SL2(R) Dynamics and Harmonic Analysis

Livio Flaminio [email protected]

Institut de Mathématiques Université de Lille

Nantes, 2-7 juillet 2017 R { } 1 0 The largest normal subgroup of SL2( ) is its centre Z = I , I = [ 0 1 ]. Hence the quotient group PSL2(R) := SL2(R)/{I } has trivial centre.

Remarquable one-parameter subgroups { [ ] } { [ ] } { } cos θ − sin θ a 0 1 u K = , A = −1 : a > 0 , N = [ 0 1 ] , sin θ cos θ θ∈R 0 a u∈R { [ ] } { } ⊥ cosh θ sinh θ 1 0 A = sinh θ cosh θ , N = [ v 1 ] , θ∈R v∈R

Introduction to SL2(R) Topology via remarkable actions

The group SL2(R) and some subgroups

Deﬁnition

The group SL2(R) is the group of 2 × 2 real matrices of determinant 1: {[ ] } R a b ∈ R − SL2( ) = c d : a, b, c, d , ad bc = 1 .

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 2 33 Remarquable one-parameter subgroups { [ ] } { [ ] } { } cos θ − sin θ a 0 1 u K = , A = −1 : a > 0 , N = [ 0 1 ] , sin θ cos θ θ∈R 0 a u∈R { [ ] } { } ⊥ cosh θ sinh θ 1 0 A = sinh θ cosh θ , N = [ v 1 ] , θ∈R v∈R

Introduction to SL2(R) Topology via remarkable actions

The group SL2(R) and some subgroups

Deﬁnition

The group SL2(R) is the group of 2 × 2 real matrices of determinant 1: {[ ] } R a b ∈ R − SL2( ) = c d : a, b, c, d , ad bc = 1 .

R { } 1 0 The largest normal subgroup of SL2( ) is its centre Z = I , I = [ 0 1 ]. Hence the quotient group PSL2(R) := SL2(R)/{I } has trivial centre.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 2 33 Introduction to SL2(R) Topology via remarkable actions

The group SL2(R) and some subgroups

Deﬁnition

The group SL2(R) is the group of 2 × 2 real matrices of determinant 1: {[ ] } R a b ∈ R − SL2( ) = c d : a, b, c, d , ad bc = 1 .

R { } 1 0 The largest normal subgroup of SL2( ) is its centre Z = I , I = [ 0 1 ]. Hence the quotient group PSL2(R) := SL2(R)/{I } has trivial centre.

Remarquable one-parameter subgroups { [ ] } { [ ] } { } cos θ − sin θ a 0 1 u K = , A = −1 : a > 0 , N = [ 0 1 ] , sin θ cos θ θ∈R 0 a u∈R { [ ] } { } ⊥ cosh θ sinh θ 1 0 A = sinh θ cosh θ , N = [ v 1 ] , θ∈R v∈R

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 2 33 • not compact (e.g. since N ≈ R is properly embedded); • connected;

• not simply connected: in fact π1(G) = Z; • a simple Lie group.

Properties

SL2(R) is

Next we introduce some fundamental actions of SL2(R) which will be further exploited later on.

Introduction to SL2(R) Topology via remarkable actions

Topology of G = SL2(R)

4 The group SL2(R) as a sub-manifold of R inherits the diﬀerentiable structure and topology from R4.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 3 33 • not compact (e.g. since N ≈ R is properly embedded); • connected;

• not simply connected: in fact π1(G) = Z; • a simple Lie group.

Next we introduce some fundamental actions of SL2(R) which will be further exploited later on.

Introduction to SL2(R) Topology via remarkable actions

Topology of G = SL2(R)

4 The group SL2(R) as a sub-manifold of R inherits the diﬀerentiable structure and topology from R4.

Properties

SL2(R) is

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 3 33 • connected;

• not simply connected: in fact π1(G) = Z; • a simple Lie group.

Next we introduce some fundamental actions of SL2(R) which will be further exploited later on.

Introduction to SL2(R) Topology via remarkable actions

Topology of G = SL2(R)

4 The group SL2(R) as a sub-manifold of R inherits the diﬀerentiable structure and topology from R4.

Properties

SL2(R) is • not compact (e.g. since N ≈ R is properly embedded);

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 3 33 • not simply connected: in fact π1(G) = Z; • a simple Lie group.

Next we introduce some fundamental actions of SL2(R) which will be further exploited later on.

Introduction to SL2(R) Topology via remarkable actions

Topology of G = SL2(R)

4 The group SL2(R) as a sub-manifold of R inherits the diﬀerentiable structure and topology from R4.

Properties

SL2(R) is • not compact (e.g. since N ≈ R is properly embedded); • connected;

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 3 33 • a simple Lie group.

Next we introduce some fundamental actions of SL2(R) which will be further exploited later on.

Introduction to SL2(R) Topology via remarkable actions

Topology of G = SL2(R)

4 The group SL2(R) as a sub-manifold of R inherits the diﬀerentiable structure and topology from R4.

Properties

SL2(R) is • not compact (e.g. since N ≈ R is properly embedded); • connected;

• not simply connected: in fact π1(G) = Z;

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 3 33 Next we introduce some fundamental actions of SL2(R) which will be further exploited later on.

Introduction to SL2(R) Topology via remarkable actions

Topology of G = SL2(R)

4 The group SL2(R) as a sub-manifold of R inherits the diﬀerentiable structure and topology from R4.

Properties

SL2(R) is • not compact (e.g. since N ≈ R is properly embedded); • connected;

• not simply connected: in fact π1(G) = Z; • a simple Lie group.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 3 33 Introduction to SL2(R) Topology via remarkable actions

Topology of G = SL2(R)

4 The group SL2(R) as a sub-manifold of R inherits the diﬀerentiable structure and topology from R4.

Properties

SL2(R) is • not compact (e.g. since N ≈ R is properly embedded); • connected;

• not simply connected: in fact π1(G) = Z; • a simple Lie group.

Next we introduce some fundamental actions of SL2(R) which will be further exploited later on.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 3 33 Consequence As R2 \{0} and N are connected, G is connected.

Remark (for the wine cellar) The linear action of G on R2 preserves the Lebesgue measure on R2 and induces a linear action on the full tensor algebra of R2.

The orbits are: {0} and R2 \{0}. The stabiliser of the point (1, 0) is the group N. Hence we have

2 R \{0} = G.(1, 0) ≈ G/ StabG [(1, 0)] = G/N.

Introduction to SL2(R) Topology via remarkable actions

Topology of G = SL2(R) Connectness

G acts linearly on R2: [ ] a b c d .(x, y) = (ax + by, cx + dy)

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 4 33 Consequence As R2 \{0} and N are connected, G is connected.

Remark (for the wine cellar) The linear action of G on R2 preserves the Lebesgue measure on R2 and induces a linear action on the full tensor algebra of R2.

Introduction to SL2(R) Topology via remarkable actions

Topology of G = SL2(R) Connectness

G acts linearly on R2: [ ] a b c d .(x, y) = (ax + by, cx + dy) The orbits are: {0} and R2 \{0}. The stabiliser of the point (1, 0) is the group N. Hence we have

2 R \{0} = G.(1, 0) ≈ G/ StabG [(1, 0)] = G/N.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 4 33 Remark (for the wine cellar) The linear action of G on R2 preserves the Lebesgue measure on R2 and induces a linear action on the full tensor algebra of R2.

Introduction to SL2(R) Topology via remarkable actions

Topology of G = SL2(R) Connectness

G acts linearly on R2: [ ] a b c d .(x, y) = (ax + by, cx + dy) The orbits are: {0} and R2 \{0}. The stabiliser of the point (1, 0) is the group N. Hence we have

2 R \{0} = G.(1, 0) ≈ G/ StabG [(1, 0)] = G/N.

Consequence As R2 \{0} and N are connected, G is connected.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 4 33 Introduction to SL2(R) Topology via remarkable actions

Topology of G = SL2(R) Connectness

G acts linearly on R2: [ ] a b c d .(x, y) = (ax + by, cx + dy) The orbits are: {0} and R2 \{0}. The stabiliser of the point (1, 0) is the group N. Hence we have

2 R \{0} = G.(1, 0) ≈ G/ StabG [(1, 0)] = G/N.

Consequence As R2 \{0} and N are connected, G is connected.

Remark (for the wine cellar) The linear action of G on R2 preserves the Lebesgue measure on R2 and induces a linear action on the full tensor algebra of R2.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 4 33 Consequence

Since StabG [−I ] = K and since the sheet S is contractible from the ﬁbration K → G → G/K ≈ S we get that K injects its π1 in G: π1(G) = π1(K) ≈ Z.

Consequence

Isomorphism PSL2(R) ≈ SO(1, 2)0.

The action preserves signature and determinant. In the coordinates [ ] Q ↔ R3 AB ↔ 2 , BC (A, B, C) the orbit G.(−I ) is the sheet of hyperboloid S = {AC − B2 = 1, A < 0}

([ ]) R3 − 2 AB The linear action on preserves the Lorentzian metric AC B = det BC .

Introduction to SL2(R) Topology via remarkable actions

Topology of G = SL2(R) Fundamental group

G (in fact PSL2(R)) acts on Q2, be the vector space of symmetric bilinear forms on R2: T (g, Q) 7→ gQg , (g ∈ G, Q ∈ Q2)

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 5 33 Consequence

Since StabG [−I ] = K and since the sheet S is contractible from the ﬁbration K → G → G/K ≈ S we get that K injects its π1 in G: π1(G) = π1(K) ≈ Z.

Consequence

Isomorphism PSL2(R) ≈ SO(1, 2)0.

In the coordinates [ ] Q ↔ R3 AB ↔ 2 , BC (A, B, C) the orbit G.(−I ) is the sheet of hyperboloid S = {AC − B2 = 1, A < 0}

([ ]) R3 − 2 AB The linear action on preserves the Lorentzian metric AC B = det BC .

Introduction to SL2(R) Topology via remarkable actions

Topology of G = SL2(R) Fundamental group

G (in fact PSL2(R)) acts on Q2, be the vector space of symmetric bilinear forms on R2: T (g, Q) 7→ gQg , (g ∈ G, Q ∈ Q2) The action preserves signature and determinant.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 5 33 Consequence

Since StabG [−I ] = K and since the sheet S is contractible from the ﬁbration K → G → G/K ≈ S we get that K injects its π1 in G: π1(G) = π1(K) ≈ Z.

Consequence

Isomorphism PSL2(R) ≈ SO(1, 2)0.

([ ]) R3 − 2 AB The linear action on preserves the Lorentzian metric AC B = det BC .

Introduction to SL2(R) Topology via remarkable actions

Topology of G = SL2(R) Fundamental group

G (in fact PSL2(R)) acts on Q2, be the vector space of symmetric bilinear forms on R2: T (g, Q) 7→ gQg , (g ∈ G, Q ∈ Q2) The action preserves signature and determinant. In the coordinates [ ] Q ↔ R3 AB ↔ 2 , BC (A, B, C) the orbit G.(−I ) is the sheet of hyperboloid S = {AC − B2 = 1, A < 0}

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 5 33 Consequence

Isomorphism PSL2(R) ≈ SO(1, 2)0.

([ ]) R3 − 2 AB The linear action on preserves the Lorentzian metric AC B = det BC .

Introduction to SL2(R) Topology via remarkable actions

Topology of G = SL2(R) Fundamental group

G (in fact PSL2(R)) acts on Q2, be the vector space of symmetric bilinear forms on R2: T (g, Q) 7→ gQg , (g ∈ G, Q ∈ Q2) The action preserves signature and determinant. In the coordinates [ ] Q ↔ R3 AB ↔ 2 , BC (A, B, C) the orbit G.(−I ) is the sheet of hyperboloid S = {AC − B2 = 1, A < 0} Consequence

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 5 33 Consequence

Isomorphism PSL2(R) ≈ SO(1, 2)0.

([ ]) R3 − 2 AB The linear action on preserves the Lorentzian metric AC B = det BC .

Introduction to SL2(R) Topology via remarkable actions

Topology of G = SL2(R) Fundamental group

G (in fact PSL2(R)) acts on Q2, be the vector space of symmetric bilinear forms on R2: T (g, Q) 7→ gQg , (g ∈ G, Q ∈ Q2) The action preserves signature and determinant. In the coordinates [ ] Q ↔ R3 AB ↔ 2 , BC (A, B, C) the orbit G.(−I ) is the sheet of hyperboloid S = {AC − B2 = 1, A < 0} Consequence

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 5 33 Consequence

Isomorphism PSL2(R) ≈ SO(1, 2)0.

Introduction to SL2(R) Topology via remarkable actions

Topology of G = SL2(R) Fundamental group

G (in fact PSL2(R)) acts on Q2, be the vector space of symmetric bilinear forms on R2: T (g, Q) 7→ gQg , (g ∈ G, Q ∈ Q2) The action preserves signature and determinant. In the coordinates [ ] Q ↔ R3 AB ↔ 2 , BC (A, B, C) the orbit G.(−I ) is the sheet of hyperboloid S = {AC − B2 = 1, A < 0} Consequence

Since StabG [−I ] = K and since the sheet S is contractible from the ﬁbration K → G → G/K ≈ S we get that K injects its π1 in G: π1(G) = π1(K) ≈ Z. ([ ]) R3 − 2 AB The linear action on preserves the Lorentzian metric AC B = det BC .

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 5 33 Introduction to SL2(R) Topology via remarkable actions

Topology of G = SL2(R) Fundamental group

Since StabG [−I ] = K and since the sheet S is contractible from the ﬁbration K → G → G/K ≈ S we get that K injects its π1 in G: π1(G) = π1(K) ≈ Z. ([ ]) R3 − 2 AB The linear action on preserves the Lorentzian metric AC B = det BC . Consequence

Isomorphism PSL2(R) ≈ SO(1, 2)0.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 5 33 Consequence

SL2(R) is a simple Lie group (no subspace of sl2(R) is stable for all the maps ad(X ): Y 7→ [X , Y ]).

Generators of the one-parameters subgroups K, A, A⊥, N and N¯ are: [ ] [ ] 0 −1 1 0 ⊥ 0 1 + 0 1 − 0 0 κ = 1 0 , h = 0 −1 , h = [ 1 0 ] , n = [ 0 0 ] , n = [ 1 0 ] . In the basis (h, n+, n−), the Lie algebra structure is completely determined by the commutation rules [h, n] = 2n, [n+, n−] = h

Introduction to SL2(R) Topology via remarkable actions

Topology of G = SL2(R) Simplicity

The Lie algebra of SL2(R) is the tangent space sl2(R) := TI G of G at the identity I . Thus {[ ] } R a b ∈ R sl2( ) = c −a : a, b, c .

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 6 33 Consequence

SL2(R) is a simple Lie group (no subspace of sl2(R) is stable for all the maps ad(X ): Y 7→ [X , Y ]).

In the basis (h, n+, n−), the Lie algebra structure is completely determined by the commutation rules [h, n] = 2n, [n+, n−] = h

Introduction to SL2(R) Topology via remarkable actions

Topology of G = SL2(R) Simplicity

The Lie algebra of SL2(R) is the tangent space sl2(R) := TI G of G at the identity I . Thus {[ ] } R a b ∈ R sl2( ) = c −a : a, b, c .

Generators of the one-parameters subgroups K, A, A⊥, N and N¯ are: [ ] [ ] 0 −1 1 0 ⊥ 0 1 + 0 1 − 0 0 κ = 1 0 , h = 0 −1 , h = [ 1 0 ] , n = [ 0 0 ] , n = [ 1 0 ] .

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 6 33 Consequence

SL2(R) is a simple Lie group (no subspace of sl2(R) is stable for all the maps ad(X ): Y 7→ [X , Y ]).

Introduction to SL2(R) Topology via remarkable actions

Topology of G = SL2(R) Simplicity

The Lie algebra of SL2(R) is the tangent space sl2(R) := TI G of G at the identity I . Thus {[ ] } R a b ∈ R sl2( ) = c −a : a, b, c .

Generators of the one-parameters subgroups K, A, A⊥, N and N¯ are: [ ] [ ] 0 −1 1 0 ⊥ 0 1 + 0 1 − 0 0 κ = 1 0 , h = 0 −1 , h = [ 1 0 ] , n = [ 0 0 ] , n = [ 1 0 ] . In the basis (h, n+, n−), the Lie algebra structure is completely determined by the commutation rules [h, n] = 2n, [n+, n−] = h

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 6 33 Introduction to SL2(R) Topology via remarkable actions

Topology of G = SL2(R) Simplicity

Generators of the one-parameters subgroups K, A, A⊥, N and N¯ are: [ ] [ ] 0 −1 1 0 ⊥ 0 1 + 0 1 − 0 0 κ = 1 0 , h = 0 −1 , h = [ 1 0 ] , n = [ 0 0 ] , n = [ 1 0 ] . In the basis (h, n+, n−), the Lie algebra structure is completely determined by the commutation rules [h, n] = 2n, [n+, n−] = h

Consequence

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 6 33 Theorem (Iwasawa decomposition for SL2(R))

Every element g ∈ SL2(R) can be written in a unique way as a product

g = kan

with k ∈ K, a ∈ A and n ∈ N. In particular SL2(R) = KAN.

Since K ∩ AN = I we obtain

2 From the linear action on R we also get a number of representations of SL2(R).

Introduction to SL2(R) More on these actions Linear action on R2 KAN decomposition

[ ] [ ] ̸ ∃ cos θ − sin θ ∈ ∃ r 0 ∈ For all (x, y) = (0, 0) ! kθ = sin θ cos θ K and ! ar = 0 r −1 A such that kθar .(1, 0) = (r cos θ, r sin θ) = (x, y).

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 7 33 2 From the linear action on R we also get a number of representations of SL2(R).

Introduction to SL2(R) More on these actions Linear action on R2 KAN decomposition

[ ] [ ] ̸ ∃ cos θ − sin θ ∈ ∃ r 0 ∈ For all (x, y) = (0, 0) ! kθ = sin θ cos θ K and ! ar = 0 r −1 A such that kθar .(1, 0) = (r cos θ, r sin θ) = (x, y). Since K ∩ AN = I we obtain

Theorem (Iwasawa decomposition for SL2(R))

Every element g ∈ SL2(R) can be written in a unique way as a product

g = kan

with k ∈ K, a ∈ A and n ∈ N. In particular SL2(R) = KAN.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 7 33 2 From the linear action on R we also get a number of representations of SL2(R).

Introduction to SL2(R) More on these actions Linear action on R2 KAN decomposition

[ ] [ ] ̸ ∃ cos θ − sin θ ∈ ∃ r 0 ∈ For all (x, y) = (0, 0) ! kθ = sin θ cos θ K and ! ar = 0 r −1 A such that kθar .(1, 0) = (r cos θ, r sin θ) = (x, y). Since K ∩ AN = I we obtain

Theorem (Iwasawa decomposition for SL2(R))

Every element g ∈ SL2(R) can be written in a unique way as a product

g = kan

with k ∈ K, a ∈ A and n ∈ N. In particular SL2(R) = KAN.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 7 33 Introduction to SL2(R) More on these actions Linear action on R2 KAN decomposition

[ ] [ ] ̸ ∃ cos θ − sin θ ∈ ∃ r 0 ∈ For all (x, y) = (0, 0) ! kθ = sin θ cos θ K and ! ar = 0 r −1 A such that kθar .(1, 0) = (r cos θ, r sin θ) = (x, y). Since K ∩ AN = I we obtain

Theorem (Iwasawa decomposition for SL2(R))

Every element g ∈ SL2(R) can be written in a unique way as a product

g = kan

with k ∈ K, a ∈ A and n ∈ N. In particular SL2(R) = KAN.

2 From the linear action on R we also get a number of representations of SL2(R).

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 7 33 For ﬁnite dimensional V , π is a smooth map. When V is a inﬁnite dimensional locally convex, complete, Hausdorﬀ TVS then we require that π : G → Ls(V )

with Ls(V ) the space of continuous operators on V with the strong topology. If V is a Hilbert, Banach or Frechet space, this is tantamount to the continuity of the map (g, v) ∈ G × V 7→ π(g)v ∈ V

Introduction to SL2(R) More on these actions Representations Interlude,1

Deﬁnition A representation of a Lie group G on C-vector space V is a continuous homomorphism π : G → Aut(V )

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 8 33 For ﬁnite dimensional V , π is a smooth map. When V is a inﬁnite dimensional locally convex, complete, Hausdorﬀ TVS then we require that π : G → Ls(V )

with Ls(V ) the space of continuous operators on V with the strong topology. If V is a Hilbert, Banach or Frechet space, this is tantamount to the continuity of the map (g, v) ∈ G × V 7→ π(g)v ∈ V

Introduction to SL2(R) More on these actions Representations Interlude,1

Deﬁnition A representation of a Lie group G on C-vector space V is a continuous homomorphism π : G → Aut(V )

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 8 33 When V is a inﬁnite dimensional locally convex, complete, Hausdorﬀ TVS then we require that π : G → Ls(V )

with Ls(V ) the space of continuous operators on V with the strong topology. If V is a Hilbert, Banach or Frechet space, this is tantamount to the continuity of the map (g, v) ∈ G × V 7→ π(g)v ∈ V

Introduction to SL2(R) More on these actions Representations Interlude,1

Deﬁnition A representation of a Lie group G on C-vector space V is a continuous homomorphism π : G → Aut(V )

For ﬁnite dimensional V , π is a smooth map.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 8 33 If V is a Hilbert, Banach or Frechet space, this is tantamount to the continuity of the map (g, v) ∈ G × V 7→ π(g)v ∈ V

Introduction to SL2(R) More on these actions Representations Interlude,1

For ﬁnite dimensional V , π is a smooth map. When V is a inﬁnite dimensional locally convex, complete, Hausdorﬀ TVS then we require that π : G → Ls(V )

with Ls(V ) the space of continuous operators on V with the strong topology.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 8 33 Introduction to SL2(R) More on these actions Representations Interlude,1

For ﬁnite dimensional V , π is a smooth map. When V is a inﬁnite dimensional locally convex, complete, Hausdorﬀ TVS then we require that π : G → Ls(V )

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 8 33 Deﬁnition A representation of a Lie group G on a TVS V is irreducible if no closed subspace W is G-invariant.

Unitary representations of SL2(R), or more generally of semi-simple (or reductive), groups split as a direct sum or direct integral of unitary irreducible representations. To some extent the goal of representation theory is the classiﬁcation of all irreducible (unitary) representations of a group G and the decomposition of (remarkable) representations (e.g. the regular representation of G on L2(G, dg)).

Introduction to SL2(R) More on these actions Representations Interlude,2

Remark The previous hypotheses on (π, V∫) are suﬃcient to allows to deﬁne

π(f ) := f (g)π(g) dg ∈ Ls(V ) G for any function f ∈ Cc (G). (Here dg is a (left)-Haar measure on G).

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 9 33 Deﬁnition A representation of a Lie group G on a TVS V is irreducible if no closed subspace W is G-invariant.

Unitary representations of SL2(R), or more generally of semi-simple (or reductive), groups split as a direct sum or direct integral of unitary irreducible representations. To some extent the goal of representation theory is the classiﬁcation of all irreducible (unitary) representations of a group G and the decomposition of (remarkable) representations (e.g. the regular representation of G on L2(G, dg)).

Introduction to SL2(R) More on these actions Representations Interlude,2

Remark The previous hypotheses on (π, V∫) are suﬃcient to allows to deﬁne

π(f ) := f (g)π(g) dg ∈ Ls(V ) G for any function f ∈ Cc (G). (Here dg is a (left)-Haar measure on G).

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 9 33 Unitary representations of SL2(R), or more generally of semi-simple (or reductive), groups split as a direct sum or direct integral of unitary irreducible representations. To some extent the goal of representation theory is the classiﬁcation of all irreducible (unitary) representations of a group G and the decomposition of (remarkable) representations (e.g. the regular representation of G on L2(G, dg)).

Introduction to SL2(R) More on these actions Representations Interlude,2

Remark The previous hypotheses on (π, V∫) are suﬃcient to allows to deﬁne

π(f ) := f (g)π(g) dg ∈ Ls(V ) G for any function f ∈ Cc (G). (Here dg is a (left)-Haar measure on G).

Deﬁnition A representation of a Lie group G on a TVS V is irreducible if no closed subspace W is G-invariant.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 9 33 Unitary representations of SL2(R), or more generally of semi-simple (or reductive), groups split as a direct sum or direct integral of unitary irreducible representations. To some extent the goal of representation theory is the classiﬁcation of all irreducible (unitary) representations of a group G and the decomposition of (remarkable) representations (e.g. the regular representation of G on L2(G, dg)).

Introduction to SL2(R) More on these actions Representations Interlude,2

Remark The previous hypotheses on (π, V∫) are suﬃcient to allows to deﬁne

Deﬁnition A representation of a Lie group G on a TVS V is irreducible if no closed subspace W is G-invariant.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 9 33 To some extent the goal of representation theory is the classiﬁcation of all irreducible (unitary) representations of a group G and the decomposition of (remarkable) representations (e.g. the regular representation of G on L2(G, dg)).

Introduction to SL2(R) More on these actions Representations Interlude,2

Remark The previous hypotheses on (π, V∫) are suﬃcient to allows to deﬁne

Deﬁnition A representation of a Lie group G on a TVS V is irreducible if no closed subspace W is G-invariant.

Unitary representations of SL2(R), or more generally of semi-simple (or reductive), groups split as a direct sum or direct integral of unitary irreducible representations.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 9 33 Introduction to SL2(R) More on these actions Representations Interlude,2

Remark The previous hypotheses on (π, V∫) are suﬃcient to allows to deﬁne

Unitary representations of SL2(R), or more generally of semi-simple (or reductive), groups split as a direct sum or direct integral of unitary irreducible representations. To some extent the goal of representation theory is the classiﬁcation of all irreducible (unitary) representations of a group G and the decomposition of (remarkable) representations (e.g. the regular representation of G on L2(G, dg)).

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 9 33 Example The irreducible representations of S 1 are (one-dimensional) characters 2πinx εn : x ∈ R/Z 7→ εn(x) := e ∈ AutC(C).

If (V , ⟨ | ⟩) is a Hilbert space and G is a compact Lie group then any representation can be made unitary by deﬁning a new product ∫ (v | w) = ⟨π(g)w | π(g)w⟩ dg G For a compact Lie group all irreducible representations are ﬁnite dimensional.

Introduction to SL2(R) More on these actions Representations Interlude,2

Deﬁnition If V is a Hilbert space a representation (π, V ) of G is unitary if π(g) is a unitary operator, for all g ∈ G.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 10 33 Example The irreducible representations of S 1 are (one-dimensional) characters 2πinx εn : x ∈ R/Z 7→ εn(x) := e ∈ AutC(C).

If (V , ⟨ | ⟩) is a Hilbert space and G is a compact Lie group then any representation can be made unitary by deﬁning a new product ∫ (v | w) = ⟨π(g)w | π(g)w⟩ dg G For a compact Lie group all irreducible representations are ﬁnite dimensional.

Introduction to SL2(R) More on these actions Representations Interlude,2

Deﬁnition If V is a Hilbert space a representation (π, V ) of G is unitary if π(g) is a unitary operator, for all g ∈ G.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 10 33 Example The irreducible representations of S 1 are (one-dimensional) characters 2πinx εn : x ∈ R/Z 7→ εn(x) := e ∈ AutC(C).

For a compact Lie group all irreducible representations are ﬁnite dimensional.

Introduction to SL2(R) More on these actions Representations Interlude,2

Deﬁnition If V is a Hilbert space a representation (π, V ) of G is unitary if π(g) is a unitary operator, for all g ∈ G.

If (V , ⟨ | ⟩) is a Hilbert space and G is a compact Lie group then any representation can be made unitary by deﬁning a new product ∫ (v | w) = ⟨π(g)w | π(g)w⟩ dg G

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 10 33 Introduction to SL2(R) More on these actions Representations Interlude,2

If (V , ⟨ | ⟩) is a Hilbert space and G is a compact Lie group then any representation can be made unitary by deﬁning a new product ∫ (v | w) = ⟨π(g)w | π(g)w⟩ dg G For a compact Lie group all irreducible representations are ﬁnite dimensional.

Example The irreducible representations of S 1 are (one-dimensional) characters 2πinx εn : x ∈ R/Z 7→ εn(x) := e ∈ AutC(C).

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 10 33 [ ] −1 a b ∈ (g.P)(x, y) = P(ax + by, cx + dy) if g = c d , P Pd

Hence SL2(R) has a ﬁnite dimensional representation in any dimension d.

As −I acts as the identity on Pd only if d is even we conclude that PSL(2, R) has ﬁnite dimensional representations only in odd dimensions. None of these representations are unitarisable as U(d) is compact.

Introduction to SL2(R) More on these actions Linear action on R2 Finite dimensional representations

2 The linear action of SL2(R) on R induces a representation on the space Pd of homogeneous polynomials P(x, y) of degree d on R2:

d d−1 d−1 d Pd = spanC{x , x y,..., xy , y }, dim Pd = d + 1

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 11 33 Hence SL2(R) has a ﬁnite dimensional representation in any dimension d.

As −I acts as the identity on Pd only if d is even we conclude that PSL(2, R) has ﬁnite dimensional representations only in odd dimensions. None of these representations are unitarisable as U(d) is compact.

Introduction to SL2(R) More on these actions Linear action on R2 Finite dimensional representations

2 The linear action of SL2(R) on R induces a representation on the space Pd of homogeneous polynomials P(x, y) of degree d on R2:

d d−1 d−1 d Pd = spanC{x , x y,..., xy , y }, dim Pd = d + 1

[ ] −1 a b ∈ (g.P)(x, y) = P(ax + by, cx + dy) if g = c d , P Pd

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 11 33 As −I acts as the identity on Pd only if d is even we conclude that PSL(2, R) has ﬁnite dimensional representations only in odd dimensions. None of these representations are unitarisable as U(d) is compact.

Introduction to SL2(R) More on these actions Linear action on R2 Finite dimensional representations

2 The linear action of SL2(R) on R induces a representation on the space Pd of homogeneous polynomials P(x, y) of degree d on R2:

d d−1 d−1 d Pd = spanC{x , x y,..., xy , y }, dim Pd = d + 1

[ ] −1 a b ∈ (g.P)(x, y) = P(ax + by, cx + dy) if g = c d , P Pd

Hence SL2(R) has a ﬁnite dimensional representation in any dimension d.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 11 33 None of these representations are unitarisable as U(d) is compact.

Introduction to SL2(R) More on these actions Linear action on R2 Finite dimensional representations

d d−1 d−1 d Pd = spanC{x , x y,..., xy , y }, dim Pd = d + 1

[ ] −1 a b ∈ (g.P)(x, y) = P(ax + by, cx + dy) if g = c d , P Pd

Hence SL2(R) has a ﬁnite dimensional representation in any dimension d.

As −I acts as the identity on Pd only if d is even we conclude that PSL(2, R) has ﬁnite dimensional representations only in odd dimensions.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 11 33 Introduction to SL2(R) More on these actions Linear action on R2 Finite dimensional representations

d d−1 d−1 d Pd = spanC{x , x y,..., xy , y }, dim Pd = d + 1

[ ] −1 a b ∈ (g.P)(x, y) = P(ax + by, cx + dy) if g = c d , P Pd

Hence SL2(R) has a ﬁnite dimensional representation in any dimension d.

As −I acts as the identity on Pd only if d is even we conclude that PSL(2, R) has ﬁnite dimensional representations only in odd dimensions. None of these representations are unitarisable as U(d) is compact.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 11 33 The representation (ρ, H) is not irreducible: (1) notice H = H+ ⊕ H− (where H even/odd functions) (2) ρ commutes with the homotheties t t t 2 (Ut f )(x, y) := (exp(itA)f )(x, y) := e f (e x, e y), f ∈ H , t ∈ R, (x, y) ∈ R . It follows that ρ commutes with the spectral projectors of ( ) ( ) 1 ∂ √ A = + 1 , r = x 2 + y 2 , i ∂r In conclusion ρ leaves invariant the generalized eigenspaces of A { } H,iν ∈ 2 2 | −1+iν − − ∈ R = f Lloc(R ) f (rx, ry) = r f (x, y), f ( x, y) = f (x, y) (ν ).

Introduction to SL2(R) More on these actions Linear action on R2 Unitary

The map (ρ(g).f )(x, y) = f (g −1(x, y)), f ∈ H = L2(R2, dxdy)

is a unitary representation of G = SL2(R).

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 12 33 (2) ρ commutes with the homotheties t t t 2 (Ut f )(x, y) := (exp(itA)f )(x, y) := e f (e x, e y), f ∈ H , t ∈ R, (x, y) ∈ R . It follows that ρ commutes with the spectral projectors of ( ) ( ) 1 ∂ √ A = + 1 , r = x 2 + y 2 , i ∂r In conclusion ρ leaves invariant the generalized eigenspaces of A { } H,iν ∈ 2 2 | −1+iν − − ∈ R = f Lloc(R ) f (rx, ry) = r f (x, y), f ( x, y) = f (x, y) (ν ).

Introduction to SL2(R) More on these actions Linear action on R2 Unitary

The map (ρ(g).f )(x, y) = f (g −1(x, y)), f ∈ H = L2(R2, dxdy)

is a unitary representation of G = SL2(R). The representation (ρ, H) is not irreducible: (1) notice H = H+ ⊕ H− (where H even/odd functions)

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 12 33 Introduction to SL2(R) More on these actions Linear action on R2 Unitary

The map (ρ(g).f )(x, y) = f (g −1(x, y)), f ∈ H = L2(R2, dxdy)

is a unitary representation of G = SL2(R). The representation (ρ, H) is not irreducible: (1) notice H = H+ ⊕ H− (where H even/odd functions) (2) ρ commutes with the homotheties t t t 2 (Ut f )(x, y) := (exp(itA)f )(x, y) := e f (e x, e y), f ∈ H , t ∈ R, (x, y) ∈ R . It follows that ρ commutes with the spectral projectors of ( ) ( ) 1 ∂ √ A = + 1 , r = x 2 + y 2 , i ∂r In conclusion ρ leaves invariant the generalized eigenspaces of A { } H,iν ∈ 2 2 | −1+iν − − ∈ R = f Lloc(R ) f (rx, ry) = r f (x, y), f ( x, y) = f (x, y) (ν ).

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 12 33 Theorem (Cartan’s polar decomposition for SL2(R))

Every element g ∈ SL2(R) can be uniquely written as a product g = k exp S with k ∈ K and S ∈ sl2(R) a traceless symmetric matrix.

rom the Cartan’s KAK decomposition the polar decomposition follows:

Introduction to SL2(R) More on these actions

Linear action on Q2 KAK decomposition

Every positive deﬁnite bilinear form of determinant( ) 1 on R2, may be conjugated by λ 0 ∈ R∗ a rotation to the diagonal bilinear form 0 λ−1 , with λ +. Thus

Theorem (Cartan’s KAK decomposition for SL2(R))

Every element g ∈ SL2(R) can be written as a product

g = k1ak2

with k1, k2 ∈ K and a ∈ A. In particular SL2(R) = KAK.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 13 33 Theorem (Cartan’s polar decomposition for SL2(R))

Every element g ∈ SL2(R) can be uniquely written as a product g = k exp S with k ∈ K and S ∈ sl2(R) a traceless symmetric matrix.

rom the Cartan’s KAK decomposition the polar decomposition follows:

Introduction to SL2(R) More on these actions

Linear action on Q2 KAK decomposition

Every positive deﬁnite bilinear form of determinant( ) 1 on R2, may be conjugated by λ 0 ∈ R∗ a rotation to the diagonal bilinear form 0 λ−1 , with λ +. Thus

Theorem (Cartan’s KAK decomposition for SL2(R))

Every element g ∈ SL2(R) can be written as a product

g = k1ak2

with k1, k2 ∈ K and a ∈ A. In particular SL2(R) = KAK.

Proof. ∈ ↔ T ∈ Q(1) The map gK G/K gg 2 is a bijection on the[ orbit] of I (the positive T λ 0 T ∈ deﬁnite[ matrices of] determinant one).[ Since gg ]= k 0 λ−1 k , (k K), we get gK = k λ1/2 0 K, that is g = k λ1/2 0 k′. 0 λ−1/2 0 λ−1/2

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 13 33 Theorem (Cartan’s polar decomposition for SL2(R))

Every element g ∈ SL2(R) can be uniquely written as a product g = k exp S with k ∈ K and S ∈ sl2(R) a traceless symmetric matrix.

rom the Cartan’s KAK decomposition the polar decomposition follows:

Introduction to SL2(R) More on these actions

Linear action on Q2 KAK decomposition

Every positive deﬁnite bilinear form of determinant( ) 1 on R2, may be conjugated by λ 0 ∈ R∗ a rotation to the diagonal bilinear form 0 λ−1 , with λ +. Thus

Theorem (Cartan’s KAK decomposition for SL2(R))

Every element g ∈ SL2(R) can be written as a product

g = k1ak2

with k1, k2 ∈ K and a ∈ A. In particular SL2(R) = KAK.

Diﬀerently from the the Iwasawa decomposition, the Cartan KAK decomposition of an element is not unique.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 13 33 Theorem (Cartan’s polar decomposition for SL2(R))

More generally the ambiguity is by an element of the normalizer of A in K: ′ −1 k1ak2 = (k1w)a (w k2). [ ] 0 1 The normalizer of A in SO(2) is generated by w = −1 0 . The group NorK (A)/ ZK (A) hence has order two and it is called the Weyl group (of SL2(R)). rom the Cartan’s KAK decomposition the polar decomposition follows:

Introduction to SL2(R) More on these actions

Linear action on Q2 KAK decomposition

Theorem (Cartan’s KAK decomposition for SL2(R))

Every element g ∈ SL2(R) can be written as a product

g = k1ak2

with k1, k2 ∈ K and a ∈ A. In particular SL2(R) = KAK.

If we ﬁx the element a, then the ambiguity is only by and element of M = {1} the centralizer of A in K: k1ak2 = (k1(−1))a((−1)k2).

[ ] 0 1 The normalizer of A in SO(2) is generated by w = −1 0 . The group NorK (A)/ ZK (A) hence has order two and it is called the Weyl group (of SL2(R)). rom the Cartan’s KAK decomposition the polar decomposition follows:

Introduction to SL2(R) More on these actions

Linear action on Q2 KAK decomposition

Theorem (Cartan’s KAK decomposition for SL2(R))

Every element g ∈ SL2(R) can be written as a product

g = k1ak2

with k1, k2 ∈ K and a ∈ A. In particular SL2(R) = KAK.

If we ﬁx the element a, then the ambiguity is only by and element of M = {1} the centralizer of A in K: k1ak2 = (k1(−1))a((−1)k2). More generally the ambiguity is by an element of the normalizer of A in K: ′ −1 k1ak2 = (k1w)a (w k2).

The group NorK (A)/ ZK (A) hence has order two and it is called the Weyl group (of SL2(R)). rom the Cartan’s KAK decomposition the polar decomposition follows:

Introduction to SL2(R) More on these actions

Linear action on Q2 KAK decomposition

Theorem (Cartan’s KAK decomposition for SL2(R))

Every element g ∈ SL2(R) can be written as a product

g = k1ak2

with k1, k2 ∈ K and a ∈ A. In particular SL2(R) = KAK.

If we ﬁx the element a, then the ambiguity is only by and element of M = {1} the centralizer of A in K: k1ak2 = (k1(−1))a((−1)k2). More generally the ambiguity is by an element of the normalizer of A in K: ′ −1 k1ak2 = (k1w)a (w k2). [ ] 0 1 The normalizer of A in SO(2) is generated by w = −1 0 .

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 13 33 Introduction to SL2(R) More on these actions

Linear action on Q2 KAK decomposition

Theorem (Cartan’s KAK decomposition for SL2(R))

Every element g ∈ SL2(R) can be written as a product

g = k1ak2

with k1, k2 ∈ K and a ∈ A. In particular SL2(R) = KAK.

From the Cartan’s KAK decomposition the polar decomposition follows:

Theorem (Cartan’s polar decomposition for SL2(R))

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 13 33 The group SL2(R) is a subgroup of SL2(C) (a real Lie group of dimension 6). 2 As the group SL2(C) acts linearly on C we also have a projective action of 1 SL2(C) on the Riemann sphere Cˆ := P (C).

Restricting this action to SL2(C) we obtain a action of SL2(R) on Cˆ, which in the aﬃne chart z = [z : 1] of Cˆ is also given by the formulas (1), where x is replaced by z ∈ C.

Introduction to SL2(R) The Poincaré plane

The action of SL2(R) on the projective line

2 The linear action of SL2(R) on R induces an action of SL2(R) on the projective line P1(R). In the aﬃne chart x = [x : 1] the action is given by

[ ] [ ] ax + b a b .x = a b .[x : 1] = [(ax + b):(cx + d)] = , x ∈ R. (1) c d c d cx + d

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 14 33 2 As the group SL2(C) acts linearly on C we also have a projective action of 1 SL2(C) on the Riemann sphere Cˆ := P (C).

Restricting this action to SL2(C) we obtain a action of SL2(R) on Cˆ, which in the aﬃne chart z = [z : 1] of Cˆ is also given by the formulas (1), where x is replaced by z ∈ C.

Introduction to SL2(R) The Poincaré plane

The action of SL2(R) on the projective line

2 The linear action of SL2(R) on R induces an action of SL2(R) on the projective line P1(R). In the aﬃne chart x = [x : 1] the action is given by

[ ] [ ] ax + b a b .x = a b .[x : 1] = [(ax + b):(cx + d)] = , x ∈ R. (1) c d c d cx + d

The group SL2(R) is a subgroup of SL2(C) (a real Lie group of dimension 6).

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 14 33 Restricting this action to SL2(C) we obtain a action of SL2(R) on Cˆ, which in the aﬃne chart z = [z : 1] of Cˆ is also given by the formulas (1), where x is replaced by z ∈ C.

Introduction to SL2(R) The Poincaré plane

The action of SL2(R) on the projective line

2 The linear action of SL2(R) on R induces an action of SL2(R) on the projective line P1(R). In the aﬃne chart x = [x : 1] the action is given by

[ ] [ ] ax + b a b .x = a b .[x : 1] = [(ax + b):(cx + d)] = , x ∈ R. (1) c d c d cx + d

The group SL2(R) is a subgroup of SL2(C) (a real Lie group of dimension 6). 2 As the group SL2(C) acts linearly on C we also have a projective action of 1 SL2(C) on the Riemann sphere Cˆ := P (C).

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 14 33 Introduction to SL2(R) The Poincaré plane

The action of SL2(R) on the projective line

[ ] [ ] ax + b a b .x = a b .[x : 1] = [(ax + b):(cx + d)] = , x ∈ R. (1) c d c d cx + d

The group SL2(R) is a subgroup of SL2(C) (a real Lie group of dimension 6). 2 As the group SL2(C) acts linearly on C we also have a projective action of 1 SL2(C) on the Riemann sphere Cˆ := P (C).

Restricting this action to SL2(C) we obtain a action of SL2(R) on Cˆ, which in the aﬃne chart z = [z : 1] of Cˆ is also given by the formulas (1), where x is replaced by z ∈ C.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 14 33 • the orbit of [i : 1] • the orbit of [−i : 1] In the aﬃne chart z = [z : 1] the latter two orbits are identiﬁed are respectively, the upper half plane H2 := {z ∈ C | ℑz > 0} and the lower half plane 2 H− := {z ∈ C | ℑz < 0} The subgroup {( ) } s t | ∈ R∗ ∈ R AN = 0 s−1 s +, t , 2 2 acts simply transitively on H and H− by the formulas ( ) s t 2 0 s−1 .( i) = st s i.

Introduction to SL2(R) The Poincaré plane

The action of SL2(R) on Cˆ

The action of SL2(R) on Cˆ has three orbits: • the orbit of [0 : 1], that is P1(R);

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 15 33 • the orbit of [−i : 1] In the aﬃne chart z = [z : 1] the latter two orbits are identiﬁed are respectively, the upper half plane H2 := {z ∈ C | ℑz > 0} and the lower half plane 2 H− := {z ∈ C | ℑz < 0} The subgroup {( ) } s t | ∈ R∗ ∈ R AN = 0 s−1 s +, t , 2 2 acts simply transitively on H and H− by the formulas ( ) s t 2 0 s−1 .( i) = st s i.

Introduction to SL2(R) The Poincaré plane

The action of SL2(R) on Cˆ

The action of SL2(R) on Cˆ has three orbits: • the orbit of [0 : 1], that is P1(R); • the orbit of [i : 1]

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 15 33 In the aﬃne chart z = [z : 1] the latter two orbits are identiﬁed are respectively, the upper half plane H2 := {z ∈ C | ℑz > 0} and the lower half plane 2 H− := {z ∈ C | ℑz < 0} The subgroup {( ) } s t | ∈ R∗ ∈ R AN = 0 s−1 s +, t , 2 2 acts simply transitively on H and H− by the formulas ( ) s t 2 0 s−1 .( i) = st s i.

Introduction to SL2(R) The Poincaré plane

The action of SL2(R) on Cˆ

The action of SL2(R) on Cˆ has three orbits: • the orbit of [0 : 1], that is P1(R); • the orbit of [i : 1] • the orbit of [−i : 1]

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 15 33 In the aﬃne chart z = [z : 1] the latter two orbits are identiﬁed are respectively, the upper half plane H2 := {z ∈ C | ℑz > 0} and the lower half plane 2 H− := {z ∈ C | ℑz < 0} The subgroup {( ) } s t | ∈ R∗ ∈ R AN = 0 s−1 s +, t , 2 2 acts simply transitively on H and H− by the formulas ( ) s t 2 0 s−1 .( i) = st s i.

Introduction to SL2(R) The Poincaré plane

The action of SL2(R) on Cˆ

The action of SL2(R) on Cˆ has three orbits: • the orbit of [0 : 1], that is P1(R); • the orbit of [i : 1] • the orbit of [−i : 1]

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 15 33 The subgroup {( ) } s t | ∈ R∗ ∈ R AN = 0 s−1 s +, t , 2 2 acts simply transitively on H and H− by the formulas ( ) s t 2 0 s−1 .( i) = st s i.

Introduction to SL2(R) The Poincaré plane

The action of SL2(R) on Cˆ

The action of SL2(R) on Cˆ has three orbits: • the orbit of [0 : 1], that is P1(R); • the orbit of [i : 1] • the orbit of [−i : 1] In the aﬃne chart z = [z : 1] the latter two orbits are identiﬁed are respectively, the upper half plane H2 := {z ∈ C | ℑz > 0} and the lower half plane 2 H− := {z ∈ C | ℑz < 0}

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 15 33 Introduction to SL2(R) The Poincaré plane

The action of SL2(R) on Cˆ

The action of SL2(R) on Cˆ has three orbits: • the orbit of [0 : 1], that is P1(R); • the orbit of [i : 1] • the orbit of [−i : 1] In the aﬃne chart z = [z : 1] the latter two orbits are identiﬁed are respectively, the upper half plane H2 := {z ∈ C | ℑz > 0} and the lower half plane 2 H− := {z ∈ C | ℑz < 0} The subgroup {( ) } s t | ∈ R∗ ∈ R AN = 0 s−1 s +, t , 2 2 acts simply transitively on H and H− by the formulas ( ) s t 2 0 s−1 .( i) = st s i.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 15 33 ∈ H2 1 The map (2) is holomorphic; its diﬀerential in z is the multiplication by (cz+d)2 . 2 ¯ The stabilizer in PSL2(R) of i ∈(H is the group) of rotations K = K/{1}. The 7→ cos θ − sin θ −2iθ diﬀerential at i of the map z sin θ cos θ .z is the multiplication by e which acts simply transitively on rays. 2 Thus there a unique (up to rescaling) SL2(R)-invariant Riemannian metric on H given

|dz|2 dx 2 + dy 2 ds2 = = , z = x + iy ∈ H2. (ℑz)2 y 2

The upper half plane H2 endowed with the above Riemannian metric is the Poincaré half plane or Poincaré hyperbolic plane. (The normalization of the metric is such that the curvature is constantly equal to −1).

Introduction to SL2(R) The Poincaré plane 2 The action of SL2(R) on H part 1

2 We shall now focus on the action of PSL2(R)) on H given by the formulas az + b ( a b ) .z = , ( a b ) ∈ SL (R), z ∈ H2. (2) c d cz + d c d 2

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 16 33 2 ¯ The stabilizer in PSL2(R) of i ∈(H is the group) of rotations K = K/{1}. The 7→ cos θ − sin θ −2iθ diﬀerential at i of the map z sin θ cos θ .z is the multiplication by e which acts simply transitively on rays. 2 Thus there a unique (up to rescaling) SL2(R)-invariant Riemannian metric on H given

|dz|2 dx 2 + dy 2 ds2 = = , z = x + iy ∈ H2. (ℑz)2 y 2

The upper half plane H2 endowed with the above Riemannian metric is the Poincaré half plane or Poincaré hyperbolic plane. (The normalization of the metric is such that the curvature is constantly equal to −1).

Introduction to SL2(R) The Poincaré plane 2 The action of SL2(R) on H part 1

2 We shall now focus on the action of PSL2(R)) on H given by the formulas az + b ( a b ) .z = , ( a b ) ∈ SL (R), z ∈ H2. (2) c d cz + d c d 2 ∈ H2 1 The map (2) is holomorphic; its diﬀerential in z is the multiplication by (cz+d)2 .

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 16 33 ( ) The 7→ cos θ − sin θ −2iθ diﬀerential at i of the map z sin θ cos θ .z is the multiplication by e which acts simply transitively on rays. 2 Thus there a unique (up to rescaling) SL2(R)-invariant Riemannian metric on H given

|dz|2 dx 2 + dy 2 ds2 = = , z = x + iy ∈ H2. (ℑz)2 y 2

The upper half plane H2 endowed with the above Riemannian metric is the Poincaré half plane or Poincaré hyperbolic plane. (The normalization of the metric is such that the curvature is constantly equal to −1).

Introduction to SL2(R) The Poincaré plane 2 The action of SL2(R) on H part 1

2 We shall now focus on the action of PSL2(R)) on H given by the formulas az + b ( a b ) .z = , ( a b ) ∈ SL (R), z ∈ H2. (2) c d cz + d c d 2 ∈ H2 1 The map (2) is holomorphic; its diﬀerential in z is the multiplication by (cz+d)2 . 2 The stabilizer in PSL2(R) of i ∈ H is the group of rotations K¯ = K/{1}.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 16 33 2 Thus there a unique (up to rescaling) SL2(R)-invariant Riemannian metric on H given

|dz|2 dx 2 + dy 2 ds2 = = , z = x + iy ∈ H2. (ℑz)2 y 2

Introduction to SL2(R) The Poincaré plane 2 The action of SL2(R) on H part 1

2 We shall now focus on the action of PSL2(R)) on H given by the formulas az + b ( a b ) .z = , ( a b ) ∈ SL (R), z ∈ H2. (2) c d cz + d c d 2 ∈ H2 1 The map (2) is holomorphic; its diﬀerential in z is the multiplication by (cz+d)2 . 2 ¯ The stabilizer in PSL2(R) of i ∈(H is the group) of rotations K = K/{1}. The 7→ cos θ − sin θ −2iθ diﬀerential at i of the map z sin θ cos θ .z is the multiplication by e which acts simply transitively on rays.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 16 33 The upper half plane H2 endowed with the above Riemannian metric is the Poincaré half plane or Poincaré hyperbolic plane. (The normalization of the metric is such that the curvature is constantly equal to −1).

Introduction to SL2(R) The Poincaré plane 2 The action of SL2(R) on H part 1

2 We shall now focus on the action of PSL2(R)) on H given by the formulas az + b ( a b ) .z = , ( a b ) ∈ SL (R), z ∈ H2. (2) c d cz + d c d 2 ∈ H2 1 The map (2) is holomorphic; its diﬀerential in z is the multiplication by (cz+d)2 . 2 ¯ The stabilizer in PSL2(R) of i ∈(H is the group) of rotations K = K/{1}. The 7→ cos θ − sin θ −2iθ diﬀerential at i of the map z sin θ cos θ .z is the multiplication by e which acts simply transitively on rays. 2 Thus there a unique (up to rescaling) SL2(R)-invariant Riemannian metric on H given

|dz|2 dx 2 + dy 2 ds2 = = , z = x + iy ∈ H2. (ℑz)2 y 2

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 16 33 Introduction to SL2(R) The Poincaré plane 2 The action of SL2(R) on H part 1

2 We shall now focus on the action of PSL2(R)) on H given by the formulas az + b ( a b ) .z = , ( a b ) ∈ SL (R), z ∈ H2. (2) c d cz + d c d 2 ∈ H2 1 The map (2) is holomorphic; its diﬀerential in z is the multiplication by (cz+d)2 . 2 ¯ The stabilizer in PSL2(R) of i ∈(H is the group) of rotations K = K/{1}. The 7→ cos θ − sin θ −2iθ diﬀerential at i of the map z sin θ cos θ .z is the multiplication by e which acts simply transitively on rays. 2 Thus there a unique (up to rescaling) SL2(R)-invariant Riemannian metric on H given

|dz|2 dx 2 + dy 2 ds2 = = , z = x + iy ∈ H2. (ℑz)2 y 2

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 16 33 The uniqueness of the decomposition Iwasawa decomposition for PSL2(R) is now equivalent to the statement that PSL2(R) acts simply transitively on the tangent unit bundle T 1H2 of the Poincaré half plane. Hence

1 2 PSL2(R) ↔ T H

g ↔ (z, v) = (g.i, dgi (i))

1 2 PSL2(R) acts on itself, hence on T H , on the left and on the right. The action on the left is the diﬀerential of isometries of H2. The action on the right gives rise to geometrical ﬂows.

Introduction to SL2(R) The Poincaré plane 2 The action of SL2(R) on H part 2

The group PSL2(R) is now, by deﬁnition of the metric the coonnected component of the identity of group of isometries of the Poincaré hyperbolic plane.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 17 33 Hence

1 2 PSL2(R) ↔ T H

g ↔ (z, v) = (g.i, dgi (i))

1 2 PSL2(R) acts on itself, hence on T H , on the left and on the right. The action on the left is the diﬀerential of isometries of H2. The action on the right gives rise to geometrical ﬂows.

Introduction to SL2(R) The Poincaré plane 2 The action of SL2(R) on H part 2

The group PSL2(R) is now, by deﬁnition of the metric the coonnected component of the identity of group of isometries of the Poincaré hyperbolic plane.

The uniqueness of the decomposition Iwasawa decomposition for PSL2(R) is now equivalent to the statement that PSL2(R) acts simply transitively on the tangent unit bundle T 1H2 of the Poincaré half plane.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 17 33 1 2 PSL2(R) acts on itself, hence on T H , on the left and on the right. The action on the left is the diﬀerential of isometries of H2. The action on the right gives rise to geometrical ﬂows.

Introduction to SL2(R) The Poincaré plane 2 The action of SL2(R) on H part 2

The group PSL2(R) is now, by deﬁnition of the metric the coonnected component of the identity of group of isometries of the Poincaré hyperbolic plane.

The uniqueness of the decomposition Iwasawa decomposition for PSL2(R) is now equivalent to the statement that PSL2(R) acts simply transitively on the tangent unit bundle T 1H2 of the Poincaré half plane. Hence

1 2 PSL2(R) ↔ T H

g ↔ (z, v) = (g.i, dgi (i))

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 17 33 Introduction to SL2(R) The Poincaré plane 2 The action of SL2(R) on H part 2

The uniqueness of the decomposition Iwasawa decomposition for PSL2(R) is now equivalent to the statement that PSL2(R) acts simply transitively on the tangent unit bundle T 1H2 of the Poincaré half plane. Hence

1 2 PSL2(R) ↔ T H

g ↔ (z, v) = (g.i, dgi (i))

1 2 PSL2(R) acts on itself, hence on T H , on the left and on the right. The action on the left is the diﬀerential of isometries of H2. The action on the right gives rise to geometrical ﬂows.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 17 33 1 2 Under the identiﬁcation PSL2(R) ↔ T H the path (at )t∈R in PSL2(R) is identiﬁed with the geodesic path in T 1H2 t t ′ {(e i, e i) = (γ(t), γ (t))}t∈R 1 2 with initial velocity i ∈ Ti H .

As isometries map geodesics to geodesics, for any g ∈ PSL2(R), the path γ¯(t) = g.γ(t) ′ 1 2 is the unique unit speed geodesic with initial velocity γ¯ (0) = dgi (i) ∈ Tg.i H . ′ 1 2 1 2 Thus the path (gat )t∈R is identiﬁed with the path (¯γ(t), γ¯ (t)) in T H , the lift to T H of the geodesic γ¯. 1 2 1 2 Using identiﬁcation PSL2(R) ↔ T H , the geodesic ﬂow on T H is ﬂow on PSL2(R) given by the map

A (g, t) ∈ PSL2(R) × R 7→ Φt (g) := gat ∈ PSL2(R).

Introduction to SL2(R) The Poincaré plane 1 2 The action of PSL2(R) on T H Geodesic ﬂow on the Poincaré plane

Parametrize the subgroup A ⊂ PSL (R) by 2 ( ) et/2 0 at = − . 0 e t/2

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 18 33 As isometries map geodesics to geodesics, for any g ∈ PSL2(R), the path γ¯(t) = g.γ(t) ′ 1 2 is the unique unit speed geodesic with initial velocity γ¯ (0) = dgi (i) ∈ Tg.i H . ′ 1 2 1 2 Thus the path (gat )t∈R is identiﬁed with the path (¯γ(t), γ¯ (t)) in T H , the lift to T H of the geodesic γ¯. 1 2 1 2 Using identiﬁcation PSL2(R) ↔ T H , the geodesic ﬂow on T H is ﬂow on PSL2(R) given by the map

A (g, t) ∈ PSL2(R) × R 7→ Φt (g) := gat ∈ PSL2(R).

Introduction to SL2(R) The Poincaré plane 1 2 The action of PSL2(R) on T H Geodesic ﬂow on the Poincaré plane

Parametrize the subgroup A ⊂ PSL (R) by 2 ( ) et/2 0 at = − . 0 e t/2

1 2 Under the identiﬁcation PSL2(R) ↔ T H the path (at )t∈R in PSL2(R) is identiﬁed with the geodesic path in T 1H2 t t ′ {(e i, e i) = (γ(t), γ (t))}t∈R 1 2 with initial velocity i ∈ Ti H .

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 18 33 ′ 1 2 1 2 Thus the path (gat )t∈R is identiﬁed with the path (¯γ(t), γ¯ (t)) in T H , the lift to T H of the geodesic γ¯. 1 2 1 2 Using identiﬁcation PSL2(R) ↔ T H , the geodesic ﬂow on T H is ﬂow on PSL2(R) given by the map

A (g, t) ∈ PSL2(R) × R 7→ Φt (g) := gat ∈ PSL2(R).

Introduction to SL2(R) The Poincaré plane 1 2 The action of PSL2(R) on T H Geodesic ﬂow on the Poincaré plane

Parametrize the subgroup A ⊂ PSL (R) by 2 ( ) et/2 0 at = − . 0 e t/2

1 2 Under the identiﬁcation PSL2(R) ↔ T H the path (at )t∈R in PSL2(R) is identiﬁed with the geodesic path in T 1H2 t t ′ {(e i, e i) = (γ(t), γ (t))}t∈R 1 2 with initial velocity i ∈ Ti H .

As isometries map geodesics to geodesics, for any g ∈ PSL2(R), the path γ¯(t) = g.γ(t) ′ 1 2 is the unique unit speed geodesic with initial velocity γ¯ (0) = dgi (i) ∈ Tg.i H .

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 18 33 1 2 1 2 Using identiﬁcation PSL2(R) ↔ T H , the geodesic ﬂow on T H is ﬂow on PSL2(R) given by the map

A (g, t) ∈ PSL2(R) × R 7→ Φt (g) := gat ∈ PSL2(R).

Parametrize the subgroup A ⊂ PSL (R) by 2 ( ) et/2 0 at = − . 0 e t/2

1 2 Under the identiﬁcation PSL2(R) ↔ T H the path (at )t∈R in PSL2(R) is identiﬁed with the geodesic path in T 1H2 t t ′ {(e i, e i) = (γ(t), γ (t))}t∈R 1 2 with initial velocity i ∈ Ti H .

As isometries map geodesics to geodesics, for any g ∈ PSL2(R), the path γ¯(t) = g.γ(t) ′ 1 2 is the unique unit speed geodesic with initial velocity γ¯ (0) = dgi (i) ∈ Tg.i H . ′ 1 2 1 2 Thus the path (gat )t∈R is identiﬁed with the path (¯γ(t), γ¯ (t)) in T H , the lift to T H of the geodesic γ¯.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 18 33 Introduction to SL2(R) The Poincaré plane 1 2 The action of PSL2(R) on T H Geodesic ﬂow on the Poincaré plane

Parametrize the subgroup A ⊂ PSL (R) by 2 ( ) et/2 0 at = − . 0 e t/2

As isometries map geodesics to geodesics, for any g ∈ PSL2(R), the path γ¯(t) = g.γ(t) ′ 1 2 is the unique unit speed geodesic with initial velocity γ¯ (0) = dgi (i) ∈ Tg.i H . ′ 1 2 1 2 Thus the path (gat )t∈R is identiﬁed with the path (¯γ(t), γ¯ (t)) in T H , the lift to T H of the geodesic γ¯. 1 2 1 2 Using identiﬁcation PSL2(R) ↔ T H , the geodesic ﬂow on T H is ﬂow on PSL2(R) given by the map

A (g, t) ∈ PSL2(R) × R 7→ Φt (g) := gat ∈ PSL2(R).

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 18 33 Then ns at = at nse−t , n¯s at = at n¯set (4) N,N R It follows that the ﬂows (Φt )t∈R on PSL2( ) deﬁned by

N N Φt (g) = gnt , Φt (g) = gn¯t satisfy A ◦ N N ◦ A A ◦ N N ◦ A Φt Φs = Φse−t Φt , Φt Φs = Φset Φt . (5) A N −t N¯ i.e. the map Φt contracts the ﬂow (Φs ) by a factor e by expands the ﬂow (Φs ) by a factor et . Equivalent formulation: A + −t + A − t − − ∓ dΦt (n ) = e n , dΦt (n ) = e n , or [ h, n ] = n .

Introduction to SL2(R) The Poincaré plane 1 2 The action of PSL2(R) on T H Horocycle ﬂows on the Poincaré plane

Parametrize N and N = wNw −1 by

1 t 1 0 nt = [ 0 1 ] , n¯t = [ t 1 ] . (3)

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 19 33 Introduction to SL2(R) The Poincaré plane 1 2 The action of PSL2(R) on T H Horocycle ﬂows on the Poincaré plane

Parametrize N and N = wNw −1 by

1 t 1 0 nt = [ 0 1 ] , n¯t = [ t 1 ] . (3)

Then ns at = at nse−t , n¯s at = at n¯set (4) N,N R It follows that the ﬂows (Φt )t∈R on PSL2( ) deﬁned by

N N Φt (g) = gnt , Φt (g) = gn¯t satisfy A ◦ N N ◦ A A ◦ N N ◦ A Φt Φs = Φse−t Φt , Φt Φs = Φset Φt . (5) A N −t N¯ i.e. the map Φt contracts the ﬂow (Φs ) by a factor e by expands the ﬂow (Φs ) by a factor et . Equivalent formulation: A + −t + A − t − − ∓ dΦt (n ) = e n , dΦt (n ) = e n , or [ h, n ] = n .

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 19 33 ( ) cos θ − sin θ 1H2 ≈ C −2iθ The element kθ := sin θ cos θ acts on Tz by multiplication by e . 7→ −1 The ﬂow of the one-parameter group g g.(kθat kθ )t is the parallel transport of a unit vector along the geodesic making an angle −2θ (anti-clockwise) with the unit vector g.

⊥ 0 1 In particular the ﬂow generated by h = [ 1 0 ] = k−π/2hkπ/2 is therefore the parallel transport along the geodesic making a right angle π/2 with the initial unit vector. It follows that on K¯ invariant function the operator 1 − (h2 + (h⊥)2) 4 is the Laplace-Beltrami operator for the Poincaré metric on H2

Introduction to SL2(R) The Poincaré plane 1 2 The action of PSL2(R) on T H Rotational ﬂow and parallel transport on the Poincaré plane

1 2 The group K¯ operates by multiplication on the right on PSL2(R) ≈ T H leaving 1H2 each ﬁber Tz stable.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 20 33 7→ −1 The ﬂow of the one-parameter group g g.(kθat kθ )t is the parallel transport of a unit vector along the geodesic making an angle −2θ (anti-clockwise) with the unit vector g.

⊥ 0 1 In particular the ﬂow generated by h = [ 1 0 ] = k−π/2hkπ/2 is therefore the parallel transport along the geodesic making a right angle π/2 with the initial unit vector. It follows that on K¯ invariant function the operator 1 − (h2 + (h⊥)2) 4 is the Laplace-Beltrami operator for the Poincaré metric on H2

Introduction to SL2(R) The Poincaré plane 1 2 The action of PSL2(R) on T H Rotational ﬂow and parallel transport on the Poincaré plane

1 2 The group K¯ operates by multiplication on the right on PSL2(R) ≈ T H leaving each ﬁber T 1H2 stable. z ( ) cos θ − sin θ 1H2 ≈ C −2iθ The element kθ := sin θ cos θ acts on Tz by multiplication by e .

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 20 33 ⊥ 0 1 In particular the ﬂow generated by h = [ 1 0 ] = k−π/2hkπ/2 is therefore the parallel transport along the geodesic making a right angle π/2 with the initial unit vector. It follows that on K¯ invariant function the operator 1 − (h2 + (h⊥)2) 4 is the Laplace-Beltrami operator for the Poincaré metric on H2

Introduction to SL2(R) The Poincaré plane 1 2 The action of PSL2(R) on T H Rotational ﬂow and parallel transport on the Poincaré plane

1 2 The group K¯ operates by multiplication on the right on PSL2(R) ≈ T H leaving each ﬁber T 1H2 stable. z ( ) cos θ − sin θ 1H2 ≈ C −2iθ The element kθ := sin θ cos θ acts on Tz by multiplication by e . 7→ −1 The ﬂow of the one-parameter group g g.(kθat kθ )t is the parallel transport of a unit vector along the geodesic making an angle −2θ (anti-clockwise) with the unit vector g.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 20 33 It follows that on K¯ invariant function the operator 1 − (h2 + (h⊥)2) 4 is the Laplace-Beltrami operator for the Poincaré metric on H2

1 2 The group K¯ operates by multiplication on the right on PSL2(R) ≈ T H leaving each ﬁber T 1H2 stable. z ( ) cos θ − sin θ 1H2 ≈ C −2iθ The element kθ := sin θ cos θ acts on Tz by multiplication by e . 7→ −1 The ﬂow of the one-parameter group g g.(kθat kθ )t is the parallel transport of a unit vector along the geodesic making an angle −2θ (anti-clockwise) with the unit vector g.

⊥ 0 1 In particular the ﬂow generated by h = [ 1 0 ] = k−π/2hkπ/2 is therefore the parallel transport along the geodesic making a right angle π/2 with the initial unit vector.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 20 33 Introduction to SL2(R) The Poincaré plane 1 2 The action of PSL2(R) on T H Rotational ﬂow and parallel transport on the Poincaré plane

1 2 The group K¯ operates by multiplication on the right on PSL2(R) ≈ T H leaving each ﬁber T 1H2 stable. z ( ) cos θ − sin θ 1H2 ≈ C −2iθ The element kθ := sin θ cos θ acts on Tz by multiplication by e . 7→ −1 The ﬂow of the one-parameter group g g.(kθat kθ )t is the parallel transport of a unit vector along the geodesic making an angle −2θ (anti-clockwise) with the unit vector g.

⊥ 0 1 In particular the ﬂow generated by h = [ 1 0 ] = k−π/2hkπ/2 is therefore the parallel transport along the geodesic making a right angle π/2 with the initial unit vector. It follows that on K¯ invariant function the operator 1 − (h2 + (h⊥)2) 4 is the Laplace-Beltrami operator for the Poincaré metric on H2

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 20 33 As SL2(R) acts on the right on Γ\ SL2(R), we have a volume form on Γ\ SL2(R) invariant under the right action SL2(R) on Γ\ SL2(R) and deﬁning a SL2(R)-invariant measure µ. 2 We get a unitary representation π of SL2(R) on L (Γ\ SL2(R), µ) by letting

2 (π(g)f )(Γx) = f (Γxg), f ∈ L (Γ\ SL2(R), µ).

which decomposes as a direct integral or sum of irreducible represenations of SL2(R). One of our goals will be to shed some light of the relation of the geometry of the surface Γ\ SL2(R) and the decomposition into irreducible representations.

Introduction to SL2(R) The Poincaré plane Hyperbolic surfaces and their tangent unit bundle

The group SL2(R) has many discrete subgroups Γ. Then the quotient Γ\ SL2(R) is a smooth manifold and the quotient Γ\H2 of the hyperbolic plane H2 is a Riemann surface, with at most some conical points. The celebrated example is the group SL2(R)

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 21 33 2 We get a unitary representation π of SL2(R) on L (Γ\ SL2(R), µ) by letting

2 (π(g)f )(Γx) = f (Γxg), f ∈ L (Γ\ SL2(R), µ).

which decomposes as a direct integral or sum of irreducible represenations of SL2(R). One of our goals will be to shed some light of the relation of the geometry of the surface Γ\ SL2(R) and the decomposition into irreducible representations.

Introduction to SL2(R) The Poincaré plane Hyperbolic surfaces and their tangent unit bundle

The group SL2(R) has many discrete subgroups Γ. Then the quotient Γ\ SL2(R) is a smooth manifold and the quotient Γ\H2 of the hyperbolic plane H2 is a Riemann surface, with at most some conical points. The celebrated example is the group SL2(R)

As SL2(R) acts on the right on Γ\ SL2(R), we have a volume form on Γ\ SL2(R) invariant under the right action SL2(R) on Γ\ SL2(R) and deﬁning a SL2(R)-invariant measure µ.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 21 33 One of our goals will be to shed some light of the relation of the geometry of the surface Γ\ SL2(R) and the decomposition into irreducible representations.

Introduction to SL2(R) The Poincaré plane Hyperbolic surfaces and their tangent unit bundle

The group SL2(R) has many discrete subgroups Γ. Then the quotient Γ\ SL2(R) is a smooth manifold and the quotient Γ\H2 of the hyperbolic plane H2 is a Riemann surface, with at most some conical points. The celebrated example is the group SL2(R)

As SL2(R) acts on the right on Γ\ SL2(R), we have a volume form on Γ\ SL2(R) invariant under the right action SL2(R) on Γ\ SL2(R) and deﬁning a SL2(R)-invariant measure µ. 2 We get a unitary representation π of SL2(R) on L (Γ\ SL2(R), µ) by letting

2 (π(g)f )(Γx) = f (Γxg), f ∈ L (Γ\ SL2(R), µ).

which decomposes as a direct integral or sum of irreducible represenations of SL2(R).

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 21 33 Introduction to SL2(R) The Poincaré plane Hyperbolic surfaces and their tangent unit bundle

As SL2(R) acts on the right on Γ\ SL2(R), we have a volume form on Γ\ SL2(R) invariant under the right action SL2(R) on Γ\ SL2(R) and deﬁning a SL2(R)-invariant measure µ. 2 We get a unitary representation π of SL2(R) on L (Γ\ SL2(R), µ) by letting

2 (π(g)f )(Γx) = f (Γxg), f ∈ L (Γ\ SL2(R), µ).

which decomposes as a direct integral or sum of irreducible represenations of SL2(R). One of our goals will be to shed some light of the relation of the geometry of the surface Γ\ SL2(R) and the decomposition into irreducible representations.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 21 33 G acts on itself by conjugation ﬁxing the identity, and form this we get an action of G by automorphims of g0, which for a linear group is given by

− Ad(g)X = gXg 1, g ∈ G, X ∈ g

The map g 7→ Ad(g) is a homomorphism of G in Aut(g0) called the adjoint representation. Then a the Lie bracket operation arises as the inﬁnitesimal version of this action:

d ϵY ad(Y )X := Ad(e )X = [Y , X ]. dϵ ϵ=0

Introduction to SL2(R) Enveloping algebra Lie algebras/Lie groups generalities The adjoint representation

n For any matrix group G (i.e. closed subgroup of GL(R )) the Lie algebra g0 of G is the tangent space g0 = TI G of G at the identity I which we may also identify with the vector space of left invariant vector ﬁelds on G. The braket [X , Y ] is the commutator of the corresponding vector ﬁelds.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 22 33 Then a the Lie bracket operation arises as the inﬁnitesimal version of this action:

d ϵY ad(Y )X := Ad(e )X = [Y , X ]. dϵ ϵ=0

Introduction to SL2(R) Enveloping algebra Lie algebras/Lie groups generalities The adjoint representation

n For any matrix group G (i.e. closed subgroup of GL(R )) the Lie algebra g0 of G is the tangent space g0 = TI G of G at the identity I which we may also identify with the vector space of left invariant vector ﬁelds on G. The braket [X , Y ] is the commutator of the corresponding vector ﬁelds. G acts on itself by conjugation ﬁxing the identity, and form this we get an action of G by automorphims of g0, which for a linear group is given by

− Ad(g)X = gXg 1, g ∈ G, X ∈ g

The map g 7→ Ad(g) is a homomorphism of G in Aut(g0) called the adjoint representation.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 22 33 Introduction to SL2(R) Enveloping algebra Lie algebras/Lie groups generalities The adjoint representation

n For any matrix group G (i.e. closed subgroup of GL(R )) the Lie algebra g0 of G is the tangent space g0 = TI G of G at the identity I which we may also identify with the vector space of left invariant vector ﬁelds on G. The braket [X , Y ] is the commutator of the corresponding vector ﬁelds. G acts on itself by conjugation ﬁxing the identity, and form this we get an action of G by automorphims of g0, which for a linear group is given by

− Ad(g)X = gXg 1, g ∈ G, X ∈ g

The map g 7→ Ad(g) is a homomorphism of G in Aut(g0) called the adjoint representation. Then a the Lie bracket operation arises as the inﬁnitesimal version of this action:

d ϵY ad(Y )X := Ad(e )X = [Y , X ]. dϵ ϵ=0

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 22 33 Introduction to SL2(R) Enveloping algebra Lie algebras/Lie groups generalities The Cartan-Killing form

For any Lie algebra g0, the bilinear form

B(X , Y ) = trace(ad(X ) ◦ ad(Y )), (X , Y ∈ g0),

is symmetric and invariant under the adjoint action

B(Ad(g)X , Ad(g)Y ) = B(X , Y ), (X , Y ∈ g0, g0 ∈ G).

For g0 = sl2(R), using the commutation relations

⊥ ⊥ ⊥ [κ, h] = 2h , [κ, h ] = −2h, [h, h ] = −2κ

we get ∗ ∗ ⊥ ∗ B = 8 [(κ )2 − (h )2 − ((h ) )2]

which is a non degenerate bilinear form of signature (1, 2). (Thus sl2(R) is the 3-dimensional lorentzian space-time).

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 23 33 This associative algebra, that includes and extends the Lie algebra g := g0 ⊗ C, is called the universal enveloping algebra of g; it is denoted U(g); it is isomorphic to the full tensor algebra on g, modulo the relations generated by the Lie algebra commutations rules. The enveloping algebra U(g) enjoys the following fundamental property: any Lie algebra homomorphism of g0 into an associative C-algebra A extends to an associative algebra homomorphism of U(g) into A. Thus category of representation of a Lie algebra g on a C-vector spaces, is equivalent to the category of U(g)-modules.

Introduction to SL2(R) Enveloping algebra Lie algebras Enveloping algebra

Elements of g0 corresponds to left invariant vector ﬁelds on G, i.e. ﬁrst order diﬀerential operators on G. Composing such operators and considering their linear span over C we obtain an C-algebra of left invariant vector diﬀerential operators on G.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 24 33 it is denoted U(g); it is isomorphic to the full tensor algebra on g, modulo the relations generated by the Lie algebra commutations rules. The enveloping algebra U(g) enjoys the following fundamental property: any Lie algebra homomorphism of g0 into an associative C-algebra A extends to an associative algebra homomorphism of U(g) into A. Thus category of representation of a Lie algebra g on a C-vector spaces, is equivalent to the category of U(g)-modules.

Introduction to SL2(R) Enveloping algebra Lie algebras Enveloping algebra

Elements of g0 corresponds to left invariant vector ﬁelds on G, i.e. ﬁrst order diﬀerential operators on G. Composing such operators and considering their linear span over C we obtain an C-algebra of left invariant vector diﬀerential operators on G.

This associative algebra, that includes and extends the Lie algebra g := g0 ⊗ C, is called the universal enveloping algebra of g;

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 24 33 it is isomorphic to the full tensor algebra on g, modulo the relations generated by the Lie algebra commutations rules. The enveloping algebra U(g) enjoys the following fundamental property: any Lie algebra homomorphism of g0 into an associative C-algebra A extends to an associative algebra homomorphism of U(g) into A. Thus category of representation of a Lie algebra g on a C-vector spaces, is equivalent to the category of U(g)-modules.

Introduction to SL2(R) Enveloping algebra Lie algebras Enveloping algebra

Elements of g0 corresponds to left invariant vector ﬁelds on G, i.e. ﬁrst order diﬀerential operators on G. Composing such operators and considering their linear span over C we obtain an C-algebra of left invariant vector diﬀerential operators on G.

This associative algebra, that includes and extends the Lie algebra g := g0 ⊗ C, is called the universal enveloping algebra of g; it is denoted U(g);

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 24 33 The enveloping algebra U(g) enjoys the following fundamental property: any Lie algebra homomorphism of g0 into an associative C-algebra A extends to an associative algebra homomorphism of U(g) into A. Thus category of representation of a Lie algebra g on a C-vector spaces, is equivalent to the category of U(g)-modules.

Introduction to SL2(R) Enveloping algebra Lie algebras Enveloping algebra

This associative algebra, that includes and extends the Lie algebra g := g0 ⊗ C, is called the universal enveloping algebra of g; it is denoted U(g); it is isomorphic to the full tensor algebra on g, modulo the relations generated by the Lie algebra commutations rules.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 24 33 Thus category of representation of a Lie algebra g on a C-vector spaces, is equivalent to the category of U(g)-modules.

Introduction to SL2(R) Enveloping algebra Lie algebras Enveloping algebra

This associative algebra, that includes and extends the Lie algebra g := g0 ⊗ C, is called the universal enveloping algebra of g; it is denoted U(g); it is isomorphic to the full tensor algebra on g, modulo the relations generated by the Lie algebra commutations rules. The enveloping algebra U(g) enjoys the following fundamental property: any Lie algebra homomorphism of g0 into an associative C-algebra A extends to an associative algebra homomorphism of U(g) into A.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 24 33 Introduction to SL2(R) Enveloping algebra Lie algebras Enveloping algebra

This associative algebra, that includes and extends the Lie algebra g := g0 ⊗ C, is called the universal enveloping algebra of g; it is denoted U(g); it is isomorphic to the full tensor algebra on g, modulo the relations generated by the Lie algebra commutations rules. The enveloping algebra U(g) enjoys the following fundamental property: any Lie algebra homomorphism of g0 into an associative C-algebra A extends to an associative algebra homomorphism of U(g) into A. Thus category of representation of a Lie algebra g on a C-vector spaces, is equivalent to the category of U(g)-modules.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 24 33 Introduction to SL2(R) Enveloping algebra Lie algebras Enveloping algebra

This associative algebra, that includes and extends the Lie algebra g := g0 ⊗ C, is called the universal enveloping algebra of g; it is denoted U(g); it is isomorphic to the full tensor algebra on g, modulo the relations generated by the Lie algebra commutations rules. The enveloping algebra U(g) enjoys the following fundamental property: any Lie algebra homomorphism of g0 into an associative C-algebra A extends to an associative algebra homomorphism of U(g) into A. Thus category of representation of a Lie algebra g on a C-vector spaces, is equivalent to the category of U(g)-modules.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 24 33 Theorem (Dixmier-Schur Lemma) If V is an irreducible U(g)-module of countable dimension then every Z ∈ Z(g) acts as a scalar on V .

In fact Ω generates the center Z(sl2(R)) of U(sl2(R)).

Introduction to SL2(R) Enveloping algebra Lie algebras Casimir element and the center of U(g)

From the invariance of the Cartan-Killing form we have that element 1 Ω = − (h2 + (h⊥)2 − κ2) ∈ U(sl (R)) 4 2

commutes with all X ∈ sl2(R) and therefore with all U(sl2(R)).

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 25 33 Theorem (Dixmier-Schur Lemma) If V is an irreducible U(g)-module of countable dimension then every Z ∈ Z(g) acts as a scalar on V .

Introduction to SL2(R) Enveloping algebra Lie algebras Casimir element and the center of U(g)

From the invariance of the Cartan-Killing form we have that element 1 Ω = − (h2 + (h⊥)2 − κ2) ∈ U(sl (R)) 4 2

commutes with all X ∈ sl2(R) and therefore with all U(sl2(R)).

In fact Ω generates the center Z(sl2(R)) of U(sl2(R)).

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 25 33 Introduction to SL2(R) Enveloping algebra Lie algebras Casimir element and the center of U(g)

From the invariance of the Cartan-Killing form we have that element 1 Ω = − (h2 + (h⊥)2 − κ2) ∈ U(sl (R)) 4 2

commutes with all X ∈ sl2(R) and therefore with all U(sl2(R)).

In fact Ω generates the center Z(sl2(R)) of U(sl2(R)).

Theorem (Dixmier-Schur Lemma) If V is an irreducible U(g)-module of countable dimension then every Z ∈ Z(g) acts as a scalar on V .

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 25 33 Theorem (Garding) If V is a “good” space, then V ∞ is dense in V .

Theorem (Harish-Chandra, Nelson) If V is a Banach space, then V ω is dense in V .

On note V ∞,(V ω), the subspace of C ∞,(C ω), vectors. Let (π, V ) be a representation of a Lie group G.

Representations Smooth and K-ﬁnite vectors Smooth vectors Geodesic ﬂow on the Poincaré plane

Deﬁnition Let (π, V ) be a representation of a Lie group G dans un TVS V . A vector v ∈ V is of class C r (r ∈ N ∪ {∞, ω}) if the map

g ∈ G 7→ π(g)v ∈ V

is C r .

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 26 33 Theorem (Garding) If V is a “good” space, then V ∞ is dense in V .

Theorem (Harish-Chandra, Nelson) If V is a Banach space, then V ω is dense in V .

On note V ∞,(V ω), the subspace of C ∞,(C ω), vectors. Let (π, V ) be a representation of a Lie group G.

Representations Smooth and K-ﬁnite vectors Smooth vectors Geodesic ﬂow on the Poincaré plane

Deﬁnition Let (π, V ) be a representation of a Lie group G dans un TVS V . A vector v ∈ V is of class C r (r ∈ N ∪ {∞, ω}) if the map

g ∈ G 7→ π(g)v ∈ V

is C r .

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 26 33 Theorem (Harish-Chandra, Nelson) If V is a Banach space, then V ω is dense in V .

Representations Smooth and K-ﬁnite vectors Smooth vectors Geodesic ﬂow on the Poincaré plane

Deﬁnition Let (π, V ) be a representation of a Lie group G dans un TVS V . A vector v ∈ V is of class C r (r ∈ N ∪ {∞, ω}) if the map

g ∈ G 7→ π(g)v ∈ V

is C r . On note V ∞,(V ω), the subspace of C ∞,(C ω), vectors. Let (π, V ) be a representation of a Lie group G. Theorem (Garding) If V is a “good” space, then V ∞ is dense in V .

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 26 33 Theorem (Harish-Chandra, Nelson) If V is a Banach space, then V ω is dense in V .

Representations Smooth and K-ﬁnite vectors Smooth vectors Geodesic ﬂow on the Poincaré plane

g ∈ G 7→ π(g)v ∈ V

is C r . On note V ∞,(V ω), the subspace of C ∞,(C ω), vectors. Let (π, V ) be a representation of a Lie group G. Theorem (Garding) If V is a “good” space, then V ∞ is dense in V .

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 26 33 Representations Smooth and K-ﬁnite vectors Smooth vectors Geodesic ﬂow on the Poincaré plane

g ∈ G 7→ π(g)v ∈ V

is C r . On note V ∞,(V ω), the subspace of C ∞,(C ω), vectors. Let (π, V ) be a representation of a Lie group G. Theorem (Garding) If V is a “good” space, then V ∞ is dense in V .

Theorem (Harish-Chandra, Nelson) If V is a Banach space, then V ω is dense in V .

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 26 33 Deﬁnition A K-ﬁnite vector for a representation (π, V ) is K-ﬁnite if its K-orbit π(K)v spans a ﬁnite dimensional vector space.

Example If V is an Hilbert space and (π, V ) a rep of G then K-ﬁnite vectors are dense in V .

The statement of the example is still valid in “good” spaces V .

On note V(K) the subspace of V of K-ﬁnite vectors.

Representations Smooth and K-ﬁnite vectors K-ﬁnite vectors

Here the appropriate setting is G a reductive linear group (a closed subgroup of GL(n, R) [of GL(n, C)] stable under g 7→ (g −1)T , [under g 7→ (g −1)†]). The K = G ∩ O(n) [ K = G ∩ O(n)].

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 27 33 Example If V is an Hilbert space and (π, V ) a rep of G then K-ﬁnite vectors are dense in V .

The statement of the example is still valid in “good” spaces V .

On note V(K) the subspace of V of K-ﬁnite vectors.

Representations Smooth and K-ﬁnite vectors K-ﬁnite vectors

Here the appropriate setting is G a reductive linear group (a closed subgroup of GL(n, R) [of GL(n, C)] stable under g 7→ (g −1)T , [under g 7→ (g −1)†]). The K = G ∩ O(n) [ K = G ∩ O(n)].

Deﬁnition A K-ﬁnite vector for a representation (π, V ) is K-ﬁnite if its K-orbit π(K)v spans a ﬁnite dimensional vector space.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 27 33 The statement of the example is still valid in “good” spaces V .

On note V(K) the subspace of V of K-ﬁnite vectors.

Representations Smooth and K-ﬁnite vectors K-ﬁnite vectors

Here the appropriate setting is G a reductive linear group (a closed subgroup of GL(n, R) [of GL(n, C)] stable under g 7→ (g −1)T , [under g 7→ (g −1)†]). The K = G ∩ O(n) [ K = G ∩ O(n)].

Deﬁnition A K-ﬁnite vector for a representation (π, V ) is K-ﬁnite if its K-orbit π(K)v spans a ﬁnite dimensional vector space.

Example If V is an Hilbert space and (π, V ) a rep of G then K-ﬁnite vectors are dense in V .

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 27 33 The statement of the example is still valid in “good” spaces V .

On note V(K) the subspace of V of K-ﬁnite vectors.

Representations Smooth and K-ﬁnite vectors K-ﬁnite vectors

Deﬁnition A K-ﬁnite vector for a representation (π, V ) is K-ﬁnite if its K-orbit π(K)v spans a ﬁnite dimensional vector space.

Example If V is an Hilbert space and (π, V ) a rep of G then K-ﬁnite vectors are dense in V .

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 27 33 On note V(K) the subspace of V of K-ﬁnite vectors.

Representations Smooth and K-ﬁnite vectors K-ﬁnite vectors

Example If V is an Hilbert space and (π, V ) a rep of G then K-ﬁnite vectors are dense in V .

The statement of the example is still valid in “good” spaces V .

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 27 33 Representations Smooth and K-ﬁnite vectors K-ﬁnite vectors

Example If V is an Hilbert space and (π, V ) a rep of G then K-ﬁnite vectors are dense in V .

The statement of the example is still valid in “good” spaces V .

On note V(K) the subspace of V of K-ﬁnite vectors.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 27 33 Example If ϵn is the character of R/2πZ deﬁned by ϵ(x) = eix then for any v ∈ V the vector ∫ n vn := Lvˆ = ϵ (θ)π(kθ)v dθ R/2piZ

n n satisﬁes π(kθ)vn = ϵ (θ)vn. Thus Lˆ(V ) is the the isotypical component of type ϵ (consider the identical embedding of Lˆ(V ) in V ).

Representations Smooth and K-ﬁnite vectors Isotypical components

Deﬁnition Fix an irreducible representation (τ, W ) of K. The isotypical component of type τ of a representation (π, V ) of G is the set of vectors v ∈ V such that for some continuous map L: W → V we have v = Lw and π(k)L = Lτ(k) for all k ∈ K.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 28 33 Representations Smooth and K-ﬁnite vectors Isotypical components

Deﬁnition Fix an irreducible representation (τ, W ) of K. The isotypical component of type τ of a representation (π, V ) of G is the set of vectors v ∈ V such that for some continuous map L: W → V we have v = Lw and π(k)L = Lτ(k) for all k ∈ K.

Example If ϵn is the character of R/2πZ deﬁned by ϵ(x) = eix then for any v ∈ V the vector ∫ n vn := Lvˆ = ϵ (θ)π(kθ)v dθ R/2piZ

n n satisﬁes π(kθ)vn = ϵ (θ)vn. Thus Lˆ(V ) is the the isotypical component of type ϵ (consider the identical embedding of Lˆ(V ) in V ).

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 28 33 Theorem (Harish-Chandra) Irreducible unitary representation of a linear reductive group are admissible.

Theorem Every K-ﬁnite vector in an admissible representation is C ∞. If V is a Banach space every K-ﬁnite vector is analytic.

Deﬁnition A representations (π, V ) of G is admissible if for any τ ∈ Kb the isotypical component Vτ ⊂ V of type τ is ﬁnite-dimensional

Representations Smooth and K-ﬁnite vectors Admissible representations

Notation The set of equivalence classes of the unitary representations of the group G is denoted by Gb.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 29 33 Theorem (Harish-Chandra) Irreducible unitary representation of a linear reductive group are admissible.

Theorem Every K-ﬁnite vector in an admissible representation is C ∞. If V is a Banach space every K-ﬁnite vector is analytic.

Representations Smooth and K-ﬁnite vectors Admissible representations

Notation The set of equivalence classes of the unitary representations of the group G is denoted by Gb.

Deﬁnition A representations (π, V ) of G is admissible if for any τ ∈ Kb the isotypical component Vτ ⊂ V of type τ is ﬁnite-dimensional

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 29 33 Theorem (Harish-Chandra) Irreducible unitary representation of a linear reductive group are admissible.

Theorem Every K-ﬁnite vector in an admissible representation is C ∞. If V is a Banach space every K-ﬁnite vector is analytic.

Representations Smooth and K-ﬁnite vectors Admissible representations

Notation The set of equivalence classes of the unitary representations of the group G is denoted by Gb.

Deﬁnition A representations (π, V ) of G is admissible if for any τ ∈ Kb the isotypical component Vτ ⊂ V of type τ is ﬁnite-dimensional

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 29 33 Theorem Every K-ﬁnite vector in an admissible representation is C ∞. If V is a Banach space every K-ﬁnite vector is analytic.

Representations Smooth and K-ﬁnite vectors Admissible representations

Theorem (Harish-Chandra) Irreducible unitary representation of a linear reductive group are admissible.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 29 33 Representations Smooth and K-ﬁnite vectors Admissible representations

Theorem (Harish-Chandra) Irreducible unitary representation of a linear reductive group are admissible.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 29 33 • A structure of U(g)-module. • A representation ρ of K. • For any k ∈ K, X ∈ g, v ∈ V : ρ(k)X ρ(k−1)v = (Ad(k)X )v.A (g, K)-module is admissible if every isotypical component of K is ﬁnite-dimensional; it is irreducible if ) and V are the only sub-modules of V .

Deﬁnition Let G be linear reductive. A (g, K)-module is a C-vector space V endowed with

Then if (π, V ) is an admissible representation of G, the vector space of K-ﬁnite vectors is a (g, K)-module

Representations Smooth and K-ﬁnite vectors (g, K)-modules and Harish-Chandra modules

Theorem Let (π, V⊕) be admissible. The space of K-ﬁnite vectors, i.e. the algebraic sum V(K) = τ∈Kb Vτ is a stable under g0, hence a U(g)-module.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 30 33 • A structure of U(g)-module. • A representation ρ of K. • For any k ∈ K, X ∈ g, v ∈ V : ρ(k)X ρ(k−1)v = (Ad(k)X )v.A (g, K)-module is admissible if every isotypical component of K is ﬁnite-dimensional; it is irreducible if ) and V are the only sub-modules of V .

Then if (π, V ) is an admissible representation of G, the vector space of K-ﬁnite vectors is a (g, K)-module

Representations Smooth and K-ﬁnite vectors (g, K)-modules and Harish-Chandra modules

Theorem Let (π, V⊕) be admissible. The space of K-ﬁnite vectors, i.e. the algebraic sum V(K) = τ∈Kb Vτ is a stable under g0, hence a U(g)-module.

Deﬁnition Let G be linear reductive. A (g, K)-module is a C-vector space V endowed with

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 30 33 • A representation ρ of K. • For any k ∈ K, X ∈ g, v ∈ V : ρ(k)X ρ(k−1)v = (Ad(k)X )v.A (g, K)-module is admissible if every isotypical component of K is ﬁnite-dimensional; it is irreducible if ) and V are the only sub-modules of V .

Then if (π, V ) is an admissible representation of G, the vector space of K-ﬁnite vectors is a (g, K)-module

Representations Smooth and K-ﬁnite vectors (g, K)-modules and Harish-Chandra modules

Theorem Let (π, V⊕) be admissible. The space of K-ﬁnite vectors, i.e. the algebraic sum V(K) = τ∈Kb Vτ is a stable under g0, hence a U(g)-module.

Deﬁnition Let G be linear reductive. A (g, K)-module is a C-vector space V endowed with • A structure of U(g)-module.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 30 33 • For any k ∈ K, X ∈ g, v ∈ V : ρ(k)X ρ(k−1)v = (Ad(k)X )v.A (g, K)-module is admissible if every isotypical component of K is ﬁnite-dimensional; it is irreducible if ) and V are the only sub-modules of V .

Representations Smooth and K-ﬁnite vectors (g, K)-modules and Harish-Chandra modules

Deﬁnition Let G be linear reductive. A (g, K)-module is a C-vector space V endowed with • A structure of U(g)-module. • A representation ρ of K.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 30 33 Then if (π, V ) is an admissible representation of G, the vector space of K-ﬁnite vectors is a (g, K)-module

Representations Smooth and K-ﬁnite vectors (g, K)-modules and Harish-Chandra modules

Deﬁnition Let G be linear reductive. A (g, K)-module is a C-vector space V endowed with • A structure of U(g)-module. • A representation ρ of K. • For any k ∈ K, X ∈ g, v ∈ V : ρ(k)X ρ(k−1)v = (Ad(k)X )v.A (g, K)-module is admissible if every isotypical component of K is ﬁnite-dimensional; it is irreducible if ) and V are the only sub-modules of V .

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 30 33 Then if (π, V ) is an admissible representation of G, the vector space of K-ﬁnite vectors is a (g, K)-module

Representations Smooth and K-ﬁnite vectors (g, K)-modules and Harish-Chandra modules

Deﬁnition Let G be linear reductive. A (g, K)-module is a C-vector space V endowed with • A structure of U(g)-module. • A representation ρ of K. • For any k ∈ K, X ∈ g, v ∈ V : ρ(k)X ρ(k−1)v = (Ad(k)X )v.A (g, K)-module is admissible if every isotypical component of K is ﬁnite-dimensional; it is irreducible if ) and V are the only sub-modules of V .

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 30 33 Representations Smooth and K-ﬁnite vectors (g, K)-modules and Harish-Chandra modules

Deﬁnition Let G be linear reductive. A (g, K)-module is a C-vector space V endowed with • A structure of U(g)-module. • A representation ρ of K. • For any k ∈ K, X ∈ g, v ∈ V : ρ(k)X ρ(k−1)v = (Ad(k)X )v.A (g, K)-module is admissible if every isotypical component of K is ﬁnite-dimensional; it is irreducible if ) and V are the only sub-modules of V .

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 30 33 The Casimir operator, up to a rescaling, is the element

1 2 Ω := − (H2 + H⊥ − Θ2) = ∆ + Θ2/2. (7) 4 which may be rewritten as 1 i 1 i Ω = − κ2 + κ + η+η− = − κ2 − κ + η−η+. 4 2 4 2

Representations Irreducible (g, K)-modules for SL2(R)

A suitable basis for sl2(R) Raising and lowering elements

We choose as generators for sl(2, C) = sl(2, R) ⊗ C the elements [ ] [ ] [ ] 0 −1 + 1 − ⊥ 1 1 −i − 1 ⊥ 1 1 i κ = 1 0 , η = 2 (H iH ) = 2 −i −1 , η = 2 (H + iH ) = 2 i −1 which satisfy the following commutations rules:

[κ, η] = 2iη, [η+, η−] = −iκ. (6)

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 31 33 Representations Irreducible (g, K)-modules for SL2(R)

A suitable basis for sl2(R) Raising and lowering elements

We choose as generators for sl(2, C) = sl(2, R) ⊗ C the elements [ ] [ ] [ ] 0 −1 + 1 − ⊥ 1 1 −i − 1 ⊥ 1 1 i κ = 1 0 , η = 2 (H iH ) = 2 −i −1 , η = 2 (H + iH ) = 2 i −1 which satisfy the following commutations rules:

[κ, η] = 2iη, [η+, η−] = −iκ. (6)

The Casimir operator, up to a rescaling, is the element

1 2 Ω := − (H2 + H⊥ − Θ2) = ∆ + Θ2/2. (7) 4 which may be rewritten as 1 i 1 i Ω = − κ2 + κ + η+η− = − κ2 − κ + η−η+. 4 2 4 2

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 31 33 + − • η vn ∈ Vn+2, and η vn ∈ Vn−2. • Set for all k ∈ N + k − k vn+2k := (η ) vn, vn−2k := (η ) vn,, V˜ = spanC{vn+2j }

Then V˜ = V and the spectrum of κ (the set of j such that Vj ≠ 0) is an interval of integers with the same parity as n. In fact for any v ( ) ( ℓ ) + − − 1 2 1 − + − 1 2 − 1 η η vℓ = λ 4 ℓ + 2 ℓ vℓ, η η vℓ = λ 4 ℓ 2 ℓ vℓ

([ ]) cos θ − sin θ iθ Let χ be the fundamental character of K: χ sin θ cos θ = e and let Vn be the isotypical subspace of V corresponding to the character χn: then

κv = inv, ∀v ∈ Vn

Suppose ∃vn ∈ Vn with vn ≠ 0. Then

Representations Irreducible (g, K)-modules for SL2(R)

Irreducible (g, K)-modules for SL2(R) Structure

Let g = sl(2, R) ⊗ C, K = SO(2). Suppose V is an irreducible admissible (g, K)-module. By the Dixmier-Schur Lemma the Casimir operator acts as a scalar λ.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 32 33 + − • η vn ∈ Vn+2, and η vn ∈ Vn−2. • Set for all k ∈ N + k − k vn+2k := (η ) vn, vn−2k := (η ) vn,, V˜ = spanC{vn+2j }

Then V˜ = V and the spectrum of κ (the set of j such that Vj ≠ 0) is an interval of integers with the same parity as n. In fact for any v ( ) ( ℓ ) + − − 1 2 1 − + − 1 2 − 1 η η vℓ = λ 4 ℓ + 2 ℓ vℓ, η η vℓ = λ 4 ℓ 2 ℓ vℓ

Suppose ∃vn ∈ Vn with vn ≠ 0. Then

Representations Irreducible (g, K)-modules for SL2(R)

Irreducible (g, K)-modules for SL2(R) Structure

Let g = sl(2, R) ⊗ C, K = SO(2). Suppose V is an irreducible admissible (g, K)-module. By the Dixmier-Schur Lemma the Casimir operator acts as a scalar λ. ([ ]) cos θ − sin θ iθ Let χ be the fundamental character of K: χ sin θ cos θ = e and let Vn be the isotypical subspace of V corresponding to the character χn: then

κv = inv, ∀v ∈ Vn

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 32 33 Then V˜ = V and the spectrum of κ (the set of j such that Vj ≠ 0) is an interval of integers with the same parity as n. In fact for any v ( ) ( ℓ ) + − − 1 2 1 − + − 1 2 − 1 η η vℓ = λ 4 ℓ + 2 ℓ vℓ, η η vℓ = λ 4 ℓ 2 ℓ vℓ

• Set for all k ∈ N + k − k vn+2k := (η ) vn, vn−2k := (η ) vn,, V˜ = spanC{vn+2j }

Representations Irreducible (g, K)-modules for SL2(R)

Irreducible (g, K)-modules for SL2(R) Structure

Let g = sl(2, R) ⊗ C, K = SO(2). Suppose V is an irreducible admissible (g, K)-module. By the Dixmier-Schur Lemma the Casimir operator acts as a scalar λ. ([ ]) cos θ − sin θ iθ Let χ be the fundamental character of K: χ sin θ cos θ = e and let Vn be the isotypical subspace of V corresponding to the character χn: then

κv = inv, ∀v ∈ Vn

Suppose ∃vn ∈ Vn with vn ≠ 0. Then + − • η vn ∈ Vn+2, and η vn ∈ Vn−2.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 32 33 Then V˜ = V and the spectrum of κ (the set of j such that Vj ≠ 0) is an interval of integers with the same parity as n. In fact for any v ( ) ( ℓ ) + − − 1 2 1 − + − 1 2 − 1 η η vℓ = λ 4 ℓ + 2 ℓ vℓ, η η vℓ = λ 4 ℓ 2 ℓ vℓ

Representations Irreducible (g, K)-modules for SL2(R)

Irreducible (g, K)-modules for SL2(R) Structure

Let g = sl(2, R) ⊗ C, K = SO(2). Suppose V is an irreducible admissible (g, K)-module. By the Dixmier-Schur Lemma the Casimir operator acts as a scalar λ. ([ ]) cos θ − sin θ iθ Let χ be the fundamental character of K: χ sin θ cos θ = e and let Vn be the isotypical subspace of V corresponding to the character χn: then

κv = inv, ∀v ∈ Vn

Suppose ∃vn ∈ Vn with vn ≠ 0. Then + − • η vn ∈ Vn+2, and η vn ∈ Vn−2. • Set for all k ∈ N + k − k vn+2k := (η ) vn, vn−2k := (η ) vn,, V˜ = spanC{vn+2j }

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 32 33 Then V˜ = V and the spectrum of κ (the set of j such that Vj ≠ 0) is an interval of integers with the same parity as n. In fact for any v ( ) ( ℓ ) + − − 1 2 1 − + − 1 2 − 1 η η vℓ = λ 4 ℓ + 2 ℓ vℓ, η η vℓ = λ 4 ℓ 2 ℓ vℓ

Representations Irreducible (g, K)-modules for SL2(R)

Irreducible (g, K)-modules for SL2(R) Structure

κv = inv, ∀v ∈ Vn

Suppose ∃vn ∈ Vn with vn ≠ 0. Then + − • η vn ∈ Vn+2, and η vn ∈ Vn−2. • Set for all k ∈ N + k − k vn+2k := (η ) vn, vn−2k := (η ) vn,, V˜ = spanC{vn+2j }

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 32 33 Representations Irreducible (g, K)-modules for SL2(R)

Irreducible (g, K)-modules for SL2(R) Structure

κv = inv, ∀v ∈ Vn

Suppose ∃vn ∈ Vn with vn ≠ 0. Then + − • η vn ∈ Vn+2, and η vn ∈ Vn−2. • Set for all k ∈ N + k − k vn+2k := (η ) vn, vn−2k := (η ) vn,, V˜ = spanC{vn+2j }

Then V˜ = V and the spectrum of κ (the set of j such that Vj ≠ 0) is an interval of integers with the same parity as n. In fact for any v ( ) ( ℓ ) + − − 1 2 1 − + − 1 2 − 1 η η vℓ = λ 4 ℓ + 2 ℓ vℓ, η η vℓ = λ 4 ℓ 2 ℓ vℓ

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 32 33 • (ℓ − 1)2 − 1 {ℓ, ℓ + 2, . . . , ℓ + 2j,... }, ℓ ≥ 1 and Ω = . 4 + This (g, K)-module is denoted Dℓ−1. • (ℓ + 1)2 − 1 {−ℓ, −ℓ − 2,..., −ℓ − 2j,... }, ℓ ≥ 1 and Ω = . 4 − This (g, K)-module is denoted Dℓ−1. • S = 2Z or S = 2Z + 1. ̸ (m−1)2−1 ∈ In this case λ = 4 for any m S.

Representations Irreducible (g, K)-modules for SL2(R)

Irreducible (g, K)-modules for SL2(R) Classiﬁcation

Four possibilities for the spectrum of −iκ • (ℓ + 1)2 − 1 {−ℓ, −ℓ + 2, . . . , ℓ − 2, ℓ}, and Ω = . 4 This (g, K)-module is of dimension ℓ + 1 and denoted F (ℓ + 1).

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 33 33 • (ℓ + 1)2 − 1 {−ℓ, −ℓ − 2,..., −ℓ − 2j,... }, ℓ ≥ 1 and Ω = . 4 − This (g, K)-module is denoted Dℓ−1. • S = 2Z or S = 2Z + 1. ̸ (m−1)2−1 ∈ In this case λ = 4 for any m S.

Representations Irreducible (g, K)-modules for SL2(R)

Irreducible (g, K)-modules for SL2(R) Classiﬁcation

Four possibilities for the spectrum of −iκ • (ℓ + 1)2 − 1 {−ℓ, −ℓ + 2, . . . , ℓ − 2, ℓ}, and Ω = . 4 This (g, K)-module is of dimension ℓ + 1 and denoted F (ℓ + 1). • (ℓ − 1)2 − 1 {ℓ, ℓ + 2, . . . , ℓ + 2j,... }, ℓ ≥ 1 and Ω = . 4 + This (g, K)-module is denoted Dℓ−1.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 33 33 • S = 2Z or S = 2Z + 1. ̸ (m−1)2−1 ∈ In this case λ = 4 for any m S.

Representations Irreducible (g, K)-modules for SL2(R)

Irreducible (g, K)-modules for SL2(R) Classiﬁcation

Four possibilities for the spectrum of −iκ • (ℓ + 1)2 − 1 {−ℓ, −ℓ + 2, . . . , ℓ − 2, ℓ}, and Ω = . 4 This (g, K)-module is of dimension ℓ + 1 and denoted F (ℓ + 1). • (ℓ − 1)2 − 1 {ℓ, ℓ + 2, . . . , ℓ + 2j,... }, ℓ ≥ 1 and Ω = . 4 + This (g, K)-module is denoted Dℓ−1. • (ℓ + 1)2 − 1 {−ℓ, −ℓ − 2,..., −ℓ − 2j,... }, ℓ ≥ 1 and Ω = . 4 − This (g, K)-module is denoted Dℓ−1.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 33 33 Representations Irreducible (g, K)-modules for SL2(R)

Irreducible (g, K)-modules for SL2(R) Classiﬁcation

Four possibilities for the spectrum of −iκ • (ℓ + 1)2 − 1 {−ℓ, −ℓ + 2, . . . , ℓ − 2, ℓ}, and Ω = . 4 This (g, K)-module is of dimension ℓ + 1 and denoted F (ℓ + 1). • (ℓ − 1)2 − 1 {ℓ, ℓ + 2, . . . , ℓ + 2j,... }, ℓ ≥ 1 and Ω = . 4 + This (g, K)-module is denoted Dℓ−1. • (ℓ + 1)2 − 1 {−ℓ, −ℓ − 2,..., −ℓ − 2j,... }, ℓ ≥ 1 and Ω = . 4 − This (g, K)-module is denoted Dℓ−1. • S = 2Z or S = 2Z + 1. ̸ (m−1)2−1 ∈ In this case λ = 4 for any m S.

L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 33 33