The Heat Equation, the Segal-Bargmann Transform and Generalizations

The Heat equation, the Segal-Bargmann transform and generalizations

Based on joint work with

• B. Krötz

• B. Ørsted

• H. Schlichtkrull

• R. Stanton

- p. 1/54 Organizations

1. The heat equation on Rn. 2. The Fock space and the Segal-Bargmann Transform. 3. Remarks and Comments. 4. Generalizations and the Restriction Principle. 5. Structure Theory. 6. Spherical Functions and the Fourier Transform. 7. The Crown and the Heat Kernel. 8. The Abel Transform and the Heat Kernel. 9. The Faraut-Gutzmer Formula and the Orbital Integral. 10. The Image of the Segal-Bargmann transform on G/K. 11. The K-invariant case (more than one section)

- p. 2/54 1. The heat equation on Rn

◮ Consider the Laplace operator

n ∂2 ∆= ∂x2 j=1 j X on Rn.

- p. 3/54 1. The heat equation on Rn

◮ Consider the Laplace operator

n ∂2 ∆= ∂x2 j=1 j X on Rn. ◮ The heat equation is the Cauchy problem

∆u(x, t) = ∂tu(x, t) lim u(x, t) = f(x) t 0+ → where we can take f L2(Rn), a distribution, a hyperfunction, or from another class of analytic∈ objects.

- p. 3/54 ◮ Formally we write u(x, t)= et∆f(x)

- p. 4/54 ◮ Formally we write

t∆ u(x, t)= e f(x)= Htf(x)

- p. 4/54 ◮ Formally we write

t∆ u(x, t)= e f(x)= Htf(x)

t∆ where e = Ht t 0 is the heat semigroup . It is a linear map { } ≥ H : L2(Rn) L2(Rn), but also a smoothing operator as we will see. t →

- p. 4/54 ◮ Formally we write

t∆ u(x, t)= e f(x)= Htf(x)

t∆ where e = Ht t 0 is the heat semigroup . It is a linear map { } ≥ H : L2(Rn) L2(Rn), but also a smoothing operator as we will see. t → ◮ To actually solve the equation, we proceed by applying the Fourier transform

n/2 ix λ f (f)= fˆ , λ (2π)− f(x)e− · dx 7→ F 7→ Z using that (∆f)(λ)= λ 2fˆ(λ) F −| | and get the simple differential equation for t uˆ(λ, t): 7→

- p. 4/54 ◮ Formally we write

t∆ u(x, t)= e f(x)= Htf(x)

n/2 ix λ f (f)= fˆ , λ (2π)− f(x)e− · dx 7→ F 7→ Z using that (∆f)(λ)= λ 2fˆ(λ) F −| | and get the simple differential equation for t uˆ(λ, t): 7→ ∂ uˆ(λ, t)= λ 2uˆ(λ, t) , uˆ(λ, 0) = fˆ(λ) . t −| |

- p. 4/54 t λ 2 ◮ The solution to this differential equation is uˆ(λ, t)= fˆ(λ)e− | |

- p. 5/54 t λ 2 ◮ The solution to this differential equation is uˆ(λ, t)= fˆ(λ)e− | | and we get the Fourier Transform Formula for the solution:

n/2 λ 2t iλ x Htf(x)=(2π)− fˆ(λ)e−| | e · dλ (0.1) Z

- p. 5/54 t λ 2 ◮ The solution to this differential equation is uˆ(λ, t)= fˆ(λ)e− | | and we get the Fourier Transform Formula for the solution:

n/2 λ 2t iλ x Htf(x)=(2π)− fˆ(λ)e−| | e · dλ (0.1) Z

◮ The heat kernel ht is the solution to the heat equation with f = δ0. Using n/2 that the δ-distribution has Fourier transform δˆ0(λ)=(2π)− we get

n λ 2t ix λ Htδ0(x)= ht(x) = (2π)− e−| | e · dλ Z =

- p. 5/54 t λ 2 ◮ The solution to this differential equation is uˆ(λ, t)= fˆ(λ)e− | | and we get the Fourier Transform Formula for the solution:

n/2 λ 2t iλ x Htf(x)=(2π)− fˆ(λ)e−| | e · dλ (0.1) Z

◮ The heat kernel ht is the solution to the heat equation with f = δ0. Using n/2 that the δ-distribution has Fourier transform δˆ0(λ)=(2π)− we get

n λ 2t ix λ Htδ0(x)= ht(x) = (2π)− e−| | e · dλ Z n/2 x 2/4t = (4πt)− e−| |

- p. 5/54 t λ 2 ◮ The solution to this differential equation is uˆ(λ, t)= fˆ(λ)e− | | and we get the Fourier Transform Formula for the solution:

n/2 λ 2t iλ x Htf(x)=(2π)− fˆ(λ)e−| | e · dλ (0.1) Z

◮ The heat kernel ht is the solution to the heat equation with f = δ0. Using n/2 that the δ-distribution has Fourier transform δˆ0(λ)=(2π)− we get

n λ 2t ix λ Htδ0(x)= ht(x) = (2π)− e−| | e · dλ Z n/2 x 2/4t = (4πt)− e−| |

n + ◮ It is clear from this formula, that R x ht(x) R has a holomorphic extension to Cn given by ∋ 7→ ∈

n/2 z2/4t 2 2 2 ht(z)=(4πt)− e− , z = z1 + ... + zn .

- p. 5/54 ◮ Note ∂ (f h )= f (∂ h )= f (∆h )=∆(f h ) t ∗ t ∗ t t ∗ t ∗ t and lim f ht = f δ0 = f t 0+ ∗ ∗ →

- p. 6/54 ◮ Note ∂ (f h )= f (∂ h )= f (∆h )=∆(f h ) t ∗ t ∗ t t ∗ t ∗ t and lim f ht = f δ0 = f t 0+ ∗ ∗ → and we get the heat kernel formula for the solution:

n/2 (x y)2/(4t) H f(x)= f h (x)=(4πt)− f(y)e− − dy (0.2) t ∗ t Z

- p. 6/54 ◮ Note ∂ (f h )= f (∂ h )= f (∆h )=∆(f h ) t ∗ t ∗ t t ∗ t ∗ t and lim f ht = f δ0 = f t 0+ ∗ ∗ → and we get the heat kernel formula for the solution:

n/2 (x y)2/(4t) H f(x)= f h (x)=(4πt)− f(y)e− − dy (0.2) t ∗ t Z

◮ We read of from (0.1) or (0.2) that the function x Htf(x) has a holomorphic extension to Cn: 7→

n/2 (z y)2/4t Htf(z) = (4πt)− f(y)e− − dy Z =

- p. 6/54 ◮ Note ∂ (f h )= f (∂ h )= f (∆h )=∆(f h ) t ∗ t ∗ t t ∗ t ∗ t and lim f ht = f δ0 = f t 0+ ∗ ∗ → and we get the heat kernel formula for the solution:

n/2 (x y)2/(4t) H f(x)= f h (x)=(4πt)− f(y)e− − dy (0.2) t ∗ t Z

◮ We read of from (0.1) or (0.2) that the function x Htf(x) has a holomorphic extension to Cn: 7→

n/2 (z y)2/4t Htf(z) = (4πt)− f(y)e− − dy Z n/2 λ2t iz λ = (2π)− fˆ(λ)e− e · dλ Z

- p. 6/54 ◮ Note ∂ (f h )= f (∂ h )= f (∆h )=∆(f h ) t ∗ t ∗ t t ∗ t ∗ t and lim f ht = f δ0 = f t 0+ ∗ ∗ → and we get the heat kernel formula for the solution:

n/2 (x y)2/(4t) H f(x)= f h (x)=(4πt)− f(y)e− − dy (0.2) t ∗ t Z

◮ We read of from (0.1) or (0.2) that the function x Htf(x) has a holomorphic extension to Cn: 7→

n/2 (z y)2/4t Htf(z) = (4πt)− f(y)e− − dy Z n/2 λ2t iz λ = (2π)− fˆ(λ)e− e · dλ Z where z λ = n z λ . · j=1 j j P - p. 6/54 The map H : L2(Rn) (Cn) is the Segal-Bargmann transform. t → O

- p. 7/54 The map H : L2(Rn) (Cn) is the Segal-Bargmann transform. t → O ◮ Note, that we have really only used the following:

- p. 7/54 The map H : L2(Rn) (Cn) is the Segal-Bargmann transform. t → O ◮ Note, that we have really only used the following: 1. We have a Fourier transform that such that (∆g)(λ)= λ 2 (g)(λ) F −| | F to derive the Fourier transform form for the solution and to ﬁnd an explicit expression for the heat kernel. iz λ 2. In using (0.1) that the exponential function λ eλ(z)= e · grows much slower than 7→ λ 2t λ fˆ(λ)e−| | 7→ to show that the solution extends to a holomorphic function on Cn.

- p. 7/54 2. The Fock space and the Segal-Bargmann Transform

- p. 8/54 2. The Fock space and the Segal-Bargmann Transform

◮ We will now describe the image of the Segal-Bargmann transform. For that we deﬁne a positive weight function by

Rn n/2 y2/2t ωt (y)= ωt(x)=(2πt)− e− = ht/2(y)

- p. 8/54 2. The Fock space and the Segal-Bargmann Transform

◮ We will now describe the image of the Segal-Bargmann transform. For that we deﬁne a positive weight function by

Rn n/2 y2/2t ωt (y)= ωt(x)=(2πt)− e− = ht/2(y) and a measure on Cn by

dµt(x + iy)= ωt(y) dxdy .

- p. 8/54 2. The Fock space and the Segal-Bargmann Transform

◮ We will now describe the image of the Segal-Bargmann transform. For that we deﬁne a positive weight function by

Rn n/2 y2/2t ωt (y)= ωt(x)=(2πt)− e− = ht/2(y) and a measure on Cn by

dµt(x + iy)= ωt(y) dxdy . Set

Cn Cn 2 2 t( )= F ( ) F t := F (x + iy) dµt < . H { ∈ O | k k Cn | | ∞} Z

- p. 8/54 Theorem 0.1 (Segal-Bargmann, 1956-1978/1961, . . . ). The following holds: 1. (Cn) is a Hilbert space with continuous point evaluation, i.e., the maps Ht (Cn) F ev (F ) = F (z) C , z Cn Ht ∋ 7→ z ∈ ∈ are continuous. In particular, with L F (x) = F (x y) and y − −n/2 −(z−w¯)2/8t Kw(z) = K(z,w) := Ht(Lw¯ht)(z)=(8πt) e ,

we have K (Cn) and F (w)=(F, K ) for all F (Cn), i.e., w ∈ Ht w ∈ Ht K(z,w) is the reproducing kernel for (Cn) Ht 2. H : L2(Rn) (Cn) is an unitary isomorphism. t → Ht n 3. If f S(R ), then f(x) = Htf(x + iy)ht(y) dy. ∈ Rn Z

- p. 9/54 Few words on the proof, but note that I will not prove the surjectivity or that Ht is an isometry.

- p. 10/54 Few words on the proof, but note that I will not prove the surjectivity or that Ht is an isometry. ◮ For (2) we simply calculate the norm on the right hand side. Note that all functions involved are positive, so that there is no problem using Fubini’s Theorem:

2 R y2/2t 2 2y λ y2/2t c H f(x + iy) dx e− dy = c H f(λ) e− · e− dλdy | t | | t | ZZ ZZ = d

2 R y2/2t 2 2y λ y2/2t c H f(x + iy) dx e− dy = c H f(λ) e− · e− dλdy | t | | t | ZZ ZZ 2 2 (2tλ2+2y λ+ y ) = c fdˆ(λ) e− · 2t dλdy | | ZZ =

2 R y2/2t 2 2y λ y2/2t c H f(x + iy) dx e− dy = c H f(λ) e− · e− dλdy | t | | t | ZZ ZZ 2 2 (2tλ2+2y λ+ y ) = c fdˆ(λ) e− · 2t dλdy | | ZZ 2 (y+2tλ)2/2t = fˆ(λ) c e− dy dλ | | Z Z =

2 R y2/2t 2 2y λ y2/2t c H f(x + iy) dx e− dy = c H f(λ) e− · e− dλdy | t | | t | ZZ ZZ 2 2 (2tλ2+2y λ+ y ) = c fdˆ(λ) e− · 2t dλdy | | ZZ 2 (y+2tλ)2/2t = fˆ(λ) c e− dy dλ | | Z Z = fˆ(λ) 2 dλ | | Z =

- p. 10/54 n/2 ◮ The proof of the inversion formula is similar. Let c = (4πt)− :

(0.1) tλ2 i(x+iy) λ Htf(x + iy)ht(y) dy = e− fˆ(λ)e · dλ ht(y) dy Z Z Z =

- p. 11/54 n/2 ◮ The proof of the inversion formula is similar. Let c = (4πt)− :

(0.1) tλ2 i(x+iy) λ Htf(x + iy)ht(y) dy = e− fˆ(λ)e · dλ ht(y) dy Z Z Z tλ2 ix λ y λ y2/4t = c e− fˆ(λ)e · e− · e− dy dλ Z Z =

- p. 11/54 n/2 ◮ The proof of the inversion formula is similar. Let c = (4πt)− :

(0.1) tλ2 i(x+iy) λ Htf(x + iy)ht(y) dy = e− fˆ(λ)e · dλ ht(y) dy Z Z Z tλ2 ix λ y λ y2/4t = c e− fˆ(λ)e · e− · e− dy dλ Z Z 2 ix λ (2tλ+y) = c fˆ(λ)e · e− 4t dy dλ Z Z =

- p. 11/54 n/2 ◮ The proof of the inversion formula is similar. Let c = (4πt)− :

(0.1) tλ2 i(x+iy) λ Htf(x + iy)ht(y) dy = e− fˆ(λ)e · dλ ht(y) dy Z Z Z tλ2 ix λ y λ y2/4t = c e− fˆ(λ)e · e− · e− dy dλ Z Z 2 ix λ (2tλ+y) = c fˆ(λ)e · e− 4t dy dλ Z Z ix λ = fˆ(λ)e · dλ Z =

- p. 11/54 n/2 ◮ The proof of the inversion formula is similar. Let c = (4πt)− :

(0.1) tλ2 i(x+iy) λ Htf(x + iy)ht(y) dy = e− fˆ(λ)e · dλ ht(y) dy Z Z Z tλ2 ix λ y λ y2/4t = c e− fˆ(λ)e · e− · e− dy dλ Z Z 2 ix λ (2tλ+y) = c fˆ(λ)e · e− 4t dy dλ Z Z ix λ = fˆ(λ)e · dλ Z = f(x) .

- p. 11/54 ◮ For the formula for the reproducing kernel we – again – assume that Ht is an unitary isomorphism.

- p. 12/54 ◮ For the formula for the reproducing kernel we – again – assume that Ht is an unitary isomorphism. ◮ So let F (Cn) and f L2(Rn) such that F = H f. Then ∈ Ht ∈ t F (w) = Htf(w) = f(x)h (x w) dx h even t − t Z = (f, Lw¯ht)L2

= (Htf, Ht(Lw¯ht)) t Ht unitary H

= (F, Ht(Lw¯ht)) t . H

= (Htf, Ht(Lw¯ht)) t Ht unitary H

= (F, Ht(Lw¯ht)) t . H ◮ Thus

K(z, w) = Ht(λ(w ¯)ht)(z) = (λ(w ¯)h ) h (z) t ∗ t = h h (z w¯) t ∗ t − = h (z w¯) the semigroup property. 2t −

- p. 12/54 3. Remarks and Comments

- p. 13/54 3. Remarks and Comments

n n ◮ Note first of all, that we can interpret C as the cotangent bundle T ∗(R ), Rn where the y-variable in z = x + iy is an element of Tx∗ . Hence the Segal-Bargmann transform is some kind of quantization. ◮ Note also, that in the definition of the norm and in the inversion formula Rn we only weight the cotangent variable y Tx∗( ) and the weights are given by the heat kernel. ∈ ◮ There are other versions of the Segal-Bargmann transform in the literature. In particular, for the physics and infinite dimensional analysis, as well as in the original works, the space L2(Rn) was replaced by the weighted L2-space L2(Rn, dνn), where

n dν (x)= ht(x)dx the heat kernel measure on Rn.

- p. 13/54 3. Remarks and Comments

n dν (x)= ht(x)dx the heat kernel measure on Rn. On the image side the measure is then

n n z 2/2t dσt (z)=(2πt)− e−| | dxdy

- p. 13/54 2 n ◮ Denote the corresponding space of L -holomorphic functions by t(C ). It is still holds, that the Segal-Bargmann transform F

L2(Rn, dν) f f h (Cn) ∋ 7→ ∗ t ∈Ft is an unitary isomorphism. This normalization has several advantages.

- p. 14/54 2 n ◮ Denote the corresponding space of L -holomorphic functions by t(C ). It is still holds, that the Segal-Bargmann transform F

L2(Rn, dν) f f h (Cn) ∋ 7→ ∗ t ∈Ft is an unitary isomorphism. This normalization has several advantages. ◮ It is very easy to ﬁnd the reproducing kernel for the space (Cn). It is Ft n z w/¯ 2t Kt(z, w)=(2πt)− e · .

- p. 14/54 2 n ◮ Denote the corresponding space of L -holomorphic functions by t(C ). It is still holds, that the Segal-Bargmann transform F

α ◮ For the proof, one notice that the polynomials ζα(z)= z are dense in n α (C ) and therefore one only has to prove z =(ζα, Kz) for all α, and this reducesF to a simple one-dimensional integral. ◮ Connection to the theory of orthogonal polynomials: There are constants cα (easy to calculate) such that cαζα α N0 is an orthogonal basis for (Cn, dσ ) and there are constants{ (again} ∈ easy to calculate) such that Ht t Ht∗(ζα)= dαhα, where hα is the Hermite polynomial.

- p. 14/54 ◮ Furthermore, the measures on both sides are probability measures, and n-fold product of the one-dimensional measures in the coordinates.

- p. 15/54 ◮ Furthermore, the measures on both sides are probability measures, and n-fold product of the one-dimensional measures in the coordinates. ◮ We can take the limit as n . Consider the projections n n n 1 → ∞ pr : R R − . This gives us isometric maps maps → n 2 n 1 n 1 2 n n n pr : L (R − , dν − ) L (R , dν ) , f f pr ∗ → 7→ ◦ and we have a sequence of commutative diagrams

prn prn+1 2 n 1 n 1 ∗ / 2 n n ∗ / 2 L (R − , dν − ) L (R , dν ) L (R∞, dν∞) ···→ ··· n−1 n ∞ Ht Ht Ht n n pr pr +1 n 1 n 1 ∗ / n n ∗ / (C − , dσ − ) t(C , dσ ) t(C∞, dσ∞) ···→ Ht t H t ··· H t

- p. 15/54 ◮ Sometimes, in particular studying the Schrödinger representation of the Heisenberg group, one uses the Segal-Bargmann transform

S : L2(Rn, dx) (Cn) . t →Ft n 2 n One of the idea is, that t(C ) is a much simpler space than L (R ) to work with. Also, the canonicalF commutation rules, the creation operator and the annulation operator have simpler form in (Cn). Ft

- p. 16/54 ◮ Sometimes, in particular studying the Schrödinger representation of the Heisenberg group, one uses the Segal-Bargmann transform

S : L2(Rn, dx) (Cn) . t →Ft n 2 n One of the idea is, that t(C ) is a much simpler space than L (R ) to work with. Also, the canonicalF commutation rules, the creation operator and the n annulation operator have simpler form in t(C ). In this case - as I will prove later - the Segal-BargmannF transform is given by: 2 n/4 1 (y2 2xy+ x ) St(f)(z)=(πt)− f(y)e− 2t − 2 dy. Z

- p. 16/54 ◮ Sometimes, in particular studying the Schrödinger representation of the Heisenberg group, one uses the Segal-Bargmann transform

- p. 16/54 4. Generalizations and the Restriction Principle.

- p. 17/54 4. Generalizations and the Restriction Principle.

◮ Before I prove the unitarity of the Segal-Bargmann transform St let me make some general remarks about the underlying idea and general principle behind a transform like St.

- p. 17/54 4. Generalizations and the Restriction Principle.

◮ Before I prove the unitarity of the Segal-Bargmann transform St let me make some general remarks about the underlying idea and general principle behind a transform like St.

◮ Let MC be a complex analytic manifold (i.e., MC = Cn) and M MC a totally real analytic submanifold. Thus the restriction map ⊂

(MC) F F (M) O ∋ 7→ |M ∈ A is injective.

- p. 17/54 4. Generalizations and the Restriction Principle.

◮ Before I prove the unitarity of the Segal-Bargmann transform St let me make some general remarks about the underlying idea and general principle behind a transform like St.

◮ Let MC be a complex analytic manifold (i.e., MC = Cn) and M MC a totally real analytic submanifold. Thus the restriction map ⊂

(MC) F F (M) O ∋ 7→ |M ∈ A is injective.

◮ Let (MC) be a Hilbert space of holomorphic function on MC such that the point-evaluationF maps F F (w) are continuous and hence given by 7→ the inner product with an element K (MC): w ∈F F (MC) : F (w)=(F, K ) ∀ ∈F w

- p. 17/54 ◮ The function K : MC MC C, K(z, w)= K (z) is reproducing kernel × → w of (MC). It satisﬁes: 1. FK is holomorphic in the ﬁrst variable and anti-holomorphic in the second variable. 2. K(z, w)= K(w, z) because

Kw(z)=(Kw, Kz)= (Kz, Kw)= Kz(w)= K(w, z) . 3. K 2 = K(w, w) 0. k wk ≥

- p. 18/54 ◮ The function K : MC MC C, K(z, w)= K (z) is reproducing kernel × → w of (MC). It satisﬁes: 1. FK is holomorphic in the ﬁrst variable and anti-holomorphic in the second variable. 2. K(z, w)= K(w, z) because

Kw(z)=(Kw, Kz)= (Kz, Kw)= Kz(w)= K(w, z) . 3. K 2 = K(w, w) 0. k wk ≥ ◮ Furthermore, the linear hull of Kx x M is dense in (MC) and hence: { } ∈ F the reproducing kernel determines (MC), F knowing K(z, w) is the same as knowing (MC). F ◮ Assume that F is orthogonal to the linear span of Kx x M . Then { } ∈ F (x)=(F, Kx) = 0 for all x M and hence F M = 0. As M is a totally real submanifold, it follows that F∈= 0. |

- p. 18/54 We now make the following assumption: There exists a measure µ on M and a holomorphic function D : MC C, D > 0, D(z) = 0, such that → |M 6

- p. 19/54 We now make the following assumption: There exists a measure µ on M and a holomorphic function D : MC C, D > 0, D(z) = 0, such that → |M 6 1. For all F (MC) we have ∈F R(F ):=(DF ) L2(M,µ) . |M ∈ 2 2. R( (MC)) is dense in L (M) (can be dropped, but then we have only a partialF isometry later). 2 3. R : (MC) L (M) is a closed operator. F →

R∗ = U√RR∗ 2 where U : L (M,dµ) (MC) is an unitary isomorphism by deﬁnition. →F

R∗ = U√RR∗ 2 where U : L (M,dµ) (MC) is an unitary isomorphism by deﬁnition. →F ◮ We call U the generalized Segal-Bargmann transform .

- p. 19/54 ◮ Note the following:

R∗f(w)=(R∗f, Kw) =(f,RKw)= f(y)D(y)K(w, y) dx F ZM and hence

RR∗f(x)= f(y)D(x)D(y)K(y, x) dµ(y) . ZM Hence RR∗ is always an integral operator.

- p. 20/54 ◮ Note the following:

R∗f(w)=(R∗f, Kw) =(f,RKw)= f(y)D(y)K(w, y) dx F ZM and hence

RR∗f(x)= f(y)D(x)D(y)K(y, x) dµ(y) . ZM Hence RR∗ is always an integral operator.

◮ Furthermore, by multiplying by U ∗, and then using that √RR∗ is self-adjoint, we get the following formula for RU and then U:

U ∗R∗ = RU = √RR∗ or 1 Uf(x)= D(x)− √RR∗(f)(x) .

holomorphic | {z }

- p. 20/54 ◮ Note the following:

R∗f(w)=(R∗f, Kw) =(f,RKw)= f(y)D(y)K(w, y) dx F ZM and hence

RR∗f(x)= f(y)D(x)D(y)K(y, x) dµ(y) . ZM Hence RR∗ is always an integral operator.

◮ Furthermore, by multiplying by U ∗, and then using that √RR∗ is self-adjoint, we get the following formula for RU and then U:

U ∗R∗ = RU = √RR∗ or 1 Uf(x)= D(x)− √RR∗(f)(x) .

holomorphic | {z } ◮ But what is √RR∗?

- p. 20/54 ◮ We apply this now to M = Rn Cn, = and D(z)= h (z). Then: ⊂ F Ft t

RR∗f(x) = f(y)ht(x)ht(y)K(x, y) dy Z n 3n/2 = 2− (πt)− f h (x) . ∗ t

- p. 21/54 ◮ We apply this now to M = Rn Cn, = and D(z)= h (z). Then: ⊂ F Ft t

RR∗f(x) = f(y)ht(x)ht(y)K(x, y) dy Z n 3n/2 = 2− (πt)− f h (x) . ∗ t ◮ As Ht t 0 is a semi-group it follows that { } ≥ n/2 3n/4 √RR f(x) = 2− (πt)− f h (x) ∗ ∗ t/2 and hence

1 n/2 3n/4 Uf(x) = h (x)− 2− (πt)− f h (x) t ∗ t2 2 n/4 1 (y2 2xy+ x ) = (πt)− f(y)e− 2t − 2 dy. Z

- p. 21/54 ◮ We apply this now to M = Rn Cn, = and D(z)= h (z). Then: ⊂ F Ft t

RR∗f(x) = f(y)ht(x)ht(y)K(x, y) dy Z n 3n/2 = 2− (πt)− f h (x) . ∗ t ◮ As Ht t 0 is a semi-group it follows that { } ≥ n/2 3n/4 √RR f(x) = 2− (πt)− f h (x) ∗ ∗ t/2 and hence

1 n/2 3n/4 Uf(x) = h (x)− 2− (πt)− f h (x) t ∗ t2 2 n/4 1 (y2 2xy+ x ) = (πt)− f(y)e− 2t − 2 dy. Z

◮ Thus U = St and St is an unitary isomorphism.

- p. 21/54 ◮ Let me know recollect the problems/phylosophy for the general case:

- p. 22/54 ◮ Let me know recollect the problems/phylosophy for the general case: ◮ If M is a Riemannian manifold, then the elliptic differential operator ∆ is well deﬁned and invariant under isometries of M. Let dσ be the volume form on M. Then the heat equation is given as before:

2 ∆u(x, t)= ∂tu(x, t) , lim u(x, t)= f(x) L (M,dσ) . t 0+ ∈ → and the solution is (by deﬁnition) given by

t∆ Htf(x)= e f(x) .

2 ∆u(x, t)= ∂tu(x, t) , lim u(x, t)= f(x) L (M,dσ) . t 0+ ∈ → and the solution is (by deﬁnition) given by

t∆ Htf(x)= e f(x) .

◮ But more importantly, there exists a function ht(x, y), the heat kernel, such that: h (x, y)= h (y, x) 0; • t t ≥ dµ (y)= h (x, y)dσ(y) is a probability measure on M; • t t If g : M M is an isometry, then h (gx, gy)= h(x, y) . • → t H f(x)= f(y)h (x, y) dσ(y); • t t ZM

- p. 22/54 ◮ But to generalize the previous results one need to ﬁnd a “natural” complexiﬁcation MC for M such that ht, and Htf extend to holomorphic functions on MC.

- p. 23/54 ◮ But to generalize the previous results one need to ﬁnd a “natural” complexiﬁcation MC for M such that ht, and Htf extend to holomorphic functions on MC.

◮ Then we have to ﬁnd a Hilbert space t(MC) (MC) such that the transform H ⊂ O 2 L (M) f H f (MC) ∋ 7→ t ∈ Ht becomes an unitary isomorphism. In particular, what is the “natural” generalization of the measure µt? ◮ There is one class of Riemannian manifolds where a natural complexiﬁcation exists. Those are the Riemannian symmetric spaces G/K, where G is a connected and semisimple Lie group. B. Hall in 1997 for compact connected Lie groups. Here • G = M GC = MC T ∗G. ⊂ ≃ Here GC is a complex Lie group with Lie algebra gC = g R C, i.e., ⊗ G = SO(n) SO(n, C) SO(n) exp X iM(n, R) X∗ = X . ⊂ ≃ × { ∈ | }

- p. 23/54 M.B. Stenzel in 1999 for symmetric spaces M = G/K, where G is • compact. Here MC = GC/KC T (G/K)∗. Here G is a compact connected Lie group, τ : G G is a non-trivial≃ involution and → K = Gτ = g G τ(g)= g { ∈ | } i.e, Sn = SO(n + 1)/SO(n). Note, that Hall’s result is a special case as G G G/G with τ(a, b)=(b, a). ≃ ×

- p. 24/54 M.B. Stenzel in 1999 for symmetric spaces M = G/K, where G is • compact. Here MC = GC/KC T (G/K)∗. Here G is a compact connected Lie group, τ : G G is a non-trivial≃ involution and → K = Gτ = g G τ(g)= g { ∈ | } i.e, Sn = SO(n + 1)/SO(n). Note, that Hall’s result is a special case as G G G/G with τ(a, b)=(b, a). ≃ × The restriction principle was formulated in general by G. Ólafsson and B. Ørsted• in 1996, but it had been applied earlier by G. Zhang and B. Ørsted in special cases.

- p. 24/54 B. Krötz, G. Ólafsson, and R. Stanton: 2005 the general case G/K where G• is non-compact and semisimple and K is a maximal compact subgroup, i.e., SL(n, R)/SO(n).

- p. 25/54 B. Krötz, G. Ólafsson, and R. Stanton: 2005 the general case G/K where G• is non-compact and semisimple and K is a maximal compact subgroup, i.e., SL(n, R)/SO(n). ◮ A different formula for the K-invariant function on G/K and generalization to arbitrary non-negative multiplicity functions by G. Ólafsson and H. Schlichtkrull (Copenhagen) in 2005, to appear in Adv. Math. ◮ One of the reasons, that it took so long to get from the compact case to the non-compact case is, that it was not so clear, what the right complexification of G/K is. It is the Akhiezer-Gindikin domain also called the complex crown which I will define in a moment. But first we will need some basic structure theory for semisimple symmetric space of the non-compact type.

- p. 25/54 5. Structure Theory

- p. 26/54 5. Structure Theory

◮ G a connected, non-compact semisimple Lie group with ﬁnite center

- p. 26/54 5. Structure Theory

◮ G a connected, non-compact semisimple Lie group with ﬁnite center ◮ K G a maximal compact subgroup, and θ : G G the corresponding Cartan⊂ involution: → K = Gθ = g G θ(g)= g . { ∈ | }

- p. 26/54 5. Structure Theory

◮ G a connected, non-compact semisimple Lie group with ﬁnite center ◮ K G a maximal compact subgroup, and θ : G G the corresponding Cartan⊂ involution: → K = Gθ = g G θ(g)= g . { ∈ | }

◮ Denote the corresponding involution on the Lie algebra g by the same letter θ and let k = X g θ(X)= X { ∈ | } p = X g θ(X)= X { ∈ | − }

- p. 26/54 5. Structure Theory

◮ G a connected, non-compact semisimple Lie group with ﬁnite center ◮ K G a maximal compact subgroup, and θ : G G the corresponding Cartan⊂ involution: → K = Gθ = g G θ(g)= g . { ∈ | }

◮ Denote the corresponding involution on the Lie algebra g by the same letter θ and let k = X g θ(X)= X { ∈ | } p = X g θ(X)= X { ∈ | − }

◮ We have the Cartan decomposition g = k p . ⊕ . - p. 26/54 1 T ◮ Our standard example is G = SL(n, R), K = SO(n) and θ(g)=(g− ) . The corresponding involution on the Lie algebra sl(n, R)= X M (R) Tr(X) = 0 { ∈ n | } is θ(X)= XT . The decomposition g = k p corresponds to the decomposition− of sl(n, R) into skew-symmetric⊕ (= k) and symmetric (= p) matrices .

- p. 27/54 1 T ◮ Our standard example is G = SL(n, R), K = SO(n) and θ(g)=(g− ) . The corresponding involution on the Lie algebra sl(n, R)= X M (R) Tr(X) = 0 { ∈ n | } is θ(X)= XT . The decomposition g = k p corresponds to the decomposition− of sl(n, R) into skew-symmetric⊕ (= k) and symmetric (= p) matrices . ◮ Recall the linear map ad(X): g g, Y [X,Y ] and deﬁne an inner product on g by → 7→ (X,Y )= Tr(ad(X), ad(θ(Y ))) −

◮ On sl(n, R) this is (X,Y ) = 2nTr(XY T ) .

◮ If X p then ad(X)∗ = ad(X), i.e., ad(X) is symmetric. ∈

- p. 27/54 ◮ Let a Rn (for some n) be a maximal abelian subspace of p, i.e., all diagonal≃ matrices with trace zero.

m = ∆ =

- p. 28/54 ◮ Let a Rn (for some n) be a maximal abelian subspace of p, i.e., all diagonal≃ matrices with trace zero. ◮ Then ad(X) X a is a commuting family of symmetric operators and has therefore{ a joint| basis∈ } for g of eigenvectors. Thus we set: gα = m = ∆ =

α g = X g ( H a)[H,X]= α(H)X , α a∗ 0 { ∈ | ∀ ∈ } ∈ \ { } m = ∆ =

α g = X g ( H a)[H,X]= α(H)X , α a∗ 0 { ∈ | ∀ ∈ } ∈ \ { } m = zk(a)= X g ( H a)[H,X] = 0 zg(a)= m a { ∈ | ∀ ∈ }⊂ ⊕ ∆ =

α g = X g ( H a)[H,X]= α(H)X , α a∗ 0 { ∈ | ∀ ∈ } ∈ \ { } m = zk(a)= X g ( H a)[H,X] = 0 zg(a)= m a { ∈ |α ∀ ∈ }⊂ ⊕ ∆ = α a∗ 0 g = 0 . { ∈ \ { }| { }}

◮ Let X a be such that α(X) = 0 for all α ∆. This is possible as set H a (∈α ∆) α(H) = 0 is a6 ﬁnite union∈ of hyperplanes and hence {nowhere∈ | dense.∃ ∈ }

α g = X g ( H a)[H,X]= α(H)X , α a∗ 0 { ∈ | ∀ ∈ } ∈ \ { } m = zk(a)= X g ( H a)[H,X] = 0 zg(a)= m a { ∈ |α ∀ ∈ }⊂ ⊕ ∆ = α a∗ 0 g = 0 . { ∈ \ { }| { }}

◮ Let X a be such that α(X) = 0 for all α ∆. This is possible as set H a (∈α ∆) α(H) = 0 is a6 ﬁnite union∈ of hyperplanes and hence {nowhere∈ | dense.∃ ∈ } ◮ Let ∆+ := α ∆ α(X) > 0 . Then – as α θ = α – we have { ∈ | } ◦ − ∆ = ∆ ˙ ( ∆+) and (∆+ + ∆+) ∆ ∆+ . ∪ − ∩ ⊂

- p. 28/54 ◮ As [gλ, gµ] gµ+λ it follows that ⊆ n := gα α ∆+ M∈ is a nilpotent Lie algebra such that [m a, n] n ⊕ ⊆

- p. 29/54 ◮ As [gλ, gµ] gµ+λ it follows that ⊆ n := gα α ∆+ M∈ is a nilpotent Lie algebra such that [m a, n] n ⊕ ⊆ α and – with n := θ(n)= α ∆g – ⊕ ∈− g = gα m a = n m a n ⊕ ⊕ ⊕ ⊕ ⊕ α ∆ M∈ the zero eigenspace = | {z }

- p. 29/54 ◮ As [gλ, gµ] gµ+λ it follows that ⊆ n := gα α ∆+ M∈ is a nilpotent Lie algebra such that [m a, n] n ⊕ ⊆ α and – with n := θ(n)= α ∆g – ⊕ ∈− g = gα m a = n m a n ⊕ ⊕ ⊕ ⊕ ⊕ α ∆ M∈ the zero eigenspace = ( (id + θ)(|gα{z) } m) a n ⊕ ⊕ ⊕ α ∆+ M∈ =k = | {z }

- p. 29/54 ◮ On the group level this corresponds to Theorem 0.2 (Iwasawa Decomposition). The map

N A K (n, a, k) nak G × × ∋ 7→ ∈ is an analytic isomorphism. We write

∈N ∈A ∈K x =n(x)a(x)k(x) for the unique decomposition of x G. In particular G/K N A. ∈ ≃ ×

- p. 30/54 ◮ On the group level this corresponds to Theorem 0.3 (Iwasawa Decomposition). The map

N A K (n, a, k) nak G × × ∋ 7→ ∈ is an analytic isomorphism. We write

∈N ∈A ∈K x =n(x)a(x)k(x) for the unique decomposition of x G. In particular G/K N A. ∈ ≃ × ◮ We assume that G GC, where Lie(GC)= g C. Then we can ⊂ ⊗ complexify all the groups under consideration and obtain NC, AC and KC. Then NCAC KC GC is open and dense but not equal to GC. Furthermore, the decomposition⊂ x = n(x)a(x)k(x) NCACKC ∈ is not unique in general.

- p. 30/54 ◮ For our standard example this corresponds to:

a = diag(x ) x = 0 { i | i } n n 1 = x R xX+ x + ... + x = 0 R − { ∈ | 1 2 n }≃ A = ∆ = ∆+ = N =

- p. 31/54 ◮ For our standard example this corresponds to:

a = diag(x ) x = 0 { i | i } n n 1 = x R xX+ x + ... + x = 0 R − { ∈ | 1 2 n }≃ A = diag(a ) ( i)a > 0, a a = 1 { i | ∀ i 1 ··· n } ∆ = ∆+ = N =

- p. 31/54 ◮ For our standard example this corresponds to:

a = diag(x ) x = 0 { i | i } n n 1 = x R xX+ x + ... + x = 0 R − { ∈ | 1 2 n }≃ A = diag(a ) ( i)a > 0, a a = 1 { i | ∀ i 1 ··· n } ∆ = setof α : diag(x ) x x , i = j ij k 7→ i − j 6 ∆+ = N =

- p. 31/54 ◮ For our standard example this corresponds to:

a = diag(x ) x = 0 { i | i } n n 1 = x R xX+ x + ... + x = 0 R − { ∈ | 1 2 n }≃ A = diag(a ) ( i)a > 0, a a = 1 { i | ∀ i 1 ··· n } ∆ = setof α : diag(x ) x x , i = j ij k 7→ i − j 6 ∆+ = α i

- p. 31/54 ◮ For our standard example this corresponds to:

a = diag(x ) x = 0 { i | i } n n 1 = x R xX+ x + ... + x = 0 R − { ∈ | 1 2 n }≃ A = diag(a ) ( i)a > 0, a a = 1 { i | ∀ i 1 ··· n } ∆ = setof α : diag(x ) x x , i = j ij k 7→ i − j 6 ∆+ = α ij : x = 0, x = 1 { ij | ij ii }

- p. 31/54 ◮ For our standard example this corresponds to:

a = diag(x ) x = 0 { i | i } n n 1 = x R xX+ x + ... + x = 0 R − { ∈ | 1 2 n }≃ A = diag(a ) ( i)a > 0, a a = 1 { i | ∀ i 1 ··· n } ∆ = setof α : diag(x ) x x , i = j ij k 7→ i − j 6 ∆+ = α ij : x = 0, x = 1 { ij | ij ii } ◮ The Iwasawa decomposition follows directly from the Gram-Schmidt orthogonalization.

- p. 31/54 ◮ For our standard example this corresponds to:

ac+bd 1 d c a b 1 0 − = c2+d2 √c2+d2 √c2+d2 √c2+d2 c d 0 1 √ 2 2 c d ! ! 0 c + d ! √c2+d2 √c2+d2 ! and this breaks down as c2 + d2 = 0.

- p. 31/54 6. Spherical Functions and the Fourier Transform

- p. 32/54 6. Spherical Functions and the Fourier Transform

◮ We will also need the Weyl group. It is the finite reflection group in O(a) generated by the reflections rα in the hyperplanes α = 0. It is denoted by W . We have W N (a)/M M = Z (a) . ≃ K K Permutation of the coordinates for our standard case. ◮ For a differential operator D : C (G/K) C (G/K) and g G, let c → c ∈ (g D)(f)= D(f L −1 f) L . · ◦ g ◦ g Then D is G-invariant if g D = D for all g G. Thus D is G-invariant if and only if D commutes with translation· ∈ D(f L )=[D(f)] L . ◦ g ◦ g Denote by D(G/K) the commutative algebra of all invariant differential operators on G/K. On Rn this is just the algebra of constant coefficient n differential operators D(R )= C[∂1,...,∂n].

- p. 32/54 ◮ For λ a∗C let ∈ λ+ρ ϕλ(x) := a(kx) dk. ZK The functions ϕλ are the spherical functionson G/K. We have ϕ = ϕ w W : λ = wµ. λ µ ⇐⇒ ∃ ∈

- p. 33/54 ◮ For λ a∗C let ∈ λ+ρ ϕλ(x) := a(kx) dk. ZK The functions ϕλ are the spherical functionson G/K. We have ϕ = ϕ w W : λ = wµ. λ µ ⇐⇒ ∃ ∈ ◮ The spherical functions are K-invariant eigenfunctions of D(G/K). In particular for the Laplace operator ∆ D(G/K): G/K ∈ ∆ ϕ =(λ2 ρ 2)ϕ G/K λ −| | λ α where mα = dim g and 1 ρ = m α. 2 α α ∆+ X∈ ◮ In the harmonic analysis of K-invariant functions on G/K they play the λ x n same role as the exponential functions eλ(x)= e · on R . We will discuss that in more details later on.

- p. 33/54 ◮ Let

- p. 34/54 ◮ Let B = M K where M = Z (A), • \ K

- p. 34/54 ◮ Let where , • B = M K M = ZK (A) \ + a∗ = λ a∗ ( α ∆ )(λ, α) > 0 it is a fundamental domain for the • + { ∈ | ∀ ∈ } W -action on a∗.

- p. 34/54 ◮ Let where , • B = M K M = ZK (A) \ + a∗ = λ a∗ ( α ∆ )(λ, α) > 0 it is a fundamental domain for the • + { ∈ | ∀ ∈ } W -action on a∗. dλ where is the Harish-Chandra -function. • dσ(b, λ)= db c(λ) 2 c(λ) c | | ◮ For f C (G/K) deﬁne the Fourier transform fˆ : B a∗C, of f by ∈ c × ρ iλ fˆ(b, λ)= f(x)a(bx) − dx. ZG/K

Theorem 0.3 (Helgason). 1. The Fourier transform extends to an unitary isomorphism : L2(G/K) L2(B a∗, dσ)+some W -invariance. F → × iλ+ρ 2. If f Cc(G/K) then f(x) = c fˆ(b, λ)a(bx) dσ. ∈ ∗ ZB×a 3. We have (∆ f)(b, λ)=(λ2 ρ2)fˆ(b, λ). F G/K −

- p. 34/54 ◮ For K-invariant functions, this reduces to the Harish-Chandra spherical Fourier transform

fˆ(λ)= f(x)ϕ iλ(x) dx. − Z and the spherical Fourier transform extends to an unitary isomorphism

2 K 2 dλ W 2 dλ L (G/K) f fˆ L (a∗, ) L (a∗ , W ) ∋ 7→ ∈ c(λ) 2 ≃ + | | c(λ) 2 | | | |

- p. 35/54 ◮ For K-invariant functions, this reduces to the Harish-Chandra spherical Fourier transform

fˆ(λ)= f(x)ϕ iλ(x) dx. − Z and the spherical Fourier transform extends to an unitary isomorphism

1 ˆ dλ f(x)= f(λ)ϕiλ(x) 2 . W a∗ c(λ) | | Z + | |

- p. 35/54 ◮ Using the Fourier transform and part (3) of Helgason’s Theorem we get the following form for the solution of the heat equation:

(λ2+ρ2)t iλ+ρ Htf(x) = e− fˆ(b, λ) a(bx) dσ(b, λ) Z = f h (x) . ∗ t Note the ρ2-shift!

- p. 36/54 ◮ Using the Fourier transform and part (3) of Helgason’s Theorem we get the following form for the solution of the heat equation:

(λ2+ρ2)t iλ+ρ Htf(x) = e− fˆ(b, λ) a(bx) dσ(b, λ) Z = f h (x) . ∗ t Note the ρ2-shift! ◮ For the heat kernel we get the expression:

- p. 36/54 ◮ Using the Fourier transform and part (3) of Helgason’s Theorem we get the following form for the solution of the heat equation:

(λ2+ρ2)t iλ+ρ Htf(x) = e− fˆ(b, λ) a(bx) dσ(b, λ) Z = f h (x) . ∗ t Note the ρ2-shift! ◮ For the heat kernel we get the expression:

◮ So, how far does x ht(x) extend? Or, how far does x ϕλ(x) extend, and what is the growth7→ of the extension? 7→

- p. 36/54 7. The Crown and the Heat Kernel

- p. 37/54 7. The Crown and the Heat Kernel

◮ We deﬁne Ω = X a ( α ∆) α(X) < π/2 W invariant polytope { ∈ | ∀ ∈ | | } − Ξ = G exp(iΩ) x GC/KC · o ⊂ where x is the base point eKC GC/KC. Then Ξ is an open G-invariant o ⊂ subset of GC/KC, the Akhiezer-Gindikin domain or complex crown . It has been studied by several group of people: Barchini, Burns + Halverscheid + Hind, Huckleberry, Krötz + Stanton, Wolf and others.

- p. 37/54 7. The Crown and the Heat Kernel

Theorem 0.4 (Krotz+Stanton,¨ ...). 1. We have Ξ NCAC x and the ⊂ · o Iwasawa projection Ξ ξ a(ξ) AC is well deﬁned and ∋ 7→ ∈ holomorphic.

2. Ξ is a maximal G-invariant domain in GC/KC such that all the joint eigenfunctions for D(G/K) extends to holomorphic functions on Ξ. - p. 37/54 ◮ It follows that the spherical functions extends to Ξ. With some extra work, involving the the growth of the spherical functions we have: Theorem 0.5 (Krotz+Stanton)¨ . The heat kernel extends to a holomorphic function on Ξ given by the same formula

1 −(|λ|2+|ρ|2)t ht(ξ) = e ϕiλ(ξ) dσ(λ) . W ∗ | | Za+

- p. 38/54 ◮ It follows that the spherical functions extends to Ξ. With some extra work, involving the the growth of the spherical functions we have: Theorem 0.5 (Krotz+Stanton)¨ . The heat kernel extends to a holomorphic function on Ξ given by the same formula

1 −(|λ|2+|ρ|2)t ht(ξ) = e ϕiλ(ξ) dσ(λ) . W ∗ | | Za+ ◮ As a consequence we have that each solution to the heat equation f h , f L2(G/K) extends to a holomorphic function ∗ t ∈ on Ξ: −1 Htf(ξ) = f(gxo)ht(g ξ) dg . ZG As before, the problem is then to determine the image of H : L2(G/K) (Ξ). t → O

- p. 38/54 8. The Abel Transform and the Heat Kernel

- p. 39/54 8. The Abel Transform and the Heat Kernel

◮ Recall that

2ρ 2ρ f(x) dx = f(na x )a− dnda = f(an x )a dnda. · o · o ZG/K ZA ZN ZA ZN For a K-invariant f function on G/K, say of compact support, deﬁne the Abel transform of f by

ρ ρ (f)(a)= a f(an) dn = a− f(na) dn (0.3) A ZN ZN The Radon Transform using the notation (exp(X))λ |= eλ({zX). Then} (f) is a W -invariant function on A. A

- p. 39/54 8. The Abel Transform and the Heat Kernel

◮ Recall that

2ρ 2ρ f(x) dx = f(na x )a− dnda = f(an x )a dnda. · o · o ZG/K ZA ZN ZA ZN For a K-invariant f function on G/K, say of compact support, deﬁne the Abel transform of f by

ρ ρ (f)(a)= a f(an) dn = a− f(na) dn (0.3) A ZN ZN The Radon Transform using the notation (exp(X))λ |= eλ({zX). Then} (f) is a W -invariant function on A. and we have A

(∆ f)=(∆ ρ 2) (f) . (0.4) A G/K A −| | A

- p. 39/54 8. The Abel Transform and the Heat Kernel

◮ Recall that

2ρ 2ρ f(x) dx = f(na x )a− dnda = f(an x )a dnda. · o · o ZG/K ZA ZN ZA ZN For a K-invariant f function on G/K, say of compact support, deﬁne the Abel transform of f by

ρ ρ (f)(a)= a f(an) dn = a− f(na) dn (0.3) A ZN ZN The Radon Transform using the notation (exp(X))λ |= eλ({zX). Then} (f) is a W -invariant function on A. and we have A

(∆ f)=(∆ ρ 2) (f) . (0.4) A G/K A −| | A

- p. 39/54 ◮ We have the following Fourier slice theorem for K-invariant functions:

fˆ(λ) = f(x)ϕ iλ(x) dx − ZG/K 1 iλ+ρ = f(x)a(k− x)− dx ZG/K ρ iλ = a− f(na x ) dn a− da · o ZA ZN = ( (f))(λ) FA A

- p. 40/54 ◮ We have the following Fourier slice theorem for K-invariant functions:

fˆ(λ) = f(x)ϕ iλ(x) dx − ZG/K 1 iλ+ρ = f(x)a(k− x)− dx ZG/K ρ iλ = a− f(na x ) dn a− da · o ZA ZN = ( (f))(λ) FA A ◮ Or = . FG/K FA ◦ A

- p. 40/54 ◮ We have the following Fourier slice theorem for K-invariant functions:

fˆ(λ) = f(x)ϕ iλ(x) dx − ZG/K 1 iλ+ρ = f(x)a(k− x)− dx ZG/K ρ iλ = a− f(na x ) dn a− da · o ZA ZN = ( (f))(λ) FA A ◮ Or = . FG/K FA ◦ A

◮ Equation (0.4) implies also that

ρ 2t 1 n/2 X 2/4t h (exp X)= e−| | − (4πt)− e−| | . t A the ρ shift a shift operator − the heat kernel on A | {z } |{z} | {z } - p. 40/54 ◮ Deﬁne 1 ψ (exp X)= ewλ(X) λ W w W | | X∈ a holomorphic function on a∗C.

- p. 41/54 ◮ Deﬁne 1 ψ (exp X)= ewλ(X) λ W w W | | X∈ a holomorphic function on a∗C.

◮ Then the equation = implies that ∗(ψ )= ϕ . The FG/K FA ◦ A A λ λ Abel/Radon transform shifts the “ﬂat” case to the ”curved” case!

- p. 41/54 ◮ Deﬁne 1 ψ (exp X)= ewλ(X) λ W w W | | X∈ a holomorphic function on a∗C.

◮ Then the equation = implies that ∗(ψ )= ϕ . The FG/K FA ◦ A A λ λ Abel/Radon transform shifts the “ﬂat” case to the ”curved” case! ◮ Deﬁne now the pseudo-differential operator D on A by:

1 1 D = − FA ◦ c(λ) 2 ◦FG/K | |

- p. 41/54 ◮ Deﬁne 1 ψ (exp X)= ewλ(X) λ W w W | | X∈ a holomorphic function on a∗C.

1 1 D = − FA ◦ c(λ) 2 ◦FG/K | | or – for “good” – W -invariant functions:

then the multiplier 1 Dh(a)= G/K (h)(λ) ψiλ(a) dλ . ∗ F c(λ) 2 a+ z }| { Z Firt the FT on G/K | |

− | {zback using} 1 FA | {z } - p. 41/54 9. The Faraut-Gutzmer Formula and the Orbital Integral

- p. 42/54 9. The Faraut-Gutzmer Formula and the Orbital Integral

◮ For sufﬁciently decreasing functions h :Ξ C we deﬁne the G-orbital integral : 2iΩ C by → Oh → i (Y )= h(g exp Y x ) dg. Oh 2 · o ZG

- p. 42/54 9. The Faraut-Gutzmer Formula and the Orbital Integral

◮ For sufficiently decreasing functions h :Ξ C we define the G-orbital integral : 2iΩ C by → Oh → i (Y )= h(g exp Y x ) dg. Oh 2 · o ZG ◮ We define now (Ξ) (Ξ) to be the space of holomorphic functions F on Ξ such that G ⊂ O F L2(G/K) |G/K ∈

- p. 42/54 9. The Faraut-Gutzmer Formula and the Orbital Integral

- p. 42/54 The following theorem is the replacement for what we used earlier:

- p. 43/54 The following theorem is the replacement for what we used earlier:

- p. 43/54 10. The Image of the Segal-Bargmann Transform

- p. 44/54 10. The Image of the Segal-Bargmann Transform

◮ We have now set every thing up to state (and prove) what the image of the Segal-Bargmann transform in this case is. Deﬁne a ρ-shifted density function by

tρ2 e n/2 Y 2/2t ω (a exp Y ) := (2πt)− e−| | . t W | | takes care of the ρ shift the density for a − |{z} | {z }

- p. 44/54 10. The Image of the Segal-Bargmann Transform

◮ We have now set every thing up to state (and prove) what the image of the Segal-Bargmann transform in this case is. Deﬁne a ρ-shifted density function by

tρ2 e n/2 Y 2/2t ω (a exp Y ) := (2πt)− e−| | . t W | | takes care of the ρ shift the density for a − ◮ Deﬁne a “norm” on (Ξ) by |{z} | {z } G 2 F t = D F 2 (exp iY )ωt(Y ) dY k k O| | Za and set (Ξ) = F (Ξ) F < . Ft { ∈ G | k kt ∞}

- p. 44/54 Theorem 0.6 (KOS)´ . The Segal-Bargmann transform is an unitary isomorphism H : L2(G/K) (Ξ) . t →Ft

- p. 45/54 Theorem 0.6 (KOS)´ . The Segal-Bargmann transform is an unitary isomorphism H : L2(G/K) (Ξ) . t →Ft ◮ What is needed in the proof is:

(H f)(b, λ) = (f h )(b, λ) FG/K t FG/K ∗ t = fˆ(b, λ)hˆt(b, λ) 2 2 = e−t(λ +ρ )fˆ(b, λ)

- p. 45/54 Theorem 0.6 (KOS)´ . The Segal-Bargmann transform is an unitary isomorphism H : L2(G/K) (Ξ) . t →Ft ◮ What is needed in the proof is:

(H f)(b, λ) = (f h )(b, λ) FG/K t FG/K ∗ t = fˆ(b, λ)hˆt(b, λ) 2 2 = e−t(λ +ρ )fˆ(b, λ)

And hence, with F = Htf:

D 2 (iY )ω (Y )dY O|F | t Z 2 2 = fˆ(b, λ) 2e−t(λ +ρ )ψ (2iY )ω (Y ) dσdY | | λ t

ZZ - p. 45/54 Then we only need that

t(λ2+ρ2) ψλ(2iY )ωt(Y )dY = e . Za

- p. 46/54 Then we only need that

t(λ2+ρ2) ψλ(2iY )ωt(Y )dY = e . Za

10. The K-invariant case

- p. 46/54 Then we only need that

t(λ2+ρ2) ψλ(2iY )ωt(Y )dY = e . Za

10. The K-invariant case

What we need ﬁrst for the K-invariant case is the following simple theorem.

- p. 46/54 Theorem0.6 We have G = KAK and the restriction map

2 K 2 1 W 2 + L (G/K) f f L (A, W − dµ) L (A ,dµ) ∋ 7→ |A ∈ | | ≃ is an unitary isomorphism.

- p. 47/54 Theorem0.6 We have G = KAK and the restriction map

2 K 2 1 W 2 + L (G/K) f f L (A, W − dµ) L (A ,dµ) ∋ 7→ |A ∈ | | ≃ is an unitary isomorphism. ◮ This reduces the analysis of K-invariant functions on G/K to analysis of W -invariant functions on the Euclidean space A a. ≃ ◮ Next we consider the effect on the Heat equation. For that let H1,...,Hn be a orthonormal basis of a and Areg = a A ( α) aα = 1 . { ∈ | ∀ 6 } ◮ Let ( , ) be a W -invariant inner product on a (and by duality on a∗). Chose h· · a be such that (X, h )= α(X), (α, β)=(H , H ), and - for α ∈ α α β α = 0 - H = 2 h . 6 α (α,α) α

- p. 47/54 Theorem0.6 We have G = KAK and the restriction map

n 2α 2 1+ e− L = ∂(Hj) + mα 2α ∂(hα) . 1 e− j=1 α ∆+ X X∈ −

- p. 47/54 Theorem 0.7 (The radial part of the Laplacian) We have

(∆f) reg = L(f reg ) |A |A K for all f C∞(G/K) . ∈

- p. 48/54 Theorem 0.7 (The radial part of the Laplacian) We have

(∆f) reg = L(f reg ) |A |A K for all f C∞(G/K) . ∈ ◮ Hence the heat equation for K-invariant functions on G/K corresponds to the Cauchy problem on Areg (or A+)

Lu(a, t) = ∂tu(a, t) (*) t 0+ u(a, t) → f(a) L2(A+,dµ) −→ ∈

- p. 48/54 Theorem 0.7 (The radial part of the Laplacian) We have

(∆f) reg = L(f reg ) |A |A K for all f C∞(G/K) . ∈ ◮ Hence the heat equation for K-invariant functions on G/K corresponds to the Cauchy problem on Areg (or A+)

- p. 48/54 Theorem 0.7 (The radial part of the Laplacian) We have

(∆f) reg = L(f reg ) |A |A K for all f C∞(G/K) . ∈ ◮ Hence the heat equation for K-invariant functions on G/K corresponds to the Cauchy problem on Areg (or A+)

Lu(a, t) = ∂tu(a, t) (*) t 0+ u(a, t) → f(a) L2(A+,dµ) −→ ∈ ◮ The important observation now is, that every thing in (*) as well as the Harish-Chandra c-function is independent of G/K, it only depends on ◮ the space a Rn, ◮ the set of roots≃ ∆ and ◮ multiplicity function m : α m ! → α ◮ So from now on m : ∆ [0, ) is a Weyl group invariant function, deﬁned on a root system ∆→in a∞ ﬁnite dimensional Euclidean space a.

- p. 48/54 Theorem 0.7 (The radial part of the Laplacian) We have

(∆f) reg = L(f reg ) |A |A K for all f C∞(G/K) . ∈ ◮ Hence the heat equation for K-invariant functions on G/K corresponds to the Cauchy problem on Areg (or A+)

- p. 48/54 ◮ What is missing is a good Fourier analysis on a with respect to the measure dµ.

- p. 49/54 ◮ What is missing is a good Fourier analysis on a with respect to the measure dµ. ◮ This theory was developed by E. Opdam and G. Heckman in a series of article, starting around 1988

◮ What they did was to deﬁne for each λ a∗C a function - the generalized hypergeometric functions - ϕ : A C using∈ the Harish-Chandra expansion λ →

ϕλ(a)= c(wλ)Ψwλ(a) w W X∈ where Ψµ is deﬁned by an inﬁnite sum involving exponentials and rational functions Γµ(λ) that depend on mα in a rational way, and hence make sense for all multiplicity functions!

- p. 49/54 ◮ What is missing is a good Fourier analysis on a with respect to the measure dµ. ◮ This theory was developed by E. Opdam and G. Heckman in a series of article, starting around 1988

◮ What they did was to deﬁne for each λ a∗C a function - the generalized hypergeometric functions - ϕ : A C using∈ the Harish-Chandra expansion λ →

◮ Note, that the properties of the spherical functions follows from the deﬁning integral formula. It was therefore a non-trivial task to prove the following using only the expansion formula: