Commutative harmonic analysis on noncommutative Lie groups Fulvio Ricci Scuola Normale Superiore, Pisa Jubilee of Fourier Analysis and Applications: A Conference Celebrating John Benedetto’s 80th Birthday University of Maryland, College Park, MD, September 20, 2019 1 / 26
Something very old 2 / 26
Something very old The following is a direct consequence of the Wiener Tauberian Theorem on the real line: Theorem Let f be a bounded holomorphic function on the unit disc ∆ . For 0 < r < 1 , let γ r ⊂ ∆ be the circle of radius r tangent to ∂ ∆ at 1 . If, for some r ∈ (0 , 1) , lim f ( z ) = ℓ , z → 1 z ∈ γ r then the same holds true for every other r. ✬✩ ✛✘ γ r ✚✙ · · 1 ✫✪ 2 / 26
Proof → i 1+ z Via the Cayley transform C : z �− 1 − z = w , the disc ∆ is replaced by the upper half plane U = { w = x + iy : y > 0 } and γ r by the horizontal line y = 1 − r r . The function g = f ◦ C − 1 is bounded and holomorphic on U . By Fatou’s theorem, g is the Poisson integral of a bounded function g 0 on ∂ U = R , p y ( x ) = 1 y g ( x + iy ) = g 0 ∗ p y ( x ) , x 2 + y 2 . π The hypothesis implies that, for a fixed y > 0, lim x →∞ g 0 ∗ p y ( x ) = ℓ . p y ( ξ ) = e − y | ξ | � = 0 and R p y ′ = 1 for every y ′ > 0, the WTT gives Since � ´ lim x →∞ g 0 ∗ p y ′ ( x ) = ℓ for all y ′ . 3 / 26
An analogue for the ball in C n The same argument can be adapted to the unit ball in C n , n ≥ 2 (A. Hulanicki, F. R., Adv. Math. 1980) proceeding as follows: • interpret the γ r (resp. the horizontal lines in U ) as the horocycles pointed at 1 ∈ ∂ ∆ (resp. at ∞ ∈ ∂ U ) in the Poincar´ e metric; 4 / 26
An analogue for the ball in C n The same argument can be adapted to the unit ball in C n , n ≥ 2 (A. Hulanicki, F. R., Adv. Math. 1980) proceeding as follows: • interpret the γ r (resp. the horizontal lines in U ) as the horocycles pointed at 1 ∈ ∂ ∆ (resp. at ∞ ∈ ∂ U ) in the Poincar´ e metric; • transform the ball B n into the Siegel domain � � w = ( w 1 , w ′ ) ∈ C × C n − 1 : Im w 1 − | w ′ | 2 > 0 U n = via a generalized Cayley transform C ; 4 / 26
An analogue for the ball in C n The same argument can be adapted to the unit ball in C n , n ≥ 2 (A. Hulanicki, F. R., Adv. Math. 1980) proceeding as follows: • interpret the γ r (resp. the horizontal lines in U ) as the horocycles pointed at 1 ∈ ∂ ∆ (resp. at ∞ ∈ ∂ U ) in the Poincar´ e metric; • transform the ball B n into the Siegel domain � � w = ( w 1 , w ′ ) ∈ C × C n − 1 : Im w 1 − | w ′ | 2 > 0 U n = via a generalized Cayley transform C ; • identify the boundary ∂ U n = { w : Im w 1 − | w ′ | 2 = 0 } with the Heisenberg group H n − 1 ; 4 / 26
An analogue for the ball in C n The same argument can be adapted to the unit ball in C n , n ≥ 2 (A. Hulanicki, F. R., Adv. Math. 1980) proceeding as follows: • interpret the γ r (resp. the horizontal lines in U ) as the horocycles pointed at 1 ∈ ∂ ∆ (resp. at ∞ ∈ ∂ U ) in the Poincar´ e metric; • transform the ball B n into the Siegel domain � � w = ( w 1 , w ′ ) ∈ C × C n − 1 : Im w 1 − | w ′ | 2 > 0 U n = via a generalized Cayley transform C ; • identify the boundary ∂ U n = { w : Im w 1 − | w ′ | 2 = 0 } with the Heisenberg group H n − 1 ; • recognize that the horocycles in U n pointed at infinity are the vertical translates of ∂ U n ; 4 / 26
An analogue for the ball in C n The same argument can be adapted to the unit ball in C n , n ≥ 2 (A. Hulanicki, F. R., Adv. Math. 1980) proceeding as follows: • interpret the γ r (resp. the horizontal lines in U ) as the horocycles pointed at 1 ∈ ∂ ∆ (resp. at ∞ ∈ ∂ U ) in the Poincar´ e metric; • transform the ball B n into the Siegel domain � � w = ( w 1 , w ′ ) ∈ C × C n − 1 : Im w 1 − | w ′ | 2 > 0 U n = via a generalized Cayley transform C ; • identify the boundary ∂ U n = { w : Im w 1 − | w ′ | 2 = 0 } with the Heisenberg group H n − 1 ; • recognize that the horocycles in U n pointed at infinity are the vertical translates of ∂ U n ; • use the n -dimensional Fatou theorem to express g = f ◦ C − 1 on the hypersurface Im w 1 − | w ′ | 2 = y > 0 as a convolution g 0 ∗ p ( n ) y , with g 0 ∈ L ∞ ( H n − 1 ) and p ( n ) the generalized Poisson kernels. y 4 / 26
Commutativity of L 1 The Wiener Tauberian Theorem (for a locally compact abelian group G ) is the statement that L 1 ( G ) has the Wiener property , i.e., a closed ideal ϕ � = 0 at all points of � G is all of L 1 ( G ). containing a function ϕ with ˆ In the above proof for ∆ this has been used with G = R and ϕ = p y . 5 / 26
Commutativity of L 1 The Wiener Tauberian Theorem (for a locally compact abelian group G ) is the statement that L 1 ( G ) has the Wiener property , i.e., a closed ideal ϕ � = 0 at all points of � G is all of L 1 ( G ). containing a function ϕ with ˆ In the above proof for ∆ this has been used with G = R and ϕ = p y . The proof in higher dimension makes use of a “rotation invariance” property of the Poisson kernels p ( n ) y . This property identifies a closed subalgebra of L 1 ( H n − 1 ) which is commutative and satisfies the Wiener property (with � G replaced by its Gelfand spectrum). 5 / 26
z -radial functions on H d The Heisenberg group H d is C d × R with product � � ( z , t ) · ( w , u ) = z + w , t + u + 2 Im � z , w � . Convolution of two functions f , g is defined as � ( z , t ) · ( w , u ) − 1 � ˆ f ∗ g ( z , t ) = f g ( w , u ) dw du . H d A function f is z - radial if it depends on | z | , t only (i.e., it is invariant under unitary transformations in the C d -component). 6 / 26
z -radial functions on H d The Heisenberg group H d is C d × R with product � � ( z , t ) · ( w , u ) = z + w , t + u + 2 Im � z , w � . Convolution of two functions f , g is defined as � ( z , t ) · ( w , u ) − 1 � ˆ f ∗ g ( z , t ) = f g ( w , u ) dw du . H d A function f is z - radial if it depends on | z | , t only (i.e., it is invariant under unitary transformations in the C d -component). If f , g are both z -radial, then f ∗ g is also z -radial and f ∗ g = g ∗ f . Due to this commutativity property, Fourier analysis of z -radial functions on H d can be done using the scalar-valued Gelfand transform of L 1 z -rad ( H d ) rather than the operator-valued group Fourier transform. 6 / 26
Commutative pairs The context described above is the simplest, but nontrivial, example of commutative pair 1 . If G is a locally compact group and K is a compact group of automorphisms of G , we say that ( G , K ) is a commutative pair if the convolution algebra L 1 K ( G ) of K -invariant functions on G is commutative. 1 The name is for this talk only. The standard notion of Gelfand pair also includes other situations that we prefer to leave aside today. 7 / 26
Commutative pairs The context described above is the simplest, but nontrivial, example of commutative pair 1 . If G is a locally compact group and K is a compact group of automorphisms of G , we say that ( G , K ) is a commutative pair if the convolution algebra L 1 K ( G ) of K -invariant functions on G is commutative. The spectrum Σ( G , K ) of L 1 K ( G ) consists of the bounded spherical functions ϕ , satisfying the equation � � ˆ ϕ ( x ) ϕ ( y ) = ϕ x · ( ky ) dk . K 1 The name is for this talk only. The standard notion of Gelfand pair also includes other situations that we prefer to leave aside today. 7 / 26
Spherical transform ˆ f ( x ) ϕ ( x − 1 ) dx G f ( ϕ ) = G 8 / 26
Spherical transform ˆ f ( x ) ϕ ( x − 1 ) dx G f ( ϕ ) = G Properties. • Riemann-Lebesgue: G : L 1 K ( G ) − → C 0 (Σ), 8 / 26
Spherical transform ˆ f ( x ) ϕ ( x − 1 ) dx G f ( ϕ ) = G Properties. • Riemann-Lebesgue: G : L 1 K ( G ) − → C 0 (Σ), • Uniqueness: G f = 0 ⇒ f = 0, 8 / 26
Spherical transform ˆ f ( x ) ϕ ( x − 1 ) dx G f ( ϕ ) = G Properties. • Riemann-Lebesgue: G : L 1 K ( G ) − → C 0 (Σ), • Uniqueness: G f = 0 ⇒ f = 0, • Plancherel formula: there exists a unique (up to scalar multiples) measure ν on Σ such that �G f � L 2 (Σ ,ν ) = � f � 2 , 8 / 26
Spherical transform ˆ f ( x ) ϕ ( x − 1 ) dx G f ( ϕ ) = G Properties. • Riemann-Lebesgue: G : L 1 K ( G ) − → C 0 (Σ), • Uniqueness: G f = 0 ⇒ f = 0, • Plancherel formula: there exists a unique (up to scalar multiples) measure ν on Σ such that �G f � L 2 (Σ ,ν ) = � f � 2 , • Inversion formula: f ( x ) = ´ Σ G f ( ϕ ) ϕ ( x ) d ν ( ϕ ), 8 / 26
Spherical transform ˆ f ( x ) ϕ ( x − 1 ) dx G f ( ϕ ) = G Properties. • Riemann-Lebesgue: G : L 1 K ( G ) − → C 0 (Σ), • Uniqueness: G f = 0 ⇒ f = 0, • Plancherel formula: there exists a unique (up to scalar multiples) measure ν on Σ such that �G f � L 2 (Σ ,ν ) = � f � 2 , • Inversion formula: f ( x ) = ´ Σ G f ( ϕ ) ϕ ( x ) d ν ( ϕ ), • Multipliers: every bounded operator T on L 2 ( G ) which commutes with left translations and with the automorphisms in K can be expressed as Tf = G − 1 ( m G f ), for a unique m ∈ L ∞ (Σ , ν ). 8 / 26
The differentiable setting Assume now that G is a connected Lie group. It is then possible to obtain interesting models of Σ as closed subsets of some Euclidean space. 9 / 26
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