Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Hypoelliptic Laplacian, index theory andthe trace formula
Jean-Michel Bismut
Université Paris-Sud, Orsay

Luminy, le 9 octobre 2018

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

1 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References1

Loop groups and coadjoint orbits

2

Heat kernels and equivariant localization

3

Rigorous proof in the Hamiltonian formalism

4

Selberg’s trace formula

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

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Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

A compact Lie group

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

3 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

A compact Lie group
G compact connected Lie group, g Lie algebra, B
G-invariant scalar product.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

3 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

A compact Lie group
G compact connected Lie group, g Lie algebra, B
G-invariant scalar product.
b = LG o S 1 .
S 1 acts on LG, LG

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

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Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

A compact Lie group
G compact connected Lie group, g Lie algebra, B
G-invariant scalar product.
b = LG o S 1 .
S 1 acts on LG, LG
b = a d + As , a ∈ R, A· ∈ g.
Lie algebra Lg
ds

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

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Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

A compact Lie group
G compact connected Lie group, g Lie algebra, B
G-invariant scalar product.
b = LG o S 1 .
S 1 acts on LG, LG
b = a d + As , a ∈ R, A· ∈ g.
Lie algebra Lgds
dIn the sequel, we take a = 1, ds+ As = connections
1
on G bundle over S .

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

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Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

A compact Lie group
G compact connected Lie group, g Lie algebra, B
G-invariant scalar product.
b = LG o S 1 .
S 1 acts on LG, LG
b = a d + As , a ∈ R, A· ∈ g.
Lie algebra Lgds
dIn the sequel, we take a = 1, ds+ As = connections
1
on G bundle over S .
Two connections gauge equivalent ⇔ they haveconjugate holonomies.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

3 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

A compact Lie group
G compact connected Lie group, g Lie algebra, B
G-invariant scalar product.
b = LG o S 1 .
S 1 acts on LG, LG
b = a d + As , a ∈ R, A· ∈ g.
Lie algebra Lgds
dIn the sequel, we take a = 1, ds+ As = connections
1
on G bundle over S .
Two connections gauge equivalent ⇔ they haveconjugate holonomies.
If g ∈ G, Og ⊂ G adjoint orbit in G.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

3 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

A compact Lie group
G compact connected Lie group, g Lie algebra, B
G-invariant scalar product.
b = LG o S 1 .
S 1 acts on LG, LG
b = a d + As , a ∈ R, A· ∈ g.
Lie algebra Lgds
dIn the sequel, we take a = 1, ds+ As = connections
1
on G bundle over S .
Two connections gauge equivalent ⇔ they haveconjugate holonomies.
If g ∈ G, Og ⊂ G adjoint orbit in G.
b = smooth paths in G
Coadjoint orbit Og ⊂ LG
connecting 1 and Og : dtd + A ↔ ġ + Ar = 0.
Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

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Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The symplectic structure

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

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Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The symplectic structure

Og inherits canonical symplectic structure.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

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Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The symplectic structure

Og inherits canonical symplectic structure.
R
If α, β ∈ Lg, ωA (DA α, DA β) = S 1 hDA α, βi.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

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Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The symplectic structure

Og inherits canonical symplectic structure.
R
If α, β ∈ Lg, ωA (DA α, DA β) = S 1 hDA α, βi.
S 1 acts symplectically on ROg and the corresponding
R
Hamiltonian is E (A) = 21 S 1 |A|2 = 12 S 1 |ġ|2 ds.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

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Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The symplectic structure

Og inherits canonical symplectic structure.
R
If α, β ∈ Lg, ωA (DA α, DA β) = S 1 hDA α, βi.
S 1 acts symplectically on ROg and the corresponding
R
Hamiltonian is E (A) = 21 S 1 |A|2 = 12 S 1 |ġ|2 ds.
e central extension of LG.
c
LG

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

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Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The symplectic structure

Og inherits canonical symplectic structure.
R
If α, β ∈ Lg, ωA (DA α, DA β) = S 1 hDA α, βi.
S 1 acts symplectically on ROg and the corresponding
R
Hamiltonian is E (A) = 21 S 1 |A|2 = 12 S 1 |ġ|2 ds.
e central extension of LG.
cLGdFull coadjoint orbit = ds + As , E (A) .

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

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Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The heat kernel on G and Atiyah-Frenkel

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

5 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The heat kernel on G and Atiyah-Frenkelpt (g) smooth kernel for exp t∆G /2 .

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

5 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The heat kernel on G and Atiyah-Frenkelpt (g) smooth kernel for exp t∆G /2 .
pt (g) Ad-invariant function.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

5 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The heat kernel on G and Atiyah-Frenkelpt (g) smooth kernel for exp t∆G /2 .
pt (g) Ad-invariant function.
 Dg
RRpath integral.
pt (g) = Og exp − S 1 |ġ|2 ds/2t t∞/2

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

5 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The heat kernel on G and Atiyah-Frenkelpt (g) smooth kernel for exp t∆G /2 .
pt (g) Ad-invariant function.
 Dg
RRpath integral.
pt (g) = Og exp − S 1 |ġ|2 ds/2t t∞/2
I. Frenkel 84: pt (g) numerator character loop group:
Kirillov-Lefschetz principle.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

5 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The heat kernel on G and Atiyah-Frenkelpt (g) smooth kernel for exp t∆G /2 .
pt (g) Ad-invariant function.
 Dg
RRpath integral.
pt (g) = Og exp − S 1 |ġ|2 ds/2t t∞/2
I. Frenkel 84: pt (g) numerator character loop group:
Kirillov-Lefschetz principle.
R
Atiyah 85: pt (g) = Og exp (− (E + ω) /t).

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

5 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The heat kernel on G and Atiyah-Frenkelpt (g) smooth kernel for exp t∆G /2 .
pt (g) Ad-invariant function.
 Dg
RRpath integral.
pt (g) = Og exp − S 1 |ġ|2 ds/2t t∞/2
I. Frenkel 84: pt (g) numerator character loop group:
Kirillov-Lefschetz principle.
R
Atiyah 85: pt (g) = Og exp (− (E + ω) /t).
By DH, BV, should localize on 1-parameter
semigroups =⇒ correct formula in terms coroot lattice(affine Weyl group).
Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

5 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Proof of DH, BV

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

6 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Proof of DH, BV
X compact Riemannian manifold, K Killing vector
field, XK = (K = 0).

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

6 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Proof of DH, BV
X compact Riemannian manifold, K Killing vector
field, XK = (K = 0).
dK = d + iK , d2K = LK .

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

6 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Proof of DH, BV
X compact Riemannian manifold, K Killing vector
field, XK = (K = 0).
dK = d + iK , d2K = LK .
Theorem (BV)
If µ form such that dK µ = 0, then

Jean-Michel Bismut

R
X

µ=

R

µ
.
XK eK (NX /X )K

Hypoelliptic Laplacian, index theory

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Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Proof of DH, BV
X compact Riemannian manifold, K Killing vector
field, XK = (K = 0).
dK = d + iK , d2K = LK .
Theorem (BV)
If µ form such that dK µ = 0, thenRX

αs = exp (−dK K 0 /2s), dK αs = 0,

Jean-Michel Bismut

µ=
R

R

µ
.
XK eK (NX /X )
K

µ=
X

R
X

αs µ.

Hypoelliptic Laplacian, index theory

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Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Proof of DH, BV
X compact Riemannian manifold, K Killing vector
field, XK = (K = 0).
dK = d + iK , d2K = LK .
Theorem (BV)
If µ form such that dK µ = 0, thenRX

αs = exp (−dK K 0 /2s), dK αs = 0,
B86 As s → 0, αs →

µ=
R

R

µ
.
XK eK (NX /X )
K

µ=
X

R
X

αs µ.

δ XK


 as a current.
N
eK NXK /X ,∇ XK /X

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

6 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Proof of DH, BV
X compact Riemannian manifold, K Killing vector
field, XK = (K = 0).
dK = d + iK , d2K = LK .
Theorem (BV)
If µ form such that dK µ = 0, thenRX

αs = exp (−dK K 0 /2s), dK αs = 0,
B86 As s → 0, αs →

µ=
R

R

µ
.
XK eK (NX /X )
K

µ=
X

R
X

αs µ.

δ XK


 as a current.
N
eK NXK /X ,∇ XK /X‘Explains’ fantastic cancellations in local index theory.
Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

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Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References‘Evaluation’ for pt (g) in Lagrangian formalism

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

7 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References‘Evaluation’ for pt (g) in Lagrangian formalism
L2 metric on Og : |δAs |2 =Jean-Michel Bismut

R
S1|δA|2 ds.

Hypoelliptic Laplacian, index theory

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Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References‘Evaluation’ for pt (g) in Lagrangian formalism
R
L2 metric on Og : |δAs |2 = S 1 |δA|2 ds.
R
K = Ȧ, |K|2 = S 1 |Ȧs |2 ds.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

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Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References‘Evaluation’ for pt (g) in Lagrangian formalism
R
L2 metric on Og : |δAs |2 = S 1 |δA|2 ds.
R
K = Ȧ, |K|2 = S 1 |Ȧs |2 ds.
Replace
DH, BV integral for pt (g) byR
exp (− (E + ω) /t) exp (−b4 dK K 0 /2).
Og

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

7 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References‘Evaluation’ for pt (g) in Lagrangian formalism
R
L2 metric on Og : |δAs |2 = S 1 |δA|2 ds.
R
K = Ȧ, |K|2 = S 1 |Ȧs |2 ds.
Replace
DH, BV integral for pt (g) byR
exp (− (E + ω) /t) exp (−b4 dK K 0 /2).
Og
R
R
E = 12 S 1 |ġ|2 ds, |K|2 = S 1 |g̈|2 ds.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

7 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References‘Evaluation’ for pt (g) in Lagrangian formalism
R
L2 metric on Og : |δAs |2 = S 1 |δA|2 ds.
R
K = Ȧ, |K|2 = S 1 |Ȧs |2 ds.
Replace
DH, BV integral for pt (g) byR
exp (− (E + ω) /t) exp (−b4 dK K 0 /2).
Og
R
R
E = 12 S 1 |ġ|2 ds, |K|2 = S 1 |g̈|2 ds.
R
4 R
E + b4 |K|2 /2 = 12 S 1 |ġ|2 ds + b2 S 1 |g̈|2 ds. . .

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

7 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References‘Evaluation’ for pt (g) in Lagrangian formalism
R
L2 metric on Og : |δAs |2 = S 1 |δA|2 ds.
R
K = Ȧ, |K|2 = S 1 |Ȧs |2 ds.
Replace
DH, BV integral for pt (g) byR
exp (− (E + ω) /t) exp (−b4 dK K 0 /2).
Og
R
R
E = 12 S 1 |ġ|2 ds, |K|2 = S 1 |g̈|2 ds.
R
4 R
E + b4 |K|2 /2 = 12 S 1 |ġ|2 ds + b2 S 1 |g̈|2 ds. . .
R
2. . . = 12 S 1 |b2 g̈ + ġ| ds.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

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Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References‘Evaluation’ for pt (g) in Lagrangian formalism
R
L2 metric on Og : |δAs |2 = S 1 |δA|2 ds.
R
K = Ȧ, |K|2 = S 1 |Ȧs |2 ds.
Replace
DH, BV integral for pt (g) byR
exp (− (E + ω) /t) exp (−b4 dK K 0 /2).
Og
R
R
E = 12 S 1 |ġ|2 ds, |K|2 = S 1 |g̈|2 ds.
R
4 R
E + b4 |K|2 /2 = 12 S 1 |ġ|2 ds + b2 S 1 |g̈|2 ds. . .
R
2. . . = 12 S 1 |b2 g̈ + ġ| ds.
. . . b → +∞ localizes integral on g̈ = 0.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

7 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References‘Evaluation’ for pt (g) in Lagrangian formalism
R
L2 metric on Og : |δAs |2 = S 1 |δA|2 ds.
R
K = Ȧ, |K|2 = S 1 |Ȧs |2 ds.
Replace
DH, BV integral for pt (g) byR
exp (− (E + ω) /t) exp (−b4 dK K 0 /2).
Og
R
R
E = 12 S 1 |ġ|2 ds, |K|2 = S 1 |g̈|2 ds.
R
4 R
E + b4 |K|2 /2 = 12 S 1 |ġ|2 ds + b2 S 1 |g̈|2 ds. . .
R
2. . . = 12 S 1 |b2 g̈ + ġ| ds.
. . . b → +∞ localizes integral on g̈ = 0.
b2 g̈ + ġ = ẇ Gaussian suggests appearance of
hypoelliptic operator.
Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

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Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The Dirac operator of KostantJean-Michel Bismut

Hypoelliptic Laplacian, index theory

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Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The Dirac operator of Kostantb Ko ∈ bDc (g) ⊗ U (g).

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

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Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The Dirac operator of Kostantb Ko ∈ bDc (g) ⊗ U (g).
b Ko elliptic first order operator on G.
D

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

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Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The Dirac operator of Kostantb Ko ∈ bDc (g) ⊗ U (g).
b Ko elliptic first order operator on G.
Db Ko,2 = ∆G − 4π 2 |ρ|2 .
D

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

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Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The Dirac operator of Kostantb Ko ∈ bDc (g) ⊗ U (g).
b Ko elliptic first order operator on G.
Db Ko,2 = ∆G − 4π 2 |ρ|2 .
D
b Ko on C ∞ (G, R) ⊗ Λ· (g∗ ).
∆G acts on C ∞ (G, R), D

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

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Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The Dirac operator of Kostantb Ko ∈ bDc (g) ⊗ U (g).
b Ko elliptic first order operator on G.
Db Ko,2 = ∆G − 4π 2 |ρ|2 .
D
b Ko on C ∞ (G, R) ⊗ Λ· (g∗ ).
∆G acts on C ∞ (G, R), D
We will delete Λ· (g∗ ) by tensoring with S · (g∗ ), anduse Bargmann isomorphism.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

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Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
Referencesde Rham, Witten, and Bargmann

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

9 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
Referencesde Rham, Witten, and Bargmann
A· (g∗ ) = S · (g∗ ) ⊗ Λ· (g∗ ) polynomial forms on g.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

9 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
Referencesde Rham, Witten, and Bargmann
A· (g∗ ) = S · (g∗ ) ⊗ Λ· (g∗ ) polynomial forms on g.
(A· (g∗ ) , dg ) algebraic de Rham complex.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

9 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
Referencesde Rham, Witten, and Bargmann
A· (g∗ ) = S · (g∗ ) ⊗ Λ· (g∗ ) polynomial forms on g.
(A· (g∗ ) , dg ) algebraic de Rham complex.
∗· ∗dg = iY , [dg , iY ] = LY = N A (g ) .

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

9 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
Referencesde Rham, Witten, and Bargmann
A· (g∗ ) = S · (g∗ ) ⊗ Λ· (g∗ ) polynomial forms on g.
(A· (g∗ ) , dg ) algebraic de Rham complex.
∗· ∗dg = iY , [dg , iY ] = LY = N A (g ) .
Poincaré lemma and Hodge theory hold.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

9 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
Referencesde Rham, Witten, and Bargmann
A· (g∗ ) = S · (g∗ ) ⊗ Λ· (g∗ ) polynomial forms on g.
(A· (g∗ ) , dg ) algebraic de Rham complex.
∗· ∗dg = iY , [dg , iY ] = LY = N A (g ) .
Poincaré lemma and Hodge theory hold.
·∗
Via Bargmann, S (g∗ ) 'L2 (g), dg + dg maps to
∗√1
dg + Y ∧ +dg + iY . . .
2Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

9 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
Referencesde Rham, Witten, and Bargmann
A· (g∗ ) = S · (g∗ ) ⊗ Λ· (g∗ ) polynomial forms on g.
(A· (g∗ ) , dg ) algebraic de Rham complex.
∗· ∗dg = iY , [dg , iY ] = LY = N A (g ) .
Poincaré lemma and Hodge theory hold.
·∗
Via Bargmann, S (g∗ ) 'L2 (g), dg + dg maps to
∗√1
dg + Y ∧ +dg + iY . . .
2·∗. . . and LY to H + N Λ (g ) . . .

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

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Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
Referencesde Rham, Witten, and Bargmann
A· (g∗ ) = S · (g∗ ) ⊗ Λ· (g∗ ) polynomial forms on g.
(A· (g∗ ) , dg ) algebraic de Rham complex.
∗· ∗dg = iY , [dg , iY ] = LY = N A (g ) .
Poincaré lemma and Hodge theory hold.
·∗
Via Bargmann, S (g∗ ) 'L2 (g), dg + dg maps to
∗√1
dg + Y ∧ +dg + iY . . .
2·∗. . . and LY to H + N Λ (g ) . . .
with H = 12 −∆g + |Y |2 − n harmonic oscillator.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

9 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
Referencesde Rham, Witten, and Bargmann
A· (g∗ ) = S · (g∗ ) ⊗ Λ· (g∗ ) polynomial forms on g.
(A· (g∗ ) , dg ) algebraic de Rham complex.
∗· ∗dg = iY , [dg , iY ] = LY = N A (g ) .
Poincaré lemma and Hodge theory hold.
·∗
Via Bargmann, S (g∗ ) 'L2 (g), dg + dg maps to
∗√1
dg + Y ∧ +dg + iY . . .
2·∗. . . and LY to H + N Λ (g ) . . .
with H = 12 −∆g + |Y |2 − n harmonic oscillator.
These operators act on C ∞ (g) ⊗ Λ· (g∗ ).
Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

9 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

A deformed Dirac operator

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

10 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

A deformed Dirac operator
Operators will act on
C ∞ (G) ⊗ A (g∗ ) ' C ∞ (G × g) ⊗ Λ· (g∗ ).

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

10 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

A deformed Dirac operator
Operators will act on
C ∞ (G) ⊗ A (g∗ ) ' C ∞ (G × g) ⊗ Λ· (g∗ ).
b Ko + 1 (dg + Y ∧ +dg∗ + iY ).
Db = Db

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

10 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

A deformed Dirac operator
Operators will act on
C ∞ (G) ⊗ A (g∗ ) ' C ∞ (G × g) ⊗ Λ· (g∗ ).
b Ko + 1 (dg + Y ∧ +dg∗ + iY ).
Db = Db

As b → 0, Db deforms 0.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

10 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

A deformed Dirac operator
Operators will act on
C ∞ (G) ⊗ A (g∗ ) ' C ∞ (G × g) ⊗ Λ· (g∗ ).
b Ko + 1 (dg + Y ∧ +dg∗ + iY ).
Db = Db

As b → 0, Db deforms 0.


1
Ko,2
2
b
Lb =
−D
+D .
2

b

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

10 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

A deformed Dirac operator
Operators will act on
C ∞ (G) ⊗ A (g∗ ) ' C ∞ (G × g) ⊗ Λ· (g∗ ).
b Ko + 1 (dg + Y ∧ +dg∗ + iY ).
Db = Db

As b → 0, Db deforms 0.


1
Ko,2
2
b
Lb =
−D
+D .
2

b


Lb = 2b12 −∆g + |Y |2 − n + 1b (∇Y + bc (ad (Y ))).

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

10 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

A deformed Dirac operator
Operators will act on
C ∞ (G) ⊗ A (g∗ ) ' C ∞ (G × g) ⊗ Λ· (g∗ ).
b Ko + 1 (dg + Y ∧ +dg∗ + iY ).
Db = Db

As b → 0, Db deforms 0.


1
Ko,2
2
b
Lb =
−D
+D .
2

b


Lb = 2b12 −∆g + |Y |2 − n + 1b (∇Y + bc (ad (Y ))).
Lb hypoelliptic by Hörmander.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

10 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

A deformed Dirac operator
Operators will act on
C ∞ (G) ⊗ A (g∗ ) ' C ∞ (G × g) ⊗ Λ· (g∗ ).
b Ko + 1 (dg + Y ∧ +dg∗ + iY ).
Db = Db

As b → 0, Db deforms 0.


1
Ko,2
2
b
Lb =
−D
+D .
2

b


Lb = 2b12 −∆g + |Y |2 − n + 1b (∇Y + bc (ad (Y ))).
Lb hypoelliptic by Hörmander.
As b → 0, by collapsing, Lb deforms 21 −∆G + c .
Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

10 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The invariance of the trace under deformation

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

11 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The invariance of the trace under deformation
Theorem (B08)
For t > 0, g1 , g2 ∈ G,

 


G
−1
.
exp
t
∆
−
c
/2
=
Tr
L
R
exp
(−tL
)
Tr Lg1 Rg−1
s
g
b
1
g
2
2

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

11 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The invariance of the trace under deformation
Theorem (B08)
For t > 0, g1 , g2 ∈ G,

 


G
−1
.
exp
t
∆
−
c
/2
=
Tr
L
R
exp
(−tL
)
Tr Lg1 Rg−1
s
g
b
1
g
2
2
Independence of b like in index theorem, and convergenceas b → 0.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

11 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The invariance of the trace under deformation
Theorem (B08)
For t > 0, g1 , g2 ∈ G,

 


G
−1
.
exp
t
∆
−
c
/2
=
Tr
L
R
exp
(−tL
)
Tr Lg1 Rg−1
s
g
b
1
g
2
2
Independence of b like in index theorem, and convergenceas b → 0.
RemarkHamiltonian counterpart to Lagrangian deformation for
DH, BV formulas.
Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

11 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The limit as b → +∞Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

12 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The limit as b → +∞gc (ad (Y )).
Lb ' 12 − ∆b4 + |Y |2 − bn2 + ∇Y + b

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

12 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The limit as b → +∞gc (ad (Y )).
Lb ' 12 − ∆b4 + |Y |2 − bn2 + ∇Y + b
As b → +∞, geodesic flow dominates.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

12 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The limit as b → +∞gc (ad (Y )).
Lb ' 12 − ∆b4 + |Y |2 − bn2 + ∇Y + b
As b → +∞, geodesic flow dominates.
Forces localization of trace on closed geodesics.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

12 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The limit as b → +∞gc (ad (Y )).
Lb ' 12 − ∆b4 + |Y |2 − bn2 + ∇Y + b
As b → +∞, geodesic flow dominates.
Forces localization of trace on closed geodesics.
We get the required formulas for the above trace.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

12 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Reductive groups and symmetric spaces

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

13 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Reductive groups and symmetric spaces

G reductive group, θ Cartan involution, B invariant
form on g.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

13 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Reductive groups and symmetric spaces

G reductive group, θ Cartan involution, B invariant
form on g.
K maximal compact subgroup, g = p ⊕ k Cartan
splitting.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

13 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Reductive groups and symmetric spaces

G reductive group, θ Cartan involution, B invariant
form on g.
K maximal compact subgroup, g = p ⊕ k Cartan
splitting.
C g = −e∗i ei Casimir operator on G, analogue of −∆G
for compact groups.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

13 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Reductive groups and symmetric spaces

G reductive group, θ Cartan involution, B invariant
form on g.
K maximal compact subgroup, g = p ⊕ k Cartan
splitting.
C g = −e∗i ei Casimir operator on G, analogue of −∆G
for compact groups.
Symbol of C g = B (ξ, ξ) positive on p, negative on k.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

13 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The symmetric space X = G/K

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

14 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The symmetric space X = G/K

X = G/K symmetric space.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

14 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The symmetric space X = G/K

X = G/K symmetric space.
ρ : K → AutE descends to vector bundle F on X.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

14 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The symmetric space X = G/K

X = G/K symmetric space.
ρ : K → AutE descends to vector bundle F on X.
C g acts on C ∞ (X, F ).

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

14 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The Kostant operator on G

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

15 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The Kostant operator on Gb Ko still acts on C ∞ (G) ⊗ Λ· (g∗ ).
D

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

15 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The Kostant operator on Gb Ko still acts on C ∞ (G) ⊗ Λ· (g∗ ).
D
b Ko,2 = −C g + c.
D

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

15 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The Kostant operator on Gb Ko still acts on C ∞ (G) ⊗ Λ· (g∗ ).
D
b Ko,2 = −C g + c.
D
g = p ⊕ k splits as direct sums of Euclidean vector
spaces.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

15 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The Kostant operator on Gb Ko still acts on C ∞ (G) ⊗ Λ· (g∗ ).
D
b Ko,2 = −C g + c.
D
g = p ⊕ k splits as direct sums of Euclidean vector
spaces.
Introduce algebraic de Rham on g, and use Bargmannon p, k separately.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

15 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The operator Db

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

16 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The operator Dbb Ko +
Db = D√ 1−1c Y p , Y k + (dp + Y p + dp∗ + iY p )
√ b

−1 k∗
+
d + iY k − dk − Y k .
bJean-Michel Bismut

Hypoelliptic Laplacian, index theory

16 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The operator Dbb Ko +
Db = D√ 1−1c Y p , Y k + (dp + Y p + dp∗ + iY p )
√ b

−1 k∗
+
d + iY k − dk − Y k .
bb Ko,2 + D2 .
Lb = 12 −DbJean-Michel Bismut

Hypoelliptic Laplacian, index theory

16 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The operator Dbb Ko +
Db = D√ 1−1c Y p , Y k + (dp + Y p + dp∗ + iY p )
√ b

−1 k∗
+
d + iY k − dk − Y k .
bb Ko,2 + D2 .
Lb = 12 −Db
Xb → X total space of T X ⊕ N = G ×K (p ⊕ k).

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

16 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The operator Dbb Ko +
Db = D√ 1−1c Y p , Y k + (dp + Y p + dp∗ + iY p )
√ b

−1 k∗
+
d + iY k − dk − Y k .
bb Ko,2 + D2 .
Lb = 12 −Db
Xb → X total space of T X ⊕ N = G ×K (p ⊕ k).
DX , LX act on C ∞ Xb, Λ· (T ∗ X ⊕ N ∗ ) ⊗ F .
bb

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

16 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The hypoelliptic Laplacian as a deformation

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

17 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The hypoelliptic Laplacian as a deformation
 N Λ· (T ∗ X⊕N ∗ ) N TX 2 1
2
T X⊕N+ |Y | − n +Y ,Y+ 2 −∆
2
b2
|2b
{z
}

1
LX
b =

Harmonic oscillator of T X⊕N

1
+
b

!
∇Y T X +bc ad Y| {z }TX−c ad YTX+ iθad YNgeodesic flow

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

17 / 32.

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The hypoelliptic Laplacian as a deformation
 N Λ· (T ∗ X⊕N ∗ ) N TX 2 1
2
T X⊕N+ |Y | − n +Y ,Y+ 2 −∆
2
b2
|2b
{z
}

1
LX
b =

Harmonic oscillator of T X⊕N

1
+
b

!
∇Y T X +bc ad Y| {z }TX−c ad YTX+ iθad YNgeodesic flow1Xb
• b → 0, LX
b → 2 C − c : X collapses to X.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

17 / 32.

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The hypoelliptic Laplacian as a deformation
 N Λ· (T ∗ X⊕N ∗ ) N TX 2 1
2
T X⊕N+ |Y | − n +Y ,Y+ 2 −∆
2
b2
|2b
{z
}

1
LX
b =

Harmonic oscillator of T X⊕N

1
+
b

!
∇Y T X +bc ad Y| {z }TX−c ad YTX+ iθad YNgeodesic flow1Xb
• b → 0, LX
b → 2 C − c : X collapses to X.
• b → +∞, geodesic flow ∇Y T X dominant ⇒ closed
geodesics.
Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

17 / 32.

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The case of locally symmetric spaces

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

18 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The case of locally symmetric spacesΓ ⊂ G cocompact torsion free.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

18 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The case of locally symmetric spacesΓ ⊂ G cocompact torsion free.
Z = Γ \ X compact locally symmetric.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

18 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

A fundamental identity

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

19 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

A fundamental identity
Theorem B11
For t > 0, b > 0,

 


∞
TrC (Z,F ) exp −t C Z − c /2 = Trs exp −tLZb .

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

19 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

A fundamental identity
Theorem B11
For t > 0, b > 0,

 


∞
TrC (Z,F ) exp −t C Z − c /2 = Trs exp −tLZb .
ProofLimit as b → 0,

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

19 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

A fundamental identity
Theorem B11
For t > 0, b > 0,

 


∞
TrC (Z,F ) exp −t C Z − c /2 = Trs exp −tLZb .
ProofLimit as b → 0, Bianchi identity
 i Z Z  h Z  Z,2
Db , Lb = Db , Db + C Z /2 = 0.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

19 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Splitting the identity

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

20 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Splitting the identity

1

This is exactly what we wanted!

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

20 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Splitting the identity12This is exactly what we wanted!
The identity splits as identity of orbital integrals.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

20 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Semisimple orbital integrals

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

21 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Semisimple orbital integralsγ ∈ G semisimple, [γ] conjugacy class.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

21 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Semisimple orbital integralsγ ∈ G semisimple, [γ] conjugacy class.
 
For t > 0, Tr[γ] exp −t C X − c /2 orbital integralof heat kernel on orbit of γ:ZI ([γ]) =TrE pXg −1 γg dg.
tZ(γ)\G

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

21 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The minimizing set

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

22 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The minimizing set•X (γ) ⊂ X minimizing set for the convex displacement
function d (x, γx).

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

22 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The minimizing set•X (γ) ⊂ X minimizing set for the convex displacement
function d (x, γx).
•X (γ) ⊂ X totally geodesic symmetric space for thecentralizer Z (γ).

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

22 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Geometric description of the orbital integral

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

23 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Geometric description of the orbital integralZ
I (γ) =

Tr γpX(Y,
γY)
r (Y ) dY.
t| {z }
NX(γ)/X
Jacobian

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

23 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Geometric description of the orbital integralZ
I (γ) =

Tr γpX(Y,
γY)
r (Y ) dY.
t| {z }
NX(γ)/X
Jacobian

x0

Y

X(γ)

γx0

γY
d(Y, γY ) ≥ C|Y | − C 0

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

23 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Geometric description of the orbital integralZ
I (γ) =

Tr γpX(Y,
γY)
r (Y ) dY.
t| {z }
NX(γ)/X
Jacobian

x0

Y

X(γ)

γx0

γY
d(Y, γY ) ≥ C|Y | − C 0

0
0 2
0
pX
t (x, x ) ≤ C exp(−C d (x, x )).
Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

23 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The heat kernel for LXb

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

24 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The heat kernel for LXb
Theorem (B11)• For b ∈]0, M ], t > 0 fixed,
Xqb,t((x, Y ) , (x0 , Y 0 ))




2
≤ C exp −C 0 d2 (x, x0 ) + |Y |2 + |Y 0 |.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

24 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The heat kernel for LXb
Theorem (B11)• For b ∈]0, M ], t > 0 fixed,
Xqb,t((x, Y ) , (x0 , Y 0 ))




2
≤ C exp −C 0 d2 (x, x0 ) + |Y |2 + |Y 0 |.
• As b → 0,X0− dim g/2qb,t((x, Y ) , (x0 , Y 0 )) → PpX
t (x, x ) π
 

1
2
0 2
exp
|Y | + |Y |P.
2Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

24 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

A second fundamental identity

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

25 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

A second fundamental identity
Theorem
For b > 0, t > 0,

 


Tr[γ] exp −t C X − c /2 = Trs [γ] exp −tLX.
b

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

25 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

A second fundamental identity
Theorem
For b > 0, t > 0,

 


Tr[γ] exp −t C X − c /2 = Trs [γ] exp −tLX.
bRemarkThe proof uses the fact that Tr[γ] is a trace on the algebraof G-invariants smooth kernels on X with Gaussian decay.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

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Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The limit as b → +∞Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

26 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The limit as b → +∞After rescaling
Y T X , Y N , as b → +∞, N of2
b4
Lb ' 2 Y , Y T X + 21 |Y |2 − ∇Y T X .
| {z }
geodesic flow

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

26 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The limit as b → +∞After rescaling
Y T X , Y N , as b → +∞, N of2
b4
Lb ' 2 Y , Y T X + 21 |Y |2 − ∇Y T X .
| {z }
geodesic flow

As b → +∞, the orbital integral localizes near X (γ)
exactly like in Lefschetz formulas.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

26 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The limit as b → +∞After rescaling
Y T X , Y N , as b → +∞, N of2
b4
Lb ' 2 Y , Y T X + 21 |Y |2 − ∇Y T X .
| {z }
geodesic flow

As b → +∞, the orbital integral localizes near X (γ)
exactly like in Lefschetz formulas.
γ = ea k −1 , a ∈ p, k ∈ K, Ad (k) a = a.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

26 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The limit as b → +∞After rescaling
Y T X , Y N , as b → +∞, N of2
b4
Lb ' 2 Y , Y T X + 21 |Y |2 − ∇Y T X .
| {z }
geodesic flow

As b → +∞, the orbital integral localizes near X (γ)
exactly like in Lefschetz formulas.
γ = ea k −1 , a ∈ p, k ∈ K, Ad (k) a = a.
Z (γ) centralizer of γ, z (γ) = p (γ) ⊕ k (γ) Lie algebraof Z (γ).

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

26 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Semisimple orbital integrals

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

27 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Semisimple orbital integrals
Theorem (B. 2011)
There is an explicit function Jγ Y0k , Y0k ∈ k (γ), such that

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

27 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Semisimple orbital integrals
Theorem (B. 2011)
There is an explicit function Jγ Y0k , Y0k ∈ k (γ), such that
Tr

[γ]



  exp − |a|2 /2t
X
exp −t C − c /2 =(2πt)p/2
Z




Jγ Y0k TrE ρE k −1 exp −iρE Y0k
k(γ)



2
exp − Y0k /2t

Jean-Michel Bismut

dY0k(2πt)q/2

Hypoelliptic Laplacian, index theory.

27 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Semisimple orbital integrals
Theorem (B. 2011)
There is an explicit function Jγ Y0k , Y0k ∈ k (γ), such that
Tr

[γ]



  exp − |a|2 /2t
X
exp −t C − c /2 =(2πt)p/2
Z




Jγ Y0k TrE ρE k −1 exp −iρE Y0k
k(γ)



2
exp − Y0k /2t

dY0k(2πt)q/2.

The formula extends to any kernel.
Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

27 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The function Jγ (Y0) , Y0k ∈ k (γ)

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

28 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

The function Jγ (Y0) , Y0k ∈ k (γ)
Definition

Jγ

Y0k





b iad Y0k |p(γ)
A

=
 
1/2
k
b
A iad Y0 k(γ)
det (1 − Ad (γ)) |z⊥01"

1
det (1 − Ad (k −1 )) |z⊥0 (γ)#1/2Ad (k −1 ) |k⊥0 (γ)
.
det 1 − exp −iad Y0k Ad (k −1 ) |p⊥0 (γ)
det 1 − exp −iad Y0k

Jean-Michel BismutHypoelliptic Laplacian, index theory

28 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Analogy with Atiyah-Bott

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

29 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Analogy with Atiyah-Bott• Compare with fixed point formulas by Atiyah-BottZ
bg (T X) chg (E) .
L (g) =A
Xg

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

29 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Analogy with Atiyah-Bott• Compare with fixed point formulas by Atiyah-BottZ
bg (T X) chg (E) .
L (g) =A
Xg

• Here T X replaced by T X

Jean-Michel Bismut

N.

Hypoelliptic Laplacian, index theory

29 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Eta invariants and the trace formula

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

30 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Eta invariants and the trace formula
If DX classical Dirac operator on X, DX,2 = C g + c.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

30 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Eta invariants and the trace formula
If DX classical Dirac operator on X, DX,2 = C g + c.

Methods do not apply to Tr[γ] DX exp −tDX,2 .

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

30 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Eta invariants and the trace formula
If DX classical Dirac operator on X, DX,2 = C g + c.

Methods do not apply to Tr[γ] DX exp −tDX,2 .
b Ko , DX on G, X closely related.
Dirac operators D

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

30 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Eta invariants and the trace formula
If DX classical Dirac operator on X, DX,2 = C g + c.

Methods do not apply to Tr[γ] DX exp −tDX,2 .
b Ko , DX on G, X closely related.
Dirac operators D
Clifford algebras bc (g) for DKo , c (p) for DX .

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

30 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Eta invariants and the trace formula
If DX classical Dirac operator on X, DX,2 = C g + c.

Methods do not apply to Tr[γ] DX exp −tDX,2 .
b Ko , DX on G, X closely related.
Dirac operators D
Clifford algebras bc (g) for DKo , c (p) for DX .
g ⊕ p = p ⊕ p ⊕ k.

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

30 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Eta invariants and the trace formula
If DX classical Dirac operator on X, DX,2 = C g + c.

Methods do not apply to Tr[γ] DX exp −tDX,2 .
b Ko , DX on G, X closely related.
Dirac operators D
Clifford algebras bc (g) for DKo , c (p) for DX .
g ⊕ p = p ⊕ p ⊕ k.
Use action of SO (2m) on p ⊕ p to extract DX fromb Ko .
D

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

30 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

Eta invariants and the trace formula
If DX classical Dirac operator on X, DX,2 = C g + c.

Methods do not apply to Tr[γ] DX exp −tDX,2 .
b Ko , DX on G, X closely related.
Dirac operators D
Clifford algebras bc (g) for DKo , c (p) for DX .
g ⊕ p = p ⊕ p ⊕ k.
Use action of SO (2m) on p ⊕ p to extract DX fromb Ko .
D
Recover results by Moscovici-Stanton on eta invariantsof locally symmetric spaces.
Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

30 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

I. B. Frenkel, Orbital theory for affine Lie algebras,
Invent. Math. 77 (1984), no. 2, 301–352. MR
MR752823 (86d:17014)
M. F. Atiyah, Circular symmetry and stationary-phaseapproximation, Astérisque (1985), no. 131, 43–59,
Colloquium in honor of Laurent Schwartz, Vol. 1(Palaiseau, 1983). MR 87h:58206
H. Moscovici and R. J. Stanton, Eta invariants of Dirac
operators on locally symmetric manifolds, Invent. Math.
95 (1989), no. 3, 629–666. MR 90b:58252

Jean-Michel Bismut

Hypoelliptic Laplacian, index theory

31 / 32

Loop groups and coadjoint orbits
Heat kernels and equivariant localization
Rigorous proof in the Hamiltonian formalism
Selberg’s trace formula
References

J.-M. Bismut, The hypoelliptic Laplacian on a compact
Lie group, J. Funct. Anal. 255 (2008), no. 9,2190–2232. MR MR2473254, Hypoelliptic Laplacian and orbital integrals,
Annals of Mathematics Studies, vol. 177, Princeton
University Press, Princeton, NJ, 2011. MR 2828080, Eta invariants and the hypoelliptic Laplacian,J. Eur. Math. Society (to appear) (2018).

Jean-Michel Bismut

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