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Introduction Groupoids PDO Index Theory Bonus

Index theory through Lie groupoids

Joint works with J.-M. Lescure and G. Skandalis.

Inspired by ideas of A. Connes.

Claire Debord

Universite Paris-Diderot Paris 7Institut de Mathematiques de Jussieu - Paris Rive Gauche

Potsdam, March 2019

Introduction : Why put groupoids into the picture ?

ý They are already there ....We have all encountered several convolution formulas.

• On a group : f ∗ g(x) =

f(y) g(y−1x) dy.

• Kernels : f ∗ g(x, y) =

f(x, z) g(z, y) dz.

These are particular cases of convolution on Lie groupoids.

ý They are already there ....

We have all encountered several convolution formulas.

f(y) g(y−1x) dy.

f(x, z) g(z, y) dz.

f(y) g(y−1x) dy.

f(x, z) g(z, y) dz.

f(y) g(y−1x) dy.

f(x, z) g(z, y) dz.

f(y) g(y−1x) dy.

f(x, z) g(z, y) dz.

ý Convolution

ý Following Gelfand’s theorem, Noncommutative geometry proposeto replace the study of a singular space by the study of a convenientC∗-algebra.

ý Convolution

ý Gelfand’s theorem

X ←→ C0(X)Locally compact space Commutative C∗-algebra

ý Convolution

ý Gelfand’s theorem

X ←→ C0(X)Locally compact space Commutative C∗-algebra

Noncommutative geometry propose to replace the study of a singularspace by the study of a convenient C∗-algebra :

Z� Singular � space

C∗(Z) = C∗(G)Noncommutative C∗-algebra

G⇒ G(0), G(0)/G ' ZGroupoid

ý Convolution

Singular geometrical spaces : space of leaves of a foliation, manifoldwith corners, stratified pseudo-manifold...

ý Convolution

Singular geometrical spaces : space of leaves of a foliation, manifoldwith corners, stratified pseudo-manifold...

Ingredients : algebra of � continuous functions � on the space ofparameter, pseudodifferential calculus, � tangent space �...

ý Convolution

Method approach - initiated by A. Connes in ’79 : get to geometrythanks to (Lie) groupoids. The C∗-algebra of a groupoid is from J.Renault ’80.

Today in this talk...

1. Groupoids and a few words on the C*-algebra of a groupoid

2. Pseudodifferential operators and analytic index

3. Constructions of Lie groupoids in connexion with index theory

1. Groupoidsand a few words on the C*-algebra of a groupoid

Groupoids

Definition

A groupoid is a small category such that every arrow is invertible.

• Set of objects G(0), set of arrows G(1) = G

• Range and source maps r, s : G→ G(0).

• x, y composable if s(x) = r(y), we obtain x · y (or xy) with sources(y) and range r(x).

• x, y, z with s(x) = r(y) and s(y) = r(z), then (x · y) · z = x · (y · z) ;

• u ∈ G(0), unit eu ∈ G with r(eu) = s(eu) = u ; er(x) · x = x and

x · es(x) = x for all x ∈ G ; identification G(0) ⊂ G ;

• Inverse : ∀x ∈ G, ∃x−1 ∈ G with r(x−1) = s(x), s(x−1) = r(x),x · x−1 = er(x) and x−1 · x = es(x).

Groupoids

Definition

Groupoids

Definition

• Range and source maps r, s : G→ G(0)

x##r(x)

Groupoids

Definition

• Range and source maps r, s : G→ G(0)

x##r(x)

• x, y composable if s(x) = r(y),we obtain x · y (or xy) withsource s(y) and range r(x). s(y)

x · y

##r(y)=s(x)

x %%r(x)

• Range and source maps r, s : G→ G(0).• x, y composable if s(x) = r(y), we obtain x · y (or xy) with sources(y) and range r(x).

• x, y, z with s(x) = r(y) and s(y) = r(z), then (x · y) · z = x · (y · z) ;• u ∈ G(0), unit eu ∈ G with r(eu) = s(eu) = u ; er(x) · x = x and

x · es(x) = x for all x ∈ G ; identification G(0) ⊂ G ;• Inverse : ∀x ∈ G, ∃x−1 ∈ G with r(x−1) = s(x), s(x−1) = r(x),x · x−1 = er(x) and x−1 · x = es(x).

Groupoids

Definition

• Associativity : x, y, z with s(x) = r(y) and s(y) = r(z), then(x · y) · z = x · (y · z) ;

Groupoids

Definition

• Units : u ∈ G(0), unit eu ∈ G with r(eu) = s(eu) = u ; er(x) ·x = x

and x · es(x) = x for all x ∈ G ; identification G(0) ⊂ G ;

Groupoids

Definition

• Set of objects G(0), set of arrows G(1) = G• Range and source maps r, s : G→ G(0).• x, y composable if s(x) = r(y), we obtain x · y (or xy) with sources(y) and range r(x).

• Inverse : ∀x ∈ G, ∃x−1 ∈ G withr(x−1) = s(x), s(x−1) = r(x), x · x−1 = er(x)

and x−1 · x = es(x).

x##r(x)

We denote : G⇒ G(0)

Groupoids

Definition

! G acts on G(0) : the orbit of x ∈ G(0) is r(s−1(x)).

Groupoids

Definition

! G acts on G(0) : the orbit of x ∈ G(0) is r(s−1(x)).

Lie Groupoids

Definition

A Lie groupoid is a groupoid G⇒ G(0) such that

• G and G(0) are manifolds ;

• All maps smooth.

2. A Lie group is a Lie groupoid.

3. A smooth vector bundle is a Lie groupoid.

4. Pair groupoid M ×Mr,s

⇒M ; s(x, y) = y, r(x, y) = x,u(x) = (x, x), (x, y) · (y, z) = (x, z) et (x, y)−1 = (y, x).

Lie Groupoids

Definition

• inclusion u 7→ eu of G(0) to G, inverse are smooth

• s, r : G→ G(0) are smooth submersionsG(2) = {(x, y); s(x) = r(y)} is a submanifold of G×G ;

• composition G(2) → G is smooth.

Lie Groupoids

Definition

• s, r : G→ G(0) are smooth submersions

G(2) = {(x, y); s(x) = r(y)} is a submanifold of G×G ;

Lie Groupoids

Definition

Lie Groupoids

Definition

Lie Groupoids

Definition

Examples

Lie Groupoids

Definition

Examples

1. A manifold M is a Lie groupoid. All maps r, s, composition...idM .

Lie Groupoids

Definition

Examples

1. A manifold is a Lie groupoid M ⇒M .

2. A Lie group is a Lie groupoid with just one unit.

Lie Groupoids

Definition

Examples

Lie Groupoids

Definition

Examples

More examples

5. Group actionSmooth action of a Lie group H on a manifold M : H nM ⇒M .

s(h, x) = x, r(h, x) = h · x, u(x) = (e, x)(k, h · x) · (h, x) = (kh, x) et (h, x)−1 = (h−1, h · x)

6. Poincare Groupoidγ a path on M , [γ] homotopy class with fixed end points of γ,s[γ] = γ(0), r[γ] = γ(1), concatenation product...

Π(M) = {[γ] | γ path on M}⇒M

For x ∈M , π1(M,x) = s−1(x)∩ r−1(x) is the fondamental group,it acts (on the right) on the universal cover s−1(x).

More examples

7. Graph of an equivalence relationThe pair groupoid M ×M ⇒M ; (x, y) · (y, z) = (x, z).

I Sub-groupoids (with M as units space) of the pair groupoidover M are exactly graphs of equivalence relations.

I Let R be an equivalence relation on M . Its graphGR = {(x, y) ∈M ×M | xRy}⇒M is a Lie groupoid when R isthe relation � being on the same leaf of a regular foliation withno holonomy. �

8. Holonomy groupoid of a regular foliation on M

Winkelnkemper - Pradines ’80

Construction of the holonomy groupoid : the � smallest � Liegroupoid with units M and the leaves of F as orbits.

A few words about Lie theory for groupoids

The Lie algebroid of the Lie groupoid G⇒M is : (AG, ], [·, ·])

• AG = KerTs|G(0) → G(0) - smooth vector bundle.

• ] = Tr : AG→ TG(0) - smooth bundle map.

• [·, ·] Lie bracket on smooth sections of AG - constructed from leftinvariant vector fields.

It satisfies for any X, Y ∈ Γ(AG), f ∈ C∞(G(0)) :

][X,Y ] = [](X), ](Y )] and [X, fY ] = f [X,Y ] + ](X)(f).Y

Examples

1. Lie algebroid of a Lie group : Lie algebra.

2. Lie algebroid of the pair groupoid M ×M ⇒M : (TM, Id, [·, ·]).3. Lie algebroid of the holonomy groupoid of a regular foliation F :

(TF , Id, [·, ·]).

! Lie third theorem fails : Lie algebroid may not be the Lie algebroidof a Lie groupoid.

The Lie algebroid of the Lie groupoid G⇒M is : (AG, ], [·, ·])• AG = KerTs|G(0) → G(0) - smooth vector bundle.

Examples

(TF , Id, [·, ·]).

Examples

(TF , Id, [·, ·]).

Examples

(TF , Id, [·, ·]).

Examples

(TF , Id, [·, ·]).

Examples

(TF , Id, [·, ·]).

Examples

2. Lie algebroid of the pair groupoid M ×M ⇒M : (TM, Id, [·, ·]).

3. Lie algebroid of the holonomy groupoid of a regular foliation F :(TF , Id, [·, ·]).

Examples

(TF , Id, [·, ·]).

Examples

(TF , Id, [·, ·]).

Convolution on a Lie groupoid G

Algebra C∞c (G) : f1 ∗ f2(x) =

∫(x1,x2)∈G; x1x2=x

f1(x1)f2(x2) dν

• The set {(x1, x2) ∈ G×G; x1x2 = x} : smooth manifoldx1 ∈ Gr(x) = {y ∈ G; r(y) = r(x)} and x2 = x−1

• dν is a smooth “Haar system”• i.e. smooth choice of a Lebesgue measure νu on every Gu.• Left invariance : for every x ∈ G, the measure νs(x) ↔ νr(x)

through diffeomorphism y 7→ x · y from Gs(x) with Gr(x).

Convolution formula f1 ∗ f2(x) =

∫Gr(x)

f1(y)f2(y−1x) dνr(x)(y).

Convolution associative by invariance of the Haar system (andFubini).

Adjoint of f ∈ C∞c (G) : function f∗ : x 7→ f(x−1).

We choose an operator nom : C∗(G) = completion of C∞c (G).

In the � good cases � : C0(G(0)/G)! C∗(G).

∫(x1,x2)∈G; x1x2=x

f1(x1)f2(x2) dν

∫Gr(x)

∫(x1,x2)∈G; x1x2=x

f1(x1)f2(x2) dν

• dν is a smooth “Haar system”• i.e. smooth choice of a Lebesgue measure νu on every Gu.

• Left invariance : for every x ∈ G, the measure νs(x) ↔ νr(x)

∫Gr(x)

∫(x1,x2)∈G; x1x2=x

f1(x1)f2(x2) dν

∫Gr(x)

∫(x1,x2)∈G; x1x2=x

f1(x1)f2(x2) dν

∫Gr(x)

∫(x1,x2)∈G; x1x2=x

f1(x1)f2(x2) dν

∫Gr(x)

∫(x1,x2)∈G; x1x2=x

f1(x1)f2(x2) dν

∫Gr(x)

∫(x1,x2)∈G; x1x2=x

f1(x1)f2(x2) dν

∫Gr(x)

∫(x1,x2)∈G; x1x2=x

f1(x1)f2(x2) dν

∫Gr(x)

Examples of groupoids and their C∗-algebras

1. G = M ×M .

Then C∗(G) = K : algebra of compact operators.Equivalent to C.

2. G = M ×B M where M → B is a submersion.C∗(G) = C(B)⊗K. Equivalent to C(B).

3. If G is a vector bundle E →M , then C∗(G) = C0(E∗) : whereE∗ is the dual bundle and C0(E∗) : functions that vanish atinfinity on (the total space of) E∗ (using Fourier transform).

4. If U ⊂ G(0) is open and saturated (i.e. s(x) ∈ U ⇔ r(x) ∈ U) andF = G(0) \ U , exact sequence :

0→ C∗(GU )→ C∗(G)→ C∗(GF )→ 0.

1. G = M ×M . Then C∗(G) = K : algebra of compact operators.Equivalent to C.

0→ C∗(GU )→ C∗(G)→ C∗(GF )→ 0.

2. Pseudodifferential operators andanalytic index

Pseudodifferential operators on a Lie groupoid G

Differential operators : (enveloping) algebra generated by sections ofthe algebroid AG = ∪x∈G(0)TGx ' the normal bundle of the inclusion

G(0) ⊂ G.

Pm(G) - pseudodifferential operators of order m ∈ Z are distributionswith singular support G(0) ⊂ G, conormal, i.e. of the form P + kwhere k ∈ C∞c (G) and

P (x) = (2π)−d∫

(A∗G)s(x)

ei〈θ(x)|ξ〉χ(x)a(s(x), ξ) dξ

• d is the dimension of the algebroid AG ;• θ : U → AG is a diffeomorphism, inverse of an exp map, from a

tubular neighbourhood of G(0) in G to the normal bundle AG.• χ is a smooth bump function (1 on G(0), and 0 outside U) ;

• a is a classical polyhomogeneous symbol a(u, ξ) ∼∑j

am−j(u, ξ)

where a` is homogeneous of order `.

• oscilatory integral

∫(A∗G)s(x)

= limR→∞

∫‖ξ‖≤R

(as a distribution).

G(0) ⊂ G.

P (x) = (2π)−d∫

(A∗G)s(x)

am−j(u, ξ)

∫(A∗G)s(x)

= limR→∞

∫‖ξ‖≤R

G(0) ⊂ G.

P (x) = (2π)−d∫

(A∗G)s(x)

tubular neighbourhood of G(0) in G to the normal bundle AG.

• χ is a smooth bump function (1 on G(0), and 0 outside U) ;

am−j(u, ξ)

∫(A∗G)s(x)

= limR→∞

∫‖ξ‖≤R

G(0) ⊂ G.

P (x) = (2π)−d∫

(A∗G)s(x)

am−j(u, ξ)

∫(A∗G)s(x)

= limR→∞

∫‖ξ‖≤R

G(0) ⊂ G.

P (x) = (2π)−d∫

(A∗G)s(x)

am−j(u, ξ)

∫(A∗G)s(x)

= limR→∞

∫‖ξ‖≤R

G(0) ⊂ G.

P (x) = (2π)−d∫

(A∗G)s(x)

am−j(u, ξ)

∫(A∗G)s(x)

= limR→∞

∫‖ξ‖≤R

G(0) ⊂ G.

P (x) = (2π)−d∫

(A∗G)s(x)

am−j(u, ξ)

∫(A∗G)s(x)

= limR→∞

∫‖ξ‖≤R

Pseudodifferential calculus

These pseudodifferential operators form a convolution ∗-algebra :

• P of < 0 order - with compact support - is in C∗(G).

• P of order ≤ 0 - with compact support - bounded.

Ψ∗(G) = completion of P0(G).

We obtain an exact sequence of C∗ algebras.

0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0 (PDO)

• SA∗G is the sphere bundle of the dual bundle to the algebroidAG.

• σ : principal symbol map (a ∼∑

a−j 7→ a0).

0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0 (PDO)

a−j 7→ a0).

0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0 (PDO)

a−j 7→ a0).

0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0 (PDO)

a−j 7→ a0).

0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0 (PDO)

a−j 7→ a0).

0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0 (PDO)

a−j 7→ a0).

0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0 (PDO)

a−j 7→ a0).

0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0 (PDO)

a−j 7→ a0).

Analytic index

0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0

Connecting map : ∂G : K∗+1(C(SA∗G))→ K∗(C∗(G)).

Consider the inclusion i : SA∗G× R∗+ ' A∗G \G(0) → A∗G.

∂G = ∂G ◦ [i]

Example

G = M ×M pair groupoid ; AG = TM ; C∗(G) = K(L2(M)).

∂G : K0(C0(T ∗M))→ K0(K) = Z is the Atiyah-Singer analytic index.

Thus ∂G : K∗(C0(A∗G))→ K∗(C∗(G)) is a generalised analytic index.

Analytic index

0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0

∂G = ∂G ◦ [i]

Example

Analytic index

0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0

∂G = ∂G ◦ [i]

Example

Analytic index

0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0

∂G = ∂G ◦ [i]

Example

G = M ×M pair groupoid ; AG = TM ;

C∗(G) = K(L2(M)).

Analytic index

0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0

∂G = ∂G ◦ [i]

Example

Analytic index

0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0

∂G = ∂G ◦ [i]

Example

Analytic index

0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0

∂G = ∂G ◦ [i]

Example

In the examples...

1. G = M ×M .

Atiyah-Singer index - with values in Z.

2. G = M ×B M . Atiyah-Singer index for families - with values inK∗(C0(B)).

3. G holonomy groupoid of a foliation (M,F). Connes’ index withvalues in K∗(C

∗(M,F)).

4. Manifolds with boundary, with corners... Corresponding indexproblems.

In the examples...

1. G = M ×M . Atiyah-Singer index - with values in Z.

∗(M,F)).

In the examples...

2. G = M ×B M .

Atiyah-Singer index for families - with values inK∗(C0(B)).

∗(M,F)).

In the examples...

∗(M,F)).

In the examples...

3. G holonomy groupoid of a foliation (M,F).

Connes’ index withvalues in K∗(C

∗(M,F)).

In the examples...

∗(M,F)).

In the examples...

∗(M,F)).

3. Constructions of Lie groupoidsin connection with index theory

or PDOs through geometry

3. Constructions of Lie groupoidsin connection with index theory

or PDOs through geometry

Connes’ tangent groupoid

Deformation to the normal cone : V ⊂M a submanifold, NMV the

normal bundle.

DNC(M,V ) = (M × R∗) t (NMV × {0}).

Natural smooth structure. Generated by :

• the natural map ϕ : DNC(M,V )→M × R is smooth(given by (x, t) 7→ (x, t) and (x, ξ, 0) 7→ (x, 0)).

• if f : M → R smooth and vanishes on V , the function

fdnc : DNC(M,V )→ R given by (x, t) 7→ f(x)

(x, ξ, 0) 7→ df(ξ) is smooth.

Remarks (D.-Skandalis)

1. This construction is functorial

2. If G is a Lie groupoid and H is a subgroupoid,DNC(G,H)⇒ DNC(G(0), H(0)) is a Lie groupoid.

normal bundle. DNC(M,V ) = (M × R∗) t (NMV × {0}).

Connes’ tangent groupoid : Analytic index without PDO’s

Definition (Connes’ tangent groupoid)

DNC(M ×M,M) (M ⊂M ×M diagonally)

= (M ×M × R∗) t (TM × {0})⇒M × R.

GT = DNC[0,1](M ×M,M) = (M ×M × (0, 1]) t (TM × {0}).

Exact sequence 0→ K⊗ C0((0, 1]) −→ C∗(GT )ev0−→ C∗(TM)→ 0.

K ⊗ C0((0, 1]) is contractible : ev0 isomorphism in K-theory.

0 // K ⊗ C0((0, 1]) // C∗(GT )ev0 //

��

C∗(TM) = C0(T ∗M) //

indavv

KTheorem (Connes) : The Atiyah-Singer analytic index is equal to∂M×M = [ev1] ◦ [ev0]−1 : K0(C0(T ∗M))→ K0(K) = Z

DNC(M ×M,M) (M ⊂M ×M diagonally)= (M ×M × R∗) t (TM × {0})⇒M × R.

GT = DNC[0,1](M ×M,M) = (M ×M × (0, 1]) t (TM × {0}).

0 // K ⊗ C0((0, 1]) // C∗(GT )ev0 //

��

C∗(TM) = C0(T ∗M) //

indavv

DNC(M ×M,M) = (M ×M × R∗) t (TM × {0})⇒M × R.

More preciselyGT = DNC[0,1](M ×M,M) = (M ×M × (0, 1]) t (TM × {0}).

0 // K ⊗ C0((0, 1]) // C∗(GT )ev0 //

��

C∗(TM) = C0(T ∗M) //

indavv

DNC(M ×M,M) = (M ×M × R∗) t (TM × {0})⇒M × R.

GT = DNC[0,1](M ×M,M) = (M ×M × (0, 1]) t (TM × {0}).

0 // K ⊗ C0((0, 1]) // C∗(GT )ev0 //

��

C∗(TM) = C0(T ∗M) //

indavv

DNC(M ×M,M) = (M ×M × R∗) t (TM × {0})⇒M × R.

GT = DNC[0,1](M ×M,M) = (M ×M × (0, 1]) t (TM × {0}).

0 // K ⊗ C0((0, 1]) // C∗(GT )ev0 //

��

C∗(TM) = C0(T ∗M) //

indavv

DNC(M ×M,M) = (M ×M × R∗) t (TM × {0})⇒M × R.

GT = DNC[0,1](M ×M,M) = (M ×M × (0, 1]) t (TM × {0}).

0 // K ⊗ C0((0, 1]) // C∗(GT )ev0 //

��

C∗(TM) = C0(T ∗M) // 0

0 // K ⊗ C0((0, 1]) // C∗(GT )ev0 //

��

C∗(TM) = C0(T ∗M) //

indavv

DNC(M ×M,M) = (M ×M × R∗) t (TM × {0})⇒M × R.

GT = DNC[0,1](M ×M,M) = (M ×M × (0, 1]) t (TM × {0}).

0 // K ⊗ C0((0, 1]) // C∗(GT )ev0 //

��

C∗(TM) = C0(T ∗M) //

indavv

Gauge adiabatic groupoid and PDOs (D.-Skandalis)

Let G⇒M be a Lie groupoid. Start with the adiabatic groupoid

Gad = DNCR+(G,M) = (G× R∗+) t (AG× {0})⇒M × R+

ä Natural � zooming action � of R∗+ :

λ(x, t) = (x, λt) for t 6= 0 and λ(u, ξ, 0) = (u, λ−1ξ, 0) elsewhere.

Gauge adiabatic groupoid :Gag = Gad oR∗+ = (G× R∗+ × R∗+) t (AG oR∗+)⇒M × R

ä Look at the evaluation map : ev0 : C∗(Gad)→ C∗(AG) ' C0(A∗G)and consider the ideal J(G) = ev−1

0 (C0(A∗G \M))

0→ C∗(G)⊗ C0(R∗+) −→ J(G)ev0−→ C0(A∗G \G(0))→ 0

Which is equivariant under the action of R∗+ and leads to

0→(C∗(G)⊗ C0(R∗+)

)oR∗+

' C∗(G)⊗K→ J(G) oR∗+⊂ C∗(Gga)

→ C0(A∗G \G(0)) oR∗+' C(S∗AG)⊗K

0 (C0(A∗G \M))

0→ C∗(G)⊗ C0(R∗+) −→ J(G)ev0−→ C0(A∗G \G(0))→ 0

0→(C∗(G)⊗ C0(R∗+)

)oR∗+

' C∗(G)⊗K→ J(G) oR∗+⊂ C∗(Gga)

→ C0(A∗G \G(0)) oR∗+' C(S∗AG)⊗K

0 (C0(A∗G \M))

0→ C∗(G)⊗ C0(R∗+) −→ J(G)ev0−→ C0(A∗G \G(0))→ 0

0→(C∗(G)⊗ C0(R∗+)

)oR∗+

' C∗(G)⊗K→ J(G) oR∗+⊂ C∗(Gga)

→ C0(A∗G \G(0)) oR∗+' C(S∗AG)⊗K

0 (C0(A∗G \M))

0→ C∗(G)⊗ C0(R∗+) −→ J(G)ev0−→ C0(A∗G \G(0))→ 0

0→(C∗(G)⊗ C0(R∗+)

)oR∗+

' C∗(G)⊗K→ J(G) oR∗+⊂ C∗(Gga)

→ C0(A∗G \G(0)) oR∗+' C(S∗AG)⊗K

0 (C0(A∗G \M))

0→ C∗(G)⊗ C0(R∗+) −→ J(G)ev0−→ C0(A∗G \G(0))→ 0

0→(C∗(G)⊗ C0(R∗+)

)oR∗+

' C∗(G)⊗K→ J(G) oR∗+⊂ C∗(Gga)

→ C0(A∗G \G(0)) oR∗+' C(S∗AG)⊗K

0 (C0(A∗G \M))

0→ C∗(G)⊗ C0(R∗+) −→ J(G)ev0−→ C0(A∗G \G(0))→ 0

0→(C∗(G)⊗ C0(R∗+)

)oR∗+

' C∗(G)⊗K→ J(G) oR∗+⊂ C∗(Gga)

→ C0(A∗G \G(0)) oR∗+' C(S∗AG)⊗K

We have an exact sequence

0→ C∗(G)⊗K −→ J(G) oR∗+ −→ C(SA∗G)⊗K → 0 (GAG)

Compare with...

0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0 (PDO)

Theorem (D.-Skandalis)

There is a natural isomorphism J(G) oR∗+ ' Ψ∗(G)⊗K.

In other words, pseudodifferential operators can be expressed asconvolution kernels on a groupoid.

Compare with...

0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0 (PDO)

Compare with...

0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0 (PDO)

Compare with...

0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0 (PDO)

Theorem (D. & Skandalis)

There is an ideal J (G) ⊂ C∞c (Gad) such that forf = (ft)t∈R+

∈ J (G) and m ∈ N let

∫ +∞

tmftdt

tand σ : (x, ξ) ∈ A∗G 7→

∫ +∞

tmf(x, tξ, 0)dt

Then P belongs to P−m(G) and its principal symbol is σ.

What does it mean : There exists a pseudodifferential operatorP ∈ P−m(G) with principal symbol σ such that if g ∈ C∞c (G) :

P ∗ g =

∫ +∞

tmft ∗ gdt

tand g ∗ P =

∫ +∞

tmg ∗ ftdt

Remark : Moreover any P ∈ P−m(G) is a Pf =

∫ +∞

tmftdt

tfor some

f = (ft)t∈R+ ∈ J (G).

∫ +∞

tmftdt

tand σ : (x, ξ) ∈ A∗G 7→

∫ +∞

tmf(x, tξ, 0)dt

What does it mean :

There exists a pseudodifferential operatorP ∈ P−m(G) with principal symbol σ such that if g ∈ C∞c (G) :

P ∗ g =

∫ +∞

tmft ∗ gdt

tand g ∗ P =

∫ +∞

tmg ∗ ftdt

∫ +∞

tmftdt

tfor some

f = (ft)t∈R+ ∈ J (G).

∫ +∞

tmftdt

tand σ : (x, ξ) ∈ A∗G 7→

∫ +∞

tmf(x, tξ, 0)dt

P ∗ g =

∫ +∞

tmft ∗ gdt

tand g ∗ P =

∫ +∞

tmg ∗ ftdt

∫ +∞

tmftdt

tfor some

f = (ft)t∈R+ ∈ J (G).

∫ +∞

tmftdt

tand σ : (x, ξ) ∈ A∗G 7→

∫ +∞

tmf(x, tξ, 0)dt

P ∗ g =

∫ +∞

tmft ∗ gdt

tand g ∗ P =

∫ +∞

tmg ∗ ftdt

∫ +∞

tmftdt

tfor some

f = (ft)t∈R+ ∈ J (G).

In conclusion...

Lie groupoids allow us

1. to generalise the analytic index ;

2. to express the analytic index in a (pseudo)differential operatorfree way ;

3. to give proofs of index theorems ;

4. to express the order 0 pseudodifferential operators in a(pseudo)differential operator free way.

In conclusion...

4. to express the order 0 pseudodifferential operators

in a(pseudo)differential operator free way.

In conclusion...

Thank you for your attention !

Analytic index for groupoids

Let G⇒M be a Lie groupoid.

Define the adiabatic groupoid Gad = DNC[0,1](G,M).Diagram

0 // C∗(G)⊗ C0((0, 1]) // C∗(Gad)ev0 //

��

C0(A∗G) //

indayy

C∗(G)

Theorem (Monthubert-Pierrot and Nistor-Weinstein-Xu)

Analytic index = [ev1] ◦ [ev0]−1.

Let G⇒M be a Lie groupoid.Define the adiabatic groupoid Gad = DNC[0,1](G,M).

Diagram

0 // C∗(G)⊗ C0((0, 1]) // C∗(Gad)ev0 //

��

C0(A∗G) //

indayy

C∗(G)

Let G⇒M be a Lie groupoid.Define the adiabatic groupoid Gad = DNC[0,1](G,M).Diagram

0 // C∗(G)⊗ C0((0, 1]) // C∗(Gad)ev0 //

��

C0(A∗G) //

indayy

C∗(G)

Let G⇒M be a Lie groupoid.Define the adiabatic groupoid Gad = DNC[0,1](G,M).Diagram

0 // C∗(G)⊗ C0((0, 1]) // C∗(Gad)ev0 //

��

C0(A∗G) //

indayy

C∗(G)

Digression : Pullback groupoid and Morita equivalence

Suppose G⇒M is a smooth groupoid and let f : N →M be asurjective submersion. The pullback groupoid of G by f is the smoothgroupoid

Gff := {(x, γ, y) ∈ N ×G×N | r(γ) = f(x), s(γ) = f(y)}⇒ N

Two smooth groupoids G⇒M and H ⇒ N are Morita equivalent ifone can find a manifold Z with two surjective submersions p : Z →Mand q : Z → N such that the pullbacks Gpp ⇒ Z and Hq

q ⇒ Z areisomorphic.

Theorem (Muhly, Renault, Williams)

The C∗-algebras of two Morita equivalent groupoids are Moritaequivalent.

Atiyah-Singer index theorem

π : E →M a vector bundle ; consider ∆E ⊂ E ×ME ⊂ E × E :

T = DNC(DNC(E × E,E ×

ME),∆E × {0}

)⇒ E × R× R

Let T � = T |E×[0,1]×[0,1] and T hom = T |E×{0}×[0,1].

Proposition (D.-Lescure-Nistor)

For j : M ↪→ Rn and E = NRnM the normal of the inclusion, τE is the

inverse of the Thom isomorphism.

Particular case The normal bundle of · ↪→ Rn is just Rn → · andleads to the Bott periodicity

τRn : K0(C∗(TRn))→ Z.

Conclusion Indt = τRn ◦ [j] ◦ τ−1E is entirely described with

(deformation) groupoids.

ME),∆E × {0}

)⇒ E × R× R

The Thom groupoid T hom = TE × {0} t TMππ×]0, 1]⇒ E × [0, 1]

and the Morita equivalence TMππ ∼ TM provides :

τE : K∗(C∗(TE)) = K∗(C0(T ∗E))→ K∗(C

∗(TM)) = K∗(C0(T ∗M))

ME),∆E × {0}

)⇒ E × R× R

The Thom groupoid T hom = TE × {0} t TMππ×]0, 1]⇒ E × [0, 1]

and the Morita equivalence TMππ ∼ TM provides :

τE : K∗(C∗(TE)) = K∗(C0(T ∗E))→ K∗(C

∗(TM)) = K∗(C0(T ∗M))

ME),∆E × {0}

)⇒ E × R× R

ME),∆E × {0}

)⇒ E × R× R

E = NRnM , T � = DNC

(DNC(E × E,E ×

ME),∆E × {0}

)|E×[0,1]×[0,1]

TM ×M

,,DNC(E × E,E ×

'(GT (M))ππ

)|E×[0,1] E × E

T hom T � GT (E)

Thom−1

TE × [0, 1] TE

Gives IndMa = IndMt [D.-Lescure-Nistor].

Can be extended to M with isolated conical singularities... and to Ma general pseudomanifold.

(DNC(E × E,E ×

ME),∆E × {0}

)|E×[0,1]×[0,1]

TM ×M

,,DNC(E × E,E ×

'(GT (M))ππ

)|E×[0,1] E × E

T hom T � GT (E)

Thom−1

TE × [0, 1] TE

(DNC(E × E,E ×

ME),∆E × {0}

)|E×[0,1]×[0,1]

TM ×M

,,DNC(E × E,E ×

'(GT (M))ππ

)|E×[0,1] E × E

T hom T � GT (E)

Thom−1

TE × [0, 1] TE

Can be extended to M with isolated conical singularities...

and to Ma general pseudomanifold.

(DNC(E × E,E ×

ME),∆E × {0}

)|E×[0,1]×[0,1]

TM ×M

,,DNC(E × E,E ×

'(GT (M))ππ

)|E×[0,1] E × E

T hom T � GT (E)

Thom−1

TE × [0, 1] TE

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