Cd (x,x ) .
Geometric formula for twisted orbital integral:Z
Tr[γσ] [Q] =TrF [γσq(f, γσf )] r(f ) df.
|{z}
p⊥
(γ)
σ
Jacobian

Bingxiao LIU

Twisted trace formula

15/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

Now

We will compute twisted orbital integral for heat
kernel on symmetric spaces.

Bingxiao LIU

Twisted trace formula

16/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Definition
Geometric interpretation

Now

We will compute twisted orbital integral for heat
kernel on symmetric spaces.

Bingxiao LIU

Twisted trace formula

16/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

Casimir operator

U g enveloping algebra ' left-inv. diff. operators on G.
C g ∈ U g Casimir operator associated with (g, B):
X
X
X
Cg = −e∗i ei = −
e2i +
e2i .
p partk part| {z }Ck

C g descends to C g,X acting on C ∞ (X, F ):C g,X = −∆X,F + C k,E .

Bingxiao LIU

Twisted trace formula

17/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

Casimir operator

U g enveloping algebra ' left-inv. diff. operators on G.
C g ∈ U g Casimir operator associated with (g, B):
X
X
X
Cg = −e∗i ei = −
e2i +
e2i .
p partk part| {z }Ck

C g descends to C g,X acting on C ∞ (X, F ):C g,X = −∆X,F + C k,E .

Bingxiao LIU

Twisted trace formula

17/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

Casimir operator

U g enveloping algebra ' left-inv. diff. operators on G.
C g ∈ U g Casimir operator associated with (g, B):
X
X
X
Cg = −e∗i ei = −
e2i +
e2i .
p partk part| {z }Ck

C g descends to C g,X acting on C ∞ (X, F ):C g,X = −∆X,F + C k,E .

Bingxiao LIU

Twisted trace formula

17/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

Casimir operator

U g enveloping algebra ' left-inv. diff. operators on G.
C g ∈ U g Casimir operator associated with (g, B):
X
X
X
Cg = −e∗i ei = −
e2i +
e2i .
p partk part| {z }Ck

C g descends to C g,X acting on C ∞ (X, F ):C g,X = −∆X,F + C k,E .

Bingxiao LIU

Twisted trace formula

17/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

Elliptic operator LX on X

LX = 21 C g,X + 12 c with c a fixed constant.
LX self-adjoint, Bochner-like Laplacian.
It commutes with action of Gσ on C ∞ (X, F ).
Goal: an explicit formula for Tr[γσ] [exp(−tLX )].
Adapt Bismut’s proof for the semisimple orbital integrals:
hypoelliptic deformation.

Bingxiao LIU

Twisted trace formula

18/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

Elliptic operator LX on X

LX = 21 C g,X + 12 c with c a fixed constant.
LX self-adjoint, Bochner-like Laplacian.
It commutes with action of Gσ on C ∞ (X, F ).
Goal: an explicit formula for Tr[γσ] [exp(−tLX )].
Adapt Bismut’s proof for the semisimple orbital integrals:
hypoelliptic deformation.

Bingxiao LIU

Twisted trace formula

18/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

Elliptic operator LX on X

LX = 21 C g,X + 12 c with c a fixed constant.
LX self-adjoint, Bochner-like Laplacian.
It commutes with action of Gσ on C ∞ (X, F ).
Goal: an explicit formula for Tr[γσ] [exp(−tLX )].
Adapt Bismut’s proof for the semisimple orbital integrals:
hypoelliptic deformation.

Bingxiao LIU

Twisted trace formula

18/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

Elliptic operator LX on X

LX = 21 C g,X + 12 c with c a fixed constant.
LX self-adjoint, Bochner-like Laplacian.
It commutes with action of Gσ on C ∞ (X, F ).
Goal: an explicit formula for Tr[γσ] [exp(−tLX )].
Adapt Bismut’s proof for the semisimple orbital integrals:
hypoelliptic deformation.

Bingxiao LIU

Twisted trace formula

18/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

Elliptic operator LX on X

LX = 21 C g,X + 12 c with c a fixed constant.
LX self-adjoint, Bochner-like Laplacian.
It commutes with action of Gσ on C ∞ (X, F ).
Goal: an explicit formula for Tr[γσ] [exp(−tLX )].
Adapt Bismut’s proof for the semisimple orbital integrals:
hypoelliptic deformation.

Bingxiao LIU

Twisted trace formula

18/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

Elliptic operator LX on X

LX = 21 C g,X + 12 c with c a fixed constant.
LX self-adjoint, Bochner-like Laplacian.
It commutes with action of Gσ on C ∞ (X, F ).
Goal: an explicit formula for Tr[γσ] [exp(−tLX )].
Adapt Bismut’s proof for the semisimple orbital integrals:
hypoelliptic deformation.

Bingxiao LIU

Twisted trace formula

18/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

Recall: Hypoelliptic deformation

LX can be deformed to a family of hypoelliptic operators LX
b |b>0on an enlarged space Xb, connecting LX on the base (as b → 0)
and the geodesic flow on tangent bundle (as b → +∞).
Certain quantity can be preserved under deformation.
Two ingredients in construction of LXb :Dirac operator of Kostant as square root of C g + c acting on
C ∞ (G, Λ· (g∗ )).
Harmonic oscillator along fiber of flat vector bundle on X.

Bingxiao LIU

Twisted trace formula

19/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

Recall: Hypoelliptic deformation

LX can be deformed to a family of hypoelliptic operators LX
b |b>0on an enlarged space Xb, connecting LX on the base (as b → 0)
and the geodesic flow on tangent bundle (as b → +∞).
Certain quantity can be preserved under deformation.
Two ingredients in construction of LXb :Dirac operator of Kostant as square root of C g + c acting on
C ∞ (G, Λ· (g∗ )).
Harmonic oscillator along fiber of flat vector bundle on X.

Bingxiao LIU

Twisted trace formula

19/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

Recall: Hypoelliptic deformation

LX can be deformed to a family of hypoelliptic operators LX
b |b>0on an enlarged space Xb, connecting LX on the base (as b → 0)
and the geodesic flow on tangent bundle (as b → +∞).
Certain quantity can be preserved under deformation.
Two ingredients in construction of LXb :Dirac operator of Kostant as square root of C g + c acting on
C ∞ (G, Λ· (g∗ )).
Harmonic oscillator along fiber of flat vector bundle on X.

Bingxiao LIU

Twisted trace formula

19/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

Recall: Hypoelliptic deformation

LX can be deformed to a family of hypoelliptic operators LX
b |b>0on an enlarged space Xb, connecting LX on the base (as b → 0)
and the geodesic flow on tangent bundle (as b → +∞).
Certain quantity can be preserved under deformation.
Two ingredients in construction of LXb :Dirac operator of Kostant as square root of C g + c acting on
C ∞ (G, Λ· (g∗ )).
Harmonic oscillator along fiber of flat vector bundle on X.

Bingxiao LIU

Twisted trace formula

19/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

Recall: Hypoelliptic deformation

LX can be deformed to a family of hypoelliptic operators LX
b |b>0on an enlarged space Xb, connecting LX on the base (as b → 0)
and the geodesic flow on tangent bundle (as b → +∞).
Certain quantity can be preserved under deformation.
Two ingredients in construction of LXb :Dirac operator of Kostant as square root of C g + c acting on
C ∞ (G, Λ· (g∗ )).
Harmonic oscillator along fiber of flat vector bundle on X.

Bingxiao LIU

Twisted trace formula

19/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

Recall: Hypoelliptic deformation

LX can be deformed to a family of hypoelliptic operators LX
b |b>0on an enlarged space Xb, connecting LX on the base (as b → 0)
and the geodesic flow on tangent bundle (as b → +∞).
Certain quantity can be preserved under deformation.
Two ingredients in construction of LXb :Dirac operator of Kostant as square root of C g + c acting on
C ∞ (G, Λ· (g∗ )).
Harmonic oscillator along fiber of flat vector bundle on X.

Bingxiao LIU

Twisted trace formula

19/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

The hypoelliptic Laplacian
N = G ×K k vector bundle on X.
Xb the total space of πb : T X ⊕ N = G ×K g → X.
Hypoelliptic Laplacian of Bismut LX
b |b>0 acts on
C ∞ (Xb, πb∗ (Λ· (T ∗ X ⊕ N ∗ ) ⊗ F )).
Explicit formula:
·

LX
b =

∗

∗

1
1
N Λ (T X⊕N )|[Y N , Y T X ]|2 + 2 (−∆T X⊕N + |Y |2 − m − n) +
2
2b |
b2
{z
}
Harmonic oscillator


1 C ∞ (T X⊕N,bπ∗ (Λ· (T ∗ X⊕N ∗ )⊗F ))+ ∇Y T X+ 0-order geometric terms .
b |{z}
generator of geodesic flow

Observation: the hypoelliptic Laplacian commutes with Gσ .
Bingxiao LIU

Twisted trace formula

20/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

The hypoelliptic Laplacian
N = G ×K k vector bundle on X.
Xb the total space of πb : T X ⊕ N = G ×K g → X.
Hypoelliptic Laplacian of Bismut LX
b |b>0 acts on
C ∞ (Xb, πb∗ (Λ· (T ∗ X ⊕ N ∗ ) ⊗ F )).
Explicit formula:
·

LX
b =

∗

∗

1
1
N Λ (T X⊕N )|[Y N , Y T X ]|2 + 2 (−∆T X⊕N + |Y |2 − m − n) +
2
2b |
b2
{z
}
Harmonic oscillator


1 C ∞ (T X⊕N,bπ∗ (Λ· (T ∗ X⊕N ∗ )⊗F ))+ ∇Y T X+ 0-order geometric terms .
b |{z}
generator of geodesic flow

Observation: the hypoelliptic Laplacian commutes with Gσ .
Bingxiao LIU

Twisted trace formula

20/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

The hypoelliptic Laplacian
N = G ×K k vector bundle on X.
Xb the total space of πb : T X ⊕ N = G ×K g → X.
Hypoelliptic Laplacian of Bismut LX
b |b>0 acts on
C ∞ (Xb, πb∗ (Λ· (T ∗ X ⊕ N ∗ ) ⊗ F )).
Explicit formula:
·

LX
b =

∗

∗

1
1
N Λ (T X⊕N )|[Y N , Y T X ]|2 + 2 (−∆T X⊕N + |Y |2 − m − n) +
2
2b |
b2
{z
}
Harmonic oscillator


1 C ∞ (T X⊕N,bπ∗ (Λ· (T ∗ X⊕N ∗ )⊗F ))+ ∇Y T X+ 0-order geometric terms .
b |{z}
generator of geodesic flow

Observation: the hypoelliptic Laplacian commutes with Gσ .
Bingxiao LIU

Twisted trace formula

20/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

The hypoelliptic Laplacian
N = G ×K k vector bundle on X.
Xb the total space of πb : T X ⊕ N = G ×K g → X.
Hypoelliptic Laplacian of Bismut LX
b |b>0 acts on
C ∞ (Xb, πb∗ (Λ· (T ∗ X ⊕ N ∗ ) ⊗ F )).
Explicit formula:
·

LX
b =

∗

∗

1
1
N Λ (T X⊕N )|[Y N , Y T X ]|2 + 2 (−∆T X⊕N + |Y |2 − m − n) +
2
2b |
b2
{z
}
Harmonic oscillator


1 C ∞ (T X⊕N,bπ∗ (Λ· (T ∗ X⊕N ∗ )⊗F ))+ ∇Y T X+ 0-order geometric terms .
b |{z}
generator of geodesic flow

Observation: the hypoelliptic Laplacian commutes with Gσ .
Bingxiao LIU

Twisted trace formula

20/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

The hypoelliptic Laplacian
N = G ×K k vector bundle on X.
Xb the total space of πb : T X ⊕ N = G ×K g → X.
Hypoelliptic Laplacian of Bismut LX
b |b>0 acts on
C ∞ (Xb, πb∗ (Λ· (T ∗ X ⊕ N ∗ ) ⊗ F )).
Explicit formula:
·

LX
b =

∗

∗

1
1
N Λ (T X⊕N )|[Y N , Y T X ]|2 + 2 (−∆T X⊕N + |Y |2 − m − n) +
2
2b |
b2
{z
}
Harmonic oscillator


1 C ∞ (T X⊕N,bπ∗ (Λ· (T ∗ X⊕N ∗ )⊗F ))+ ∇Y T X+ 0-order geometric terms .
b |{z}
generator of geodesic flow

Observation: the hypoelliptic Laplacian commutes with Gσ .
Bingxiao LIU

Twisted trace formula

20/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

The hypoelliptic Laplacian
N = G ×K k vector bundle on X.
Xb the total space of πb : T X ⊕ N = G ×K g → X.
Hypoelliptic Laplacian of Bismut LX
b |b>0 acts on
C ∞ (Xb, πb∗ (Λ· (T ∗ X ⊕ N ∗ ) ⊗ F )).
Explicit formula:
·

LX
b =

∗

∗

1
1
N Λ (T X⊕N )|[Y N , Y T X ]|2 + 2 (−∆T X⊕N + |Y |2 − m − n) +
2
2b |
b2
{z
}
Harmonic oscillator


1 C ∞ (T X⊕N,bπ∗ (Λ· (T ∗ X⊕N ∗ )⊗F ))+ ∇Y T X+ 0-order geometric terms .
b |{z}
generator of geodesic flow

Observation: the hypoelliptic Laplacian commutes with Gσ .
Bingxiao LIU

Twisted trace formula

20/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

Twisted orbital supertrace(B. 2011) For b > 0, t > 0, exp(−tLXb ) has smooth kernel
Xqb,t((x, Y ), (x0 , Y 0 )).
For t > 0, b ∈ ]0, M ]:X
qb,t((x, Y ), (x0 , Y 0 )) ≤ C exp − C 0 (d2 (x, x0 ) + |Y |2 + |Y 0 |2 ) .
Twisted orbital supertrace: Trs [γσ] [exp(−tLXb )].
Trs [γσ] [exp(−tLX
b )]
Z
·
∗
∗
X
= (f,Y )∈ Trs Λ (T X⊕N )⊗F [γσqb,t((f, Y ), γσ(f, Y ))]r(f )df dY.
p⊥
σ (γ)×g

Bingxiao LIU

Twisted trace formula

21/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

Twisted orbital supertrace(B. 2011) For b > 0, t > 0, exp(−tLXb ) has smooth kernel
Xqb,t((x, Y ), (x0 , Y 0 )).
For t > 0, b ∈ ]0, M ]:X
qb,t((x, Y ), (x0 , Y 0 )) ≤ C exp − C 0 (d2 (x, x0 ) + |Y |2 + |Y 0 |2 ) .
Twisted orbital supertrace: Trs [γσ] [exp(−tLXb )].
Trs [γσ] [exp(−tLX
b )]
Z
·
∗
∗
X
= (f,Y )∈ Trs Λ (T X⊕N )⊗F [γσqb,t((f, Y ), γσ(f, Y ))]r(f )df dY.
p⊥
σ (γ)×g

Bingxiao LIU

Twisted trace formula

21/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

Twisted orbital supertrace(B. 2011) For b > 0, t > 0, exp(−tLXb ) has smooth kernel
Xqb,t((x, Y ), (x0 , Y 0 )).
For t > 0, b ∈ ]0, M ]:X
qb,t((x, Y ), (x0 , Y 0 )) ≤ C exp − C 0 (d2 (x, x0 ) + |Y |2 + |Y 0 |2 ) .
Twisted orbital supertrace: Trs [γσ] [exp(−tLXb )].
Trs [γσ] [exp(−tLX
b )]
Z
·
∗
∗
X
= (f,Y )∈ Trs Λ (T X⊕N )⊗F [γσqb,t((f, Y ), γσ(f, Y ))]r(f )df dY.
p⊥
σ (γ)×g

Bingxiao LIU

Twisted trace formula

21/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

Twisted orbital supertrace(B. 2011) For b > 0, t > 0, exp(−tLXb ) has smooth kernel
Xqb,t((x, Y ), (x0 , Y 0 )).
For t > 0, b ∈ ]0, M ]:X
qb,t((x, Y ), (x0 , Y 0 )) ≤ C exp − C 0 (d2 (x, x0 ) + |Y |2 + |Y 0 |2 ) .
Twisted orbital supertrace: Trs [γσ] [exp(−tLXb )].
Trs [γσ] [exp(−tLX
b )]
Z
·
∗
∗
X
= (f,Y )∈ Trs Λ (T X⊕N )⊗F [γσqb,t((f, Y ), γσ(f, Y ))]r(f )df dY.
p⊥
σ (γ)×g

Bingxiao LIU

Twisted trace formula

21/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

A fundamental identity
Theorem (Bismut 2011 for σ = 1, L. 2018)
For b > 0, t > 0,Tr[γσ] [exp(−tLX )] = Trs [γσ] [exp(−tLXb )].
Trs [γσ] [·] supertrace on algebra of Gσ -invariant kernels.
Right-hand side does not depend on b > 0 like McKean-Singer
index theorem.
(B. 2011) As b → 0,
X
qb,t((x, Y ), (x0 , Y 0 )) → P

1πBingxiao LIU20 2
1
pX (x, x0 )e− 2 |Y | +|Y |(m+n)/2 t

Twisted trace formula



P.

22/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

A fundamental identity
Theorem (Bismut 2011 for σ = 1, L. 2018)
For b > 0, t > 0,Tr[γσ] [exp(−tLX )] = Trs [γσ] [exp(−tLXb )].
Trs [γσ] [·] supertrace on algebra of Gσ -invariant kernels.
Right-hand side does not depend on b > 0 like McKean-Singer
index theorem.
(B. 2011) As b → 0,
X
qb,t((x, Y ), (x0 , Y 0 )) → P

1πBingxiao LIU20 2
1
pX (x, x0 )e− 2 |Y | +|Y |(m+n)/2 t

Twisted trace formula



P.

22/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

A fundamental identity
Theorem (Bismut 2011 for σ = 1, L. 2018)
For b > 0, t > 0,Tr[γσ] [exp(−tLX )] = Trs [γσ] [exp(−tLXb )].
Trs [γσ] [·] supertrace on algebra of Gσ -invariant kernels.
Right-hand side does not depend on b > 0 like McKean-Singer
index theorem.
(B. 2011) As b → 0,
X
qb,t((x, Y ), (x0 , Y 0 )) → P

1πBingxiao LIU20 2
1
pX (x, x0 )e− 2 |Y | +|Y |(m+n)/2 t

Twisted trace formula



P.

22/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

A fundamental identity
Theorem (Bismut 2011 for σ = 1, L. 2018)
For b > 0, t > 0,Tr[γσ] [exp(−tLX )] = Trs [γσ] [exp(−tLXb )].
Trs [γσ] [·] supertrace on algebra of Gσ -invariant kernels.
Right-hand side does not depend on b > 0 like McKean-Singer
index theorem.
(B. 2011) As b → 0,
X
qb,t((x, Y ), (x0 , Y 0 )) → P

1πBingxiao LIU20 2
1
pX (x, x0 )e− 2 |Y | +|Y |(m+n)/2 t

Twisted trace formula



P.

22/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

A fundamental identity
Theorem (Bismut 2011 for σ = 1, L. 2018)
For b > 0, t > 0,Tr[γσ] [exp(−tLX )] = Trs [γσ] [exp(−tLXb )].
Trs [γσ] [·] supertrace on algebra of Gσ -invariant kernels.
Right-hand side does not depend on b > 0 like McKean-Singer
index theorem.
(B. 2011) As b → 0,
X
qb,t((x, Y ), (x0 , Y 0 )) → P

1πBingxiao LIU20 2
1
pX (x, x0 )e− 2 |Y | +|Y |(m+n)/2 t

Twisted trace formula



P.

22/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

Making b → +∞As b → +∞, generator of geodesic flow ∇Y T X dominates.
For t > 0: ϕt (x, Y T X ) = γσ(x, Y T X ) ⇒ x ∈ X(γσ).
Evaluation of Trs [γσ] [exp(−tLXb )] can be localized near X(γσ).
X
After rescaling near X(γσ), qb,t((x, Y ), γσ(x, Y )) near X(γσ)
converges to heat kernel of a model operator as b → +∞involving the geometry of normal bundle NX(γσ)/X .

‘Rescaled’ twisted orbital supertrace of the model operator =Jγσ (Y0k ) with Y0k ∈ kσ (γ).

Bingxiao LIU

Twisted trace formula

23/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

Making b → +∞As b → +∞, generator of geodesic flow ∇Y T X dominates.
For t > 0: ϕt (x, Y T X ) = γσ(x, Y T X ) ⇒ x ∈ X(γσ).
Evaluation of Trs [γσ] [exp(−tLXb )] can be localized near X(γσ).
X
After rescaling near X(γσ), qb,t((x, Y ), γσ(x, Y )) near X(γσ)
converges to heat kernel of a model operator as b → +∞involving the geometry of normal bundle NX(γσ)/X .

‘Rescaled’ twisted orbital supertrace of the model operator =Jγσ (Y0k ) with Y0k ∈ kσ (γ).

Bingxiao LIU

Twisted trace formula

23/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

Making b → +∞As b → +∞, generator of geodesic flow ∇Y T X dominates.
For t > 0: ϕt (x, Y T X ) = γσ(x, Y T X ) ⇒ x ∈ X(γσ).
Evaluation of Trs [γσ] [exp(−tLXb )] can be localized near X(γσ).
X
After rescaling near X(γσ), qb,t((x, Y ), γσ(x, Y )) near X(γσ)
converges to heat kernel of a model operator as b → +∞involving the geometry of normal bundle NX(γσ)/X .

‘Rescaled’ twisted orbital supertrace of the model operator =Jγσ (Y0k ) with Y0k ∈ kσ (γ).

Bingxiao LIU

Twisted trace formula

23/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

Making b → +∞As b → +∞, generator of geodesic flow ∇Y T X dominates.
For t > 0: ϕt (x, Y T X ) = γσ(x, Y T X ) ⇒ x ∈ X(γσ).
Evaluation of Trs [γσ] [exp(−tLXb )] can be localized near X(γσ).
X
After rescaling near X(γσ), qb,t((x, Y ), γσ(x, Y )) near X(γσ)
converges to heat kernel of a model operator as b → +∞involving the geometry of normal bundle NX(γσ)/X .

‘Rescaled’ twisted orbital supertrace of the model operator =Jγσ (Y0k ) with Y0k ∈ kσ (γ).

Bingxiao LIU

Twisted trace formula

23/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

Making b → +∞As b → +∞, generator of geodesic flow ∇Y T X dominates.
For t > 0: ϕt (x, Y T X ) = γσ(x, Y T X ) ⇒ x ∈ X(γσ).
Evaluation of Trs [γσ] [exp(−tLXb )] can be localized near X(γσ).
X
After rescaling near X(γσ), qb,t((x, Y ), γσ(x, Y )) near X(γσ)
converges to heat kernel of a model operator as b → +∞involving the geometry of normal bundle NX(γσ)/X .

‘Rescaled’ twisted orbital supertrace of the model operator =Jγσ (Y0k ) with Y0k ∈ kσ (γ).

Bingxiao LIU

Twisted trace formula

23/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

Making b → +∞As b → +∞, generator of geodesic flow ∇Y T X dominates.
For t > 0: ϕt (x, Y T X ) = γσ(x, Y T X ) ⇒ x ∈ X(γσ).
Evaluation of Trs [γσ] [exp(−tLXb )] can be localized near X(γσ).
X
After rescaling near X(γσ), qb,t((x, Y ), γσ(x, Y )) near X(γσ)
converges to heat kernel of a model operator as b → +∞involving the geometry of normal bundle NX(γσ)/X .

‘Rescaled’ twisted orbital supertrace of the model operator =Jγσ (Y0k ) with Y0k ∈ kσ (γ).

Bingxiao LIU

Twisted trace formula

23/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

The function Jγσ (Y0k )Depends on the action of γσ on g = p ⊕ k.
Formula of Jγσ (Y0k ), Y0k ∈ kσ (γ)
Jγσ (Y0k ) =kb
A(iad(Y10 )|pσ (γ) )kb| det(1 − Ad(γσ))|z⊥|1/2 A(iad(Y0 )|kσ (γ) )
0
k



2
det(1 − e−iad(Y0 ) Ad(k −1 σ))|k⊥1
σ,0 (γ).
kdet(1 − Ad(k −1 σ))|z⊥det(1 − e−iad(Y0 ) Ad(k −1 σ))|p⊥ (γ)σ,0 (γ)1σ,0Bingxiao LIU

Twisted trace formula

24/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

The function Jγσ (Y0k )Depends on the action of γσ on g = p ⊕ k.
Formula of Jγσ (Y0k ), Y0k ∈ kσ (γ)
Jγσ (Y0k ) =kb
A(iad(Y10 )|pσ (γ) )kb| det(1 − Ad(γσ))|z⊥|1/2 A(iad(Y0 )|kσ (γ) )
0
k



2
det(1 − e−iad(Y0 ) Ad(k −1 σ))|k⊥1
σ,0 (γ).
kdet(1 − Ad(k −1 σ))|z⊥det(1 − e−iad(Y0 ) Ad(k −1 σ))|p⊥ (γ)σ,0 (γ)1σ,0Bingxiao LIU

Twisted trace formula

24/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

The function Jγσ (Y0k )Depends on the action of γσ on g = p ⊕ k.
Formula of Jγσ (Y0k ), Y0k ∈ kσ (γ)
Jγσ (Y0k ) =kb
A(iad(Y10 )|pσ (γ) )kb| det(1 − Ad(γσ))|z⊥|1/2 A(iad(Y0 )|kσ (γ) )
0
k



2
det(1 − e−iad(Y0 ) Ad(k −1 σ))|k⊥1
σ,0 (γ).
kdet(1 − Ad(k −1 σ))|z⊥det(1 − e−iad(Y0 ) Ad(k −1 σ))|p⊥ (γ)σ,0 (γ)1σ,0Bingxiao LIU

Twisted trace formula

24/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

An explicit geometric formula
Put p = dim pσ (γ), q = dim kσ (γ), r = p + q.

Theorem A
For t > 0, we haveTr[γσ] [exp(−tLX )] =Z

exp(−|a|2 /2t)×(2πt)p/2
k 2

Jγσ (Y0k )TrE [ρE (k −1 σ) exp(−iρE (Y0k ))]e−|Y0 | /2tkσ (γ)

Bingxiao LIU

Twisted trace formuladY0k.
(2πt)q/2

25/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

An explicit geometric formula
Put p = dim pσ (γ), q = dim kσ (γ), r = p + q.

Theorem A
For t > 0, we haveTr[γσ] [exp(−tLX )] =Z

exp(−|a|2 /2t)×(2πt)p/2
k 2

Jγσ (Y0k )TrE [ρE (k −1 σ) exp(−iρE (Y0k ))]e−|Y0 | /2tkσ (γ)

Bingxiao LIU

Twisted trace formuladY0k.
(2πt)q/2

25/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Casimir operator
Hypoelliptic Laplacian
An explicit geometric formula

An explicit geometric formula
Put p = dim pσ (γ), q = dim kσ (γ), r = p + q.

Theorem A
For t > 0, we haveTr[γσ] [exp(−tLX )] =Z

exp(−|a|2 /2t)×(2πt)p/2
k 2

Jγσ (Y0k )TrE [ρE (k −1 σ) exp(−iρE (Y0k ))]e−|Y0 | /2tkσ (γ)

Bingxiao LIU

Twisted trace formuladY0k.
(2πt)q/2

25/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

Geometric settingρE : Gσ → E, F = Gσ ×K σ E ' X × E with flat connection ∇F .
Σσ acts on F → X preserving ∇F .
Γ ⊂ G discrete, cocompact, torsion-free subgroup such that
σ(Γ) = Γ.
Z = Γ\X compact smooth manifold.
Σσ acts on F → Z.
DZ, F Hodge-de Rham operator:
DZ, F = dF + dF,∗ .

Bingxiao LIU

Twisted trace formula

26/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

Geometric settingρE : Gσ → E, F = Gσ ×K σ E ' X × E with flat connection ∇F .
Σσ acts on F → X preserving ∇F .
Γ ⊂ G discrete, cocompact, torsion-free subgroup such that
σ(Γ) = Γ.
Z = Γ\X compact smooth manifold.
Σσ acts on F → Z.
DZ, F Hodge-de Rham operator:
DZ, F = dF + dF,∗ .

Bingxiao LIU

Twisted trace formula

26/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

Geometric settingρE : Gσ → E, F = Gσ ×K σ E ' X × E with flat connection ∇F .
Σσ acts on F → X preserving ∇F .
Γ ⊂ G discrete, cocompact, torsion-free subgroup such that
σ(Γ) = Γ.
Z = Γ\X compact smooth manifold.
Σσ acts on F → Z.
DZ, F Hodge-de Rham operator:
DZ, F = dF + dF,∗ .

Bingxiao LIU

Twisted trace formula

26/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

Geometric settingρE : Gσ → E, F = Gσ ×K σ E ' X × E with flat connection ∇F .
Σσ acts on F → X preserving ∇F .
Γ ⊂ G discrete, cocompact, torsion-free subgroup such that
σ(Γ) = Γ.
Z = Γ\X compact smooth manifold.
Σσ acts on F → Z.
DZ, F Hodge-de Rham operator:
DZ, F = dF + dF,∗ .

Bingxiao LIU

Twisted trace formula

26/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

Geometric settingρE : Gσ → E, F = Gσ ×K σ E ' X × E with flat connection ∇F .
Σσ acts on F → X preserving ∇F .
Γ ⊂ G discrete, cocompact, torsion-free subgroup such that
σ(Γ) = Γ.
Z = Γ\X compact smooth manifold.
Σσ acts on F → Z.
DZ, F Hodge-de Rham operator:
DZ, F = dF + dF,∗ .

Bingxiao LIU

Twisted trace formula

26/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

Geometric settingρE : Gσ → E, F = Gσ ×K σ E ' X × E with flat connection ∇F .
Σσ acts on F → X preserving ∇F .
Γ ⊂ G discrete, cocompact, torsion-free subgroup such that
σ(Γ) = Γ.
Z = Γ\X compact smooth manifold.
Σσ acts on F → Z.
DZ, F Hodge-de Rham operator:
DZ, F = dF + dF,∗ .

Bingxiao LIU

Twisted trace formula

26/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

Geometric settingρE : Gσ → E, F = Gσ ×K σ E ' X × E with flat connection ∇F .
Σσ acts on F → X preserving ∇F .
Γ ⊂ G discrete, cocompact, torsion-free subgroup such that
σ(Γ) = Γ.
Z = Γ\X compact smooth manifold.
Σσ acts on F → Z.
DZ, F Hodge-de Rham operator:
DZ, F = dF + dF,∗ .

Bingxiao LIU

Twisted trace formula

26/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

Equivariant analytic torsion

If s ∈ C, Re s large enough, put⊥ϑσ (s) = −Trs N σ[DZ,F,2 ]−s Pker
DZ,F .
Equivariant Ray-Singer torsion
Tσ (F ) =We have

1 ∂ϑσ
(0).
2 ∂s

1 Z,F,2
D= LZ + A
2

Bingxiao LIU

Twisted trace formula

27/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

Equivariant analytic torsion

If s ∈ C, Re s large enough, put⊥ϑσ (s) = −Trs N σ[DZ,F,2 ]−s Pker
DZ,F .
Equivariant Ray-Singer torsion
Tσ (F ) =We have

1 ∂ϑσ
(0).
2 ∂s

1 Z,F,2
D= LZ + A
2

Bingxiao LIU

Twisted trace formula

27/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

Equivariant analytic torsion

If s ∈ C, Re s large enough, put⊥ϑσ (s) = −Trs N σ[DZ,F,2 ]−s Pker
DZ,F .
Equivariant Ray-Singer torsion
Tσ (F ) =We have

1 ∂ϑσ
(0).
2 ∂s

1 Z,F,2
D= LZ + A
2

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Twisted trace formula

27/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

Equivariant analytic torsion

If s ∈ C, Re s large enough, put⊥ϑσ (s) = −Trs N σ[DZ,F,2 ]−s Pker
DZ,F .
Equivariant Ray-Singer torsion
Tσ (F ) =We have

1 ∂ϑσ
(0).
2 ∂s

1 Z,F,2
D= LZ + A
2

Bingxiao LIU

Twisted trace formula

27/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

A family of representations of Gσ
U compact form of G with Lie algebra u = k ⊕ ip.
Unitary trick: representations of G can be constructed from
representations of U .
U σ = U o Σσ .
Take λ fixed by σ, irreducible representation of U extends torepresentation of U σ .
This gives representation of Gσ .
For d ∈ N∗ , representation Ed of Gσ with highest weight dλ:
Ed = H (0,0) (Mλ , Ldλ ).

Bingxiao LIU

Twisted trace formula

28/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

A family of representations of Gσ
U compact form of G with Lie algebra u = k ⊕ ip.
Unitary trick: representations of G can be constructed from
representations of U .
U σ = U o Σσ .
Take λ fixed by σ, irreducible representation of U extends torepresentation of U σ .
This gives representation of Gσ .
For d ∈ N∗ , representation Ed of Gσ with highest weight dλ:
Ed = H (0,0) (Mλ , Ldλ ).

Bingxiao LIU

Twisted trace formula

28/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

A family of representations of Gσ
U compact form of G with Lie algebra u = k ⊕ ip.
Unitary trick: representations of G can be constructed from
representations of U .
U σ = U o Σσ .
Take λ fixed by σ, irreducible representation of U extends torepresentation of U σ .
This gives representation of Gσ .
For d ∈ N∗ , representation Ed of Gσ with highest weight dλ:
Ed = H (0,0) (Mλ , Ldλ ).

Bingxiao LIU

Twisted trace formula

28/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

A family of representations of Gσ
U compact form of G with Lie algebra u = k ⊕ ip.
Unitary trick: representations of G can be constructed from
representations of U .
U σ = U o Σσ .
Take λ fixed by σ, irreducible representation of U extends torepresentation of U σ .
This gives representation of Gσ .
For d ∈ N∗ , representation Ed of Gσ with highest weight dλ:
Ed = H (0,0) (Mλ , Ldλ ).

Bingxiao LIU

Twisted trace formula

28/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

A family of representations of Gσ
U compact form of G with Lie algebra u = k ⊕ ip.
Unitary trick: representations of G can be constructed from
representations of U .
U σ = U o Σσ .
Take λ fixed by σ, irreducible representation of U extends torepresentation of U σ .
This gives representation of Gσ .
For d ∈ N∗ , representation Ed of Gσ with highest weight dλ:
Ed = H (0,0) (Mλ , Ldλ ).

Bingxiao LIU

Twisted trace formula

28/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

A family of representations of Gσ
U compact form of G with Lie algebra u = k ⊕ ip.
Unitary trick: representations of G can be constructed from
representations of U .
U σ = U o Σσ .
Take λ fixed by σ, irreducible representation of U extends torepresentation of U σ .
This gives representation of Gσ .
For d ∈ N∗ , representation Ed of Gσ with highest weight dλ:
Ed = H (0,0) (Mλ , Ldλ ).

Bingxiao LIU

Twisted trace formula

28/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

A family of representations of Gσ
U compact form of G with Lie algebra u = k ⊕ ip.
Unitary trick: representations of G can be constructed from
representations of U .
U σ = U o Σσ .
Take λ fixed by σ, irreducible representation of U extends torepresentation of U σ .
This gives representation of Gσ .
For d ∈ N∗ , representation Ed of Gσ with highest weight dλ:
Ed = H (0,0) (Mλ , Ldλ ).

Bingxiao LIU

Twisted trace formula

28/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

Nondegeneracy condition and spectral gap(Ed , ρEd ) defines a family of flat vector bundles Fd on Z
equipped with action of σ.
Nondegeneracy condition (BMZ):µ(Mλ ) ∩ k∗ = ∅ ←→ WU λ ∩ t∗K = ∅.
By BMZ, Müller-Pfaff, if λ is non-degenerate, one has spectral
gap
DZ,Fd ,2 ≥ cd2 − C.
For d large enough: H · (Z, Fd ) = 0.

Bingxiao LIU

Twisted trace formula

29/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

Nondegeneracy condition and spectral gap(Ed , ρEd ) defines a family of flat vector bundles Fd on Z
equipped with action of σ.
Nondegeneracy condition (BMZ):µ(Mλ ) ∩ k∗ = ∅ ←→ WU λ ∩ t∗K = ∅.
By BMZ, Müller-Pfaff, if λ is non-degenerate, one has spectral
gap
DZ,Fd ,2 ≥ cd2 − C.
For d large enough: H · (Z, Fd ) = 0.

Bingxiao LIU

Twisted trace formula

29/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

Nondegeneracy condition and spectral gap(Ed , ρEd ) defines a family of flat vector bundles Fd on Z
equipped with action of σ.
Nondegeneracy condition (BMZ):µ(Mλ ) ∩ k∗ = ∅ ←→ WU λ ∩ t∗K = ∅.
By BMZ, Müller-Pfaff, if λ is non-degenerate, one has spectral
gap
DZ,Fd ,2 ≥ cd2 − C.
For d large enough: H · (Z, Fd ) = 0.

Bingxiao LIU

Twisted trace formula

29/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

Nondegeneracy condition and spectral gap(Ed , ρEd ) defines a family of flat vector bundles Fd on Z
equipped with action of σ.
Nondegeneracy condition (BMZ):µ(Mλ ) ∩ k∗ = ∅ ←→ WU λ ∩ t∗K = ∅.
By BMZ, Müller-Pfaff, if λ is non-degenerate, one has spectral
gap
DZ,Fd ,2 ≥ cd2 − C.
For d large enough: H · (Z, Fd ) = 0.

Bingxiao LIU

Twisted trace formula

29/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

Nondegeneracy condition and spectral gap(Ed , ρEd ) defines a family of flat vector bundles Fd on Z
equipped with action of σ.
Nondegeneracy condition (BMZ):µ(Mλ ) ∩ k∗ = ∅ ←→ WU λ ∩ t∗K = ∅.
By BMZ, Müller-Pfaff, if λ is non-degenerate, one has spectral
gap
DZ,Fd ,2 ≥ cd2 − C.
For d large enough: H · (Z, Fd ) = 0.

Bingxiao LIU

Twisted trace formula

29/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

Asymptotics of equivariant analytic torsion

A fixed point formula:
Theorem B
Under the non-deg. condition, as d → +∞,d−m(σ)−1 Tσ (Fd )
X
X

j max
+ o(1).
=
Vol(Γ ∩ Zσ (γ)\X(γσ))
sj (γσ)d [Wγσ]{z}|
| {z }
certain elliptic
jclasses [γ]σfixed pts of σ in Z

Bingxiao LIU

oscillating

Twisted trace formula

30/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

Asymptotics of equivariant analytic torsion

A fixed point formula:
Theorem B
Under the non-deg. condition, as d → +∞,d−m(σ)−1 Tσ (Fd )
X
X

j max
+ o(1).
=
Vol(Γ ∩ Zσ (γ)\X(γσ))
sj (γσ)d [Wγσ]{z}|
| {z }
certain elliptic
jclasses [γ]σfixed pts of σ in Z

Bingxiao LIU

oscillating

Twisted trace formula

30/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

Asymptotics of equivariant analytic torsion

A fixed point formula:
Theorem B
Under the non-deg. condition, as d → +∞,d−m(σ)−1 Tσ (Fd )
X
X

j max
+ o(1).
=
Vol(Γ ∩ Zσ (γ)\X(γσ))
sj (γσ)d [Wγσ]{z}|
| {z }
certain elliptic
jclasses [γ]σfixed pts of σ in Z

Bingxiao LIU

oscillating

Twisted trace formula

30/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

Non-elliptic elements do not contribute
Z 1/d
Mellin transform:

Z +∞
···+
· · ·.
0
1/d| {z } | {z }main termO(e−cd )Selberg’s twisted trace formula:m
Trs [(N − )σ exp(−tDZ,Fd ,2 /2)] =2Xm
Vol(Γ ∩ Zσ (γ)\X(γσ))Tr[γσ] [(N − ) exp(−tDX,Fd ,2 )].
2[γ]σ ∈Cσ (Γ)

Using spectral gap and small time estimates on heat kernel,
contribution of non-elliptic classes = O(e−cd ).

Bingxiao LIU

Twisted trace formula

31/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

Non-elliptic elements do not contribute
Z 1/d
Mellin transform:

Z +∞
···+
· · ·.
0
1/d| {z } | {z }main termO(e−cd )Selberg’s twisted trace formula:m
Trs [(N − )σ exp(−tDZ,Fd ,2 /2)] =2Xm
Vol(Γ ∩ Zσ (γ)\X(γσ))Tr[γσ] [(N − ) exp(−tDX,Fd ,2 )].
2[γ]σ ∈Cσ (Γ)

Using spectral gap and small time estimates on heat kernel,
contribution of non-elliptic classes = O(e−cd ).

Bingxiao LIU

Twisted trace formula

31/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

Non-elliptic elements do not contribute
Z 1/d
Mellin transform:

Z +∞
···+
· · ·.
0
1/d| {z } | {z }main termO(e−cd )Selberg’s twisted trace formula:m
Trs [(N − )σ exp(−tDZ,Fd ,2 /2)] =2Xm
Vol(Γ ∩ Zσ (γ)\X(γσ))Tr[γσ] [(N − ) exp(−tDX,Fd ,2 )].
2[γ]σ ∈Cσ (Γ)

Using spectral gap and small time estimates on heat kernel,
contribution of non-elliptic classes = O(e−cd ).

Bingxiao LIU

Twisted trace formula

31/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

Non-elliptic elements do not contribute
Z 1/d
Mellin transform:

Z +∞
···+
· · ·.
0
1/d| {z } | {z }main termO(e−cd )Selberg’s twisted trace formula:m
Trs [(N − )σ exp(−tDZ,Fd ,2 /2)] =2Xm
Vol(Γ ∩ Zσ (γ)\X(γσ))Tr[γσ] [(N − ) exp(−tDX,Fd ,2 )].
2[γ]σ ∈Cσ (Γ)

Using spectral gap and small time estimates on heat kernel,
contribution of non-elliptic classes = O(e−cd ).

Bingxiao LIU

Twisted trace formula

31/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

Elliptic classes

If rσ (γ) = rk(Zσ (γ)0 ) − rk(Kσ (γ)0 ) 6= 1, the corresponding
twisted orbital integral vanishes (extension of a result of
Moscovici-Stanton (1991)).
Only certain elliptic classes (with rσ (γ) = 1) contributes toleading term of asymptotics.

Bingxiao LIU

Twisted trace formula

32/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

Elliptic classes

If rσ (γ) = rk(Zσ (γ)0 ) − rk(Kσ (γ)0 ) 6= 1, the corresponding
twisted orbital integral vanishes (extension of a result of
Moscovici-Stanton (1991)).
Only certain elliptic classes (with rσ (γ) = 1) contributes toleading term of asymptotics.

Bingxiao LIU

Twisted trace formula

32/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

Elliptic classes

If rσ (γ) = rk(Zσ (γ)0 ) − rk(Kσ (γ)0 ) 6= 1, the corresponding
twisted orbital integral vanishes (extension of a result of
Moscovici-Stanton (1991)).
Only certain elliptic classes (with rσ (γ) = 1) contributes toleading term of asymptotics.

Bingxiao LIU

Twisted trace formula

32/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

When rσ (γ) = 1
If rσ (γ) = 1, as d → +∞d−n(γσ)−1 (1 + 2t



∂
m
)Trs [γσ] N −exp(−tDX,Fd ,2 /2d2 )
∂t
2
X
=2
sj (γσ)d [djt ]max + O(d−1 ).
{z }|joscillating terms

Where djt are locally computable real diff. forms on X(γσ) used todefine the W -invariants.
May assume γ = k −1 ∈ K.
Apply Theorem A to Trs [γσ] [·]:dpt
Trs [γσ] [·] =exp( 2 . . . )d(2πt)p/2
Z
·
∗
y
y
dy
Jγσ ( )Trs Λ (p ) [· · · ( )]χEd (γσeiy/d ) exp(−|y|2 /2t)
.
q/2
d
d
(2πt)
y∈kσ (γ)
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Twisted trace formula

33/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

When rσ (γ) = 1
If rσ (γ) = 1, as d → +∞d−n(γσ)−1 (1 + 2t



∂
m
)Trs [γσ] N −exp(−tDX,Fd ,2 /2d2 )
∂t
2
X
=2
sj (γσ)d [djt ]max + O(d−1 ).
{z }|joscillating terms

Where djt are locally computable real diff. forms on X(γσ) used todefine the W -invariants.
May assume γ = k −1 ∈ K.
Apply Theorem A to Trs [γσ] [·]:dpt
Trs [γσ] [·] =exp( 2 . . . )d(2πt)p/2
Z
·
∗
y
y
dy
Jγσ ( )Trs Λ (p ) [· · · ( )]χEd (γσeiy/d ) exp(−|y|2 /2t)
.
q/2
d
d
(2πt)
y∈kσ (γ)
Bingxiao LIU

Twisted trace formula

33/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

When rσ (γ) = 1
If rσ (γ) = 1, as d → +∞d−n(γσ)−1 (1 + 2t



∂
m
)Trs [γσ] N −exp(−tDX,Fd ,2 /2d2 )
∂t
2
X
=2
sj (γσ)d [djt ]max + O(d−1 ).
{z }|joscillating terms

Where djt are locally computable real diff. forms on X(γσ) used todefine the W -invariants.
May assume γ = k −1 ∈ K.
Apply Theorem A to Trs [γσ] [·]:dpt
Trs [γσ] [·] =exp( 2 . . . )d(2πt)p/2
Z
·
∗
y
y
dy
Jγσ ( )Trs Λ (p ) [· · · ( )]χEd (γσeiy/d ) exp(−|y|2 /2t)
.
q/2
d
d
(2πt)
y∈kσ (γ)
Bingxiao LIU

Twisted trace formula

33/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

When rσ (γ) = 1
If rσ (γ) = 1, as d → +∞d−n(γσ)−1 (1 + 2t



∂
m
)Trs [γσ] N −exp(−tDX,Fd ,2 /2d2 )
∂t
2
X
=2
sj (γσ)d [djt ]max + O(d−1 ).
{z }|joscillating terms

Where djt are locally computable real diff. forms on X(γσ) used todefine the W -invariants.
May assume γ = k −1 ∈ K.
Apply Theorem A to Trs [γσ] [·]:dpt
Trs [γσ] [·] =exp( 2 . . . )d(2πt)p/2
Z
·
∗
y
y
dy
Jγσ ( )Trs Λ (p ) [· · · ( )]χEd (γσeiy/d ) exp(−|y|2 /2t)
.
q/2
d
d
(2πt)
y∈kσ (γ)
Bingxiao LIU

Twisted trace formula

33/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

When rσ (γ) = 1: continued
γσMλ fixed point set of γσ on Mλ :γσ

Mλ = ∪j γσ Mλj .

j(y) on uσ (γ).
Uσ (γ) acts on Lλ |γσ M j : DH integral Rγσλsj (γσ) represents action of γσ on Lλ |γσ M j .
λKirillov formula (cf. Berline-Vergne 1985): for y ∈ uσ (γ), as
d → +∞,XjχEd (γσey/d ) = dn(γσ)
sj (γσ)d Rγσ(y) + O(dn(γσ)−1 ).
jZσ (γ) → X(γσ) with a flat connection form ω zσ (γ) .
j
BMZ: Rγσgives djt (under condition rσ (γ) = 1).

Bingxiao LIU

Twisted trace formula

34/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

When rσ (γ) = 1: continued
γσMλ fixed point set of γσ on Mλ :γσ

Mλ = ∪j γσ Mλj .

j(y) on uσ (γ).
Uσ (γ) acts on Lλ |γσ M j : DH integral Rγσλsj (γσ) represents action of γσ on Lλ |γσ M j .
λKirillov formula (cf. Berline-Vergne 1985): for y ∈ uσ (γ), as
d → +∞,XjχEd (γσey/d ) = dn(γσ)
sj (γσ)d Rγσ(y) + O(dn(γσ)−1 ).
jZσ (γ) → X(γσ) with a flat connection form ω zσ (γ) .
j
BMZ: Rγσgives djt (under condition rσ (γ) = 1).

Bingxiao LIU

Twisted trace formula

34/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

When rσ (γ) = 1: continued
γσMλ fixed point set of γσ on Mλ :γσ

Mλ = ∪j γσ Mλj .

j(y) on uσ (γ).
Uσ (γ) acts on Lλ |γσ M j : DH integral Rγσλsj (γσ) represents action of γσ on Lλ |γσ M j .
λKirillov formula (cf. Berline-Vergne 1985): for y ∈ uσ (γ), as
d → +∞,XjχEd (γσey/d ) = dn(γσ)
sj (γσ)d Rγσ(y) + O(dn(γσ)−1 ).
jZσ (γ) → X(γσ) with a flat connection form ω zσ (γ) .
j
BMZ: Rγσgives djt (under condition rσ (γ) = 1).

Bingxiao LIU

Twisted trace formula

34/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

When rσ (γ) = 1: continued
γσMλ fixed point set of γσ on Mλ :γσ

Mλ = ∪j γσ Mλj .

j(y) on uσ (γ).
Uσ (γ) acts on Lλ |γσ M j : DH integral Rγσλsj (γσ) represents action of γσ on Lλ |γσ M j .
λKirillov formula (cf. Berline-Vergne 1985): for y ∈ uσ (γ), as
d → +∞,XjχEd (γσey/d ) = dn(γσ)
sj (γσ)d Rγσ(y) + O(dn(γσ)−1 ).
jZσ (γ) → X(γσ) with a flat connection form ω zσ (γ) .
j
BMZ: Rγσgives djt (under condition rσ (γ) = 1).

Bingxiao LIU

Twisted trace formula

34/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

When rσ (γ) = 1: continued
γσMλ fixed point set of γσ on Mλ :γσ

Mλ = ∪j γσ Mλj .

j(y) on uσ (γ).
Uσ (γ) acts on Lλ |γσ M j : DH integral Rγσλsj (γσ) represents action of γσ on Lλ |γσ M j .
λKirillov formula (cf. Berline-Vergne 1985): for y ∈ uσ (γ), as
d → +∞,XjχEd (γσey/d ) = dn(γσ)
sj (γσ)d Rγσ(y) + O(dn(γσ)−1 ).
jZσ (γ) → X(γσ) with a flat connection form ω zσ (γ) .
j
BMZ: Rγσgives djt (under condition rσ (γ) = 1).

Bingxiao LIU

Twisted trace formula

34/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

When rσ (γ) = 1: continued
γσMλ fixed point set of γσ on Mλ :γσ

Mλ = ∪j γσ Mλj .

j(y) on uσ (γ).
Uσ (γ) acts on Lλ |γσ M j : DH integral Rγσλsj (γσ) represents action of γσ on Lλ |γσ M j .
λKirillov formula (cf. Berline-Vergne 1985): for y ∈ uσ (γ), as
d → +∞,XjχEd (γσey/d ) = dn(γσ)
sj (γσ)d Rγσ(y) + O(dn(γσ)−1 ).
jZσ (γ) → X(γσ) with a flat connection form ω zσ (γ) .
j
BMZ: Rγσgives djt (under condition rσ (γ) = 1).

Bingxiao LIU

Twisted trace formula

34/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

When rσ (γ) = 1: continued
γσMλ fixed point set of γσ on Mλ :γσ

Mλ = ∪j γσ Mλj .

j(y) on uσ (γ).
Uσ (γ) acts on Lλ |γσ M j : DH integral Rγσλsj (γσ) represents action of γσ on Lλ |γσ M j .
λKirillov formula (cf. Berline-Vergne 1985): for y ∈ uσ (γ), as
d → +∞,XjχEd (γσey/d ) = dn(γσ)
sj (γσ)d Rγσ(y) + O(dn(γσ)−1 ).
jZσ (γ) → X(γσ) with a flat connection form ω zσ (γ) .
j
BMZ: Rγσgives djt (under condition rσ (γ) = 1).

Bingxiao LIU

Twisted trace formula

34/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

W -invariants for fixed point set
For λ non-degenerate, W -invariants associated with Zσ (γ) are
j
Wγσ
=−

Z +∞
0

djt

dt
.
t

They are local invariants on fixed point set of σ in Z.
Theorem B
As d → +∞,d−m(σ)−1 Tσ (Fd )
X
X

j max
=
Vol(Γ ∩ Zσ (γ)\X(γσ))
sj (γσ)d [Wγσ]+ o(1).
jcertain elliptic
classes [γ]σBingxiao LIU

Twisted trace formula

35/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

W -invariants for fixed point set
For λ non-degenerate, W -invariants associated with Zσ (γ) are
j
Wγσ
=−

Z +∞
0

djt

dt
.
t

They are local invariants on fixed point set of σ in Z.
Theorem B
As d → +∞,d−m(σ)−1 Tσ (Fd )
X
X

j max
=
Vol(Γ ∩ Zσ (γ)\X(γσ))
sj (γσ)d [Wγσ]+ o(1).
jcertain elliptic
classes [γ]σBingxiao LIU

Twisted trace formula

35/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

W -invariants for fixed point set
For λ non-degenerate, W -invariants associated with Zσ (γ) are
j
Wγσ
=−

Z +∞
0

djt

dt
.
t

They are local invariants on fixed point set of σ in Z.
Theorem B
As d → +∞,d−m(σ)−1 Tσ (Fd )
X
X

j max
=
Vol(Γ ∩ Zσ (γ)\X(γσ))
sj (γσ)d [Wγσ]+ o(1).
jcertain elliptic
classes [γ]σBingxiao LIU

Twisted trace formula

35/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

W -invariants for fixed point set
For λ non-degenerate, W -invariants associated with Zσ (γ) are
j
Wγσ
=−

Z +∞
0

djt

dt
.
t

They are local invariants on fixed point set of σ in Z.
Theorem B
As d → +∞,d−m(σ)−1 Tσ (Fd )
X
X

j max
=
Vol(Γ ∩ Zσ (γ)\X(γσ))
sj (γσ)d [Wγσ]+ o(1).
jcertain elliptic
classes [γ]σBingxiao LIU

Twisted trace formula

35/36

Overview
Definition of twisted orbital integrals
An explicit geometric formula
Asymptotics of the equivariant Ray-Singer torsion

Equivariant analytic torsion
A family of flat vector bundles
Asymptotics of equivariant Ray-Singer torsion

Thank you.

Bingxiao LIU

Twisted trace formula

36/36