Dunkl operators and Clifford algebras II

Translation operator for the Clifford Research Group Department of Mathematical Analysis Ghent University Hong Kong, March, 2011

Translation operator for the Hermite polynomials Translation operator for the Dunkl translation

Translation operator for the Summary previous lecture: Data: root system R in R m, encoding finite reflection group G multiplicity function k : R C

Translation operator for the Summary previous lecture: Data: root system R in R m, encoding finite reflection group G multiplicity function k : R C Dunkl operators T i, i = 1,..., m T i f (x) = xi f (x) + f (x) f (σ α (x)) k α α i α, x α R + Dunkl Laplacian Euler operator k = E = m i=1 T 2 i m x i xi i=1

Translation operator for the Define the following parameter: µ = 1 2 k x 2 = m + 2 α R + k α C Then we have the following operator identities: Theorem The operators k, x 2 and E + µ/2 generate the Lie algebra sl 2 [ k, x 2] = 4(E + µ 2 ) [ k, E + µ ] [ 2 x 2, E + µ ] 2 = 2 k = 2 x 2

Translation operator for the This lecture: Can we define Hermite polynomials related to the Dunkl operators? Is there a related quantum system? Same question for Fourier transform

Translation operator for the Outline Hermite polynomials Translation operator for the Dunkl translation

Translation operator for the Dunkl harmonics H l space of l-homogeneous null-solutions of k weight w k (x) = α R + α, x 2kα Then orthogonality: Theorem Let H l and H n be Dunkl harmonics of different degree. Then one has H l (x)h n (x)w k (x)dσ(x) = 0. S m 1

Translation operator for the Let {H (n) l }, n = 0,..., dim H l be ONB of H l.

Translation operator for the Let {H (n) l }, n = 0,..., dim H l be ONB of H l. Then Definition The Hermite polynomials related to this basis are given by ψ j,l,n := D j H (n) l, j = 0, 1,... with D := k + 4 x 2 2(2E + µ) (Natural generalization of rank one case)

Translation operator for the Theorem (Rodrigues formula) The Hermite polynomials take the form ψ j,l,n = exp( x 2 /2)( k x 2 + 2E + µ) j exp( x 2 /2)H l = exp( x 2 )( k ) j exp( x 2 )H l. Theorem (Differential equation) ψ j,l,n is a solution of the following PDE: [ k 2E]ψ j,l,n = 2(2j + l)ψ j,l,n.

Translation operator for the Note that ψ j,l,n = f j,l ( x 2 )H (n) l (x)

Translation operator for the Note that More precisely ψ j,l,n = f j,l ( x 2 )H (n) l (x) Theorem The Hermite polynomials can be written in terms of the generalized Laguerre polynomials as with L α t (x) = ψ j,l,n = c j L µ 2 +l 1 j t i=0 and c j a normalization constant. ( x 2 )H (n) (x), Γ(t + α + 1) i!(t i)!γ(i + α + 1) ( x)i. l

Translation operator for the Orthogonality Define Hermite functions by φ j,l,n := ψ j,l,n exp( x 2 /2)

Translation operator for the Orthogonality Define Hermite functions by φ j,l,n := ψ j,l,n exp( x 2 /2) Theorem One has R m φ j1,l 1,n 1 φ j2,l 2,n 2 w k (x)dx δ j1 j 2 δ l1 l 2 δ n1 n 2 Proof: split integral in spherical and radial part; use orthogonality of Laguerre polynomials.

Translation operator for the Why so important?

Translation operator for the Why so important? Proposition The Hermite polynomials {ψ j,l,n } form a basis for the space of all polynomials P. It can be proven that {φ j,l,n } is dense in both L 2 (R m, w k (x)dx) S(R m )

Translation operator for the Alternative definition: Let {φ ν, ν Z m +} be a basis of P such that φ ν P ν.

Translation operator for the Alternative definition: Let {φ ν, ν Z m +} be a basis of P such that φ ν P ν. Definition The generalized Hermite polynomials {H ν, ν Z m +} associated with {φ ν } on R m are given by H ν (x) := e k/4 φ ν (x) = ν /2 n=0 ( 1) n 4 n n! n k φ ν(x). Rösler M., Generalized Hermite polynomials and the heat equation for Dunkl operators. Comm. Math. Phys. 192, 3 (1998), 519 542.

Translation operator for the Reduces to previous definition by choosing as basis for P polynomials of the form x 2j H (l) n i.e. e k/4 x 2j H (l) n ψ j,l,n

Translation operator for the Outline Hermite polynomials Translation operator for the Dunkl translation

Translation operator for the Recall quantum harmonic oscillator in R m : 2 ψ + x 2 2 ψ = Eψ (PDE, E is eigenvalue called energy)

Translation operator for the Recall quantum harmonic oscillator in R m : 2 ψ + x 2 2 ψ = Eψ (PDE, E is eigenvalue called energy) This invites us to consider: k 2 ψ + x 2 2 ψ = Eψ (Replace Laplacian by Dunkl Laplacian!) This new equation contains differential AND difference terms

Translation operator for the These QM systems derive from actual physical problems Class of systems = Calogero-Moser-Sutherland (CMS) models m identical particles on line or circle external potential pairwise interaction T. H. Baker and P. J. Forrester, The Calogero-Sutherland model and generalized classical polynomials, Comm. Math. Phys. 188 (1997) 175 216. J.F. van Diejen and L. Vinet, Calogero-Sutherland-Moser Models (CRM Series in Mathematical Physics, Springer-Verlag, 2000).

Translation operator for the Theorem The Hermite functions {φ j,l,n } form a complete set of eigenfunctions of H := 1 2 ( k x 2 ) satisfying Hφ j,l,n = ( µ 2 + 2j + l ) φ j,l,n. Proof: use the sl 2 relations. Consequences: complete decomposition of Hilbert space L 2 (R m, w k (x)dx) into H-eigenspaces alternative proof of orthogonality of {φ j,l,n }

Translation operator for the Some more references: Explicit examples of physical systems and reduction to Dunkl case: C.F. Dunkl, Reflection groups in analysis and applications. Japan. J. Math. 3 (2008), 215 246. M. Rösler, Dunkl operators: theory and applications. Lecture Notes in Math., 1817, Orthogonal polynomials and special functions, Leuven, 2002, (Springer, Berlin, 2003) 93 135. Online: arxiv:math/0210366. Study of Hermite polynomials in superspace; extensive comparison with Dunkl case K. Coulembier, H. De Bie and F. Sommen Orthogonality of Hermite polynomials in superspace and Mehler type formulae, Accepted in Proc. LMS, arxiv:1002.1118.

Outline Hermite polynomials Translation operator for the Hermite polynomials Translation operator for the Dunkl translation

Translation operator for the The classical Fourier transform Definition: F(f )(y) = (2π) m 2 R m e i x,y f (x)dx Here, K(x, y) = e i x,y is the unique solution of the system xj K(x, y) = iy j K(x, y), K(0, y) = 1 j = 1,..., m

Translation operator for the The (k > 0) Consider the system T xj K(x, y) = iy j K(x, y), K(0, y) = 1 j = 1,..., m One proves that this system has a unique solution ( K(x, y) = V k e i x,y )

Translation operator for the The (k > 0) Consider the system T xj K(x, y) = iy j K(x, y), K(0, y) = 1 j = 1,..., m One proves that this system has a unique solution ( K(x, y) = V k e i x,y ) Then Definition The is defined by F k (f )(y) = c k R m K(x, y)f (x)w k (x)dx with w k (x)dx the G-invariant measure and c k a constant.

Translation operator for the No explicit expression of K(x, y) known, except special cases

Translation operator for the No explicit expression of K(x, y) known, except special cases Properties: K(x, y) 1, for all x, y R m F k well-defined on L 1 (R m, w k (x)) K(x, y) = K(y, x) K(g x, g y) = K(x, y) for all g G de Jeu, M.F.E. The. Invent. Math. 113 (1993), 147 162.

Translation operator for the Proposition Let f S(R m ). Then F k (T xj f ) = iy j F k (f ) F k (x j f ) = it yj F k (f ). Moreover, F k leaves S(R m ) invariant.

Translation operator for the Proposition Let f S(R m ). Then F k (T xj f ) = iy j F k (f ) F k (x j f ) = it yj F k (f ). Moreover, F k leaves S(R m ) invariant. Proof: use T xj K(x, y) = iy j K(x, y) and T j f, g = f, T j g

Translation operator for the Spectrum of : acts nicely on Hermite functions {φ j,l,n } Theorem One has F k φ j,l,n = i 2j+l φ j,l,n

Translation operator for the Spectrum of : acts nicely on Hermite functions {φ j,l,n } Theorem One has F k φ j,l,n = i 2j+l φ j,l,n Proof: Recall φ j,l,n = ( k x 2 + 2E + µ) j exp( x 2 /2)H l Compute F k (H l e x 2 /2 )

Translation operator for the Bochner formula? of f ( x )H l with H l H l

Translation operator for the Bochner formula? of f ( x )H l with H l H l Theorem Let H l H l and f ( x ) of suitable decay. Then one has with F k (f ( x )H l )(y) = c l H l (y)f l+µ/2 1 (f )( y ) F α (f )(s) := + 0 f (r)(rs) α J α (rs)r 2α+1 dr the Hankel transform and c l a constant only depending on l.

Translation operator for the Heisenberg uncertainty principle: S(R m ) is dense in L 2 (R m, w k ), so F k extends to L 2

Translation operator for the Heisenberg uncertainty principle: S(R m ) is dense in L 2 (R m, w k ), so F k extends to L 2 Theorem Let f L 2 (R m, w k (x)). Then x f 2 x F k (f ) 2 µ 2 f 2 2. Again consequence of Hermite functions being solutions of CMS system

Translation operator for the There exists interesting other expression for Dunkl TF on S(R m ): F k = e iπ 4 ( k+ x 2 µ)

Translation operator for the There exists interesting other expression for Dunkl TF on S(R m ): F k = e iπ 4 ( k+ x 2 µ) Indeed, check that using the CMS system: e iπ 4 ( k+ x 2 µ) φ j,l,n = i 2j+k φ j,l,n 1 2 ( k x 2 )φ j,l,n = ( µ 2 + 2j + l ) φ j,l,n. Ben Saïd S. On the integrability of a representation of sl(2, R). J. Funct. Anal. 250 (2007), 249 264.

Translation operator for the The formulation is ideal for further generalizations F k = e iπ 4 ( k x 2 µ) One only needs operators generating sl 2 See S. Ben Saïd, T. Kobayashi and B. Ørsted, Laguerre semigroup and Dunkl operators. Preprint: arxiv:0907.3749, 74 pages. H. De Bie, B. Orsted, P. Somberg and V. Soucek, The Clifford deformation of the Hermite semigroup. Preprint, 27 pages, arxiv:1101.5551.

Outline Hermite polynomials Translation operator for the Dunkl translation Hermite polynomials Translation operator for the Dunkl translation

Translation operator for the Dunkl translation Convolution for the classical Fourier transform Definition of convolution: (f g)(x) = f (x y)g(y)dy, R m Crucial to prove inversion of the FT for other function spaces than S(R m ) (take g the heat kernel approximation of delta distribution)? similar approach for

Translation operator for the Dunkl translation Convolution depends on the translation operator τ y : f (x) f (x y) Under the Fourier transform, τ y satisfies F (τ y f (x)) (z) = e i z,y F(f )(z) use as definition in case of Dunkl TF very powerful technique! M. Rösler, A positive radial product formula for the Dunkl kernel. Trans. Amer. Math. Soc. 355 (2003), 2413 2438. S. Thangavelu and Y. Xu, Convolution operator and maximal function for the. J. Anal. Math. 97 (2005), 25 55.

Translation operator for the Dunkl translation Formal definition: Definition The translation operator for the is defined by τ y f (x) = K( x, z)k(y, z)f k (f )(z)dz. R m If f S(R m ), then also τ y f (x) Very complicated to compute τ y f (x)

Translation operator for the Dunkl translation Main result: Theorem Let f ( x ) be a radial function. Then τ y (f )(x) = V k (f ( x y )).

Translation operator for the Dunkl translation Main result: Theorem Let f ( x ) be a radial function. Then τ y (f )(x) = V k (f ( x y )). Proof: Note that Dunkl TF of radial function is again radial Then use decomposition K(x, y) = l=0 d l Jl+µ/2 1 ( x y )V k (C µ/2 1 l ( x, y )) combined with orthogonality of Gegenbauers and addition formula for Bessel function

Translation operator for the Dunkl translation Many difficult problems and open questions related to translation operator: explicit formula for specific G? boundedness of translation on certain function spaces translation of non-radial functions, explicit examples? positivity? (in general: NO) etc.