Matrix valued orthogonal polynomials related to SU( N + 1), their algebras of differential operators and the corresponding curves

Matrix valued orthogonal polynomials related to SU(N+ 1), their algebras of differential operators and the corresponding curves a Recent Trends in Constructive Approximation Theory Universidad Carlos III de Madrid. September 1, 2006 Manuel Dom´ınguez de la Iglesia Departamento de An´alisis Matem´atico, Universidad de Sevilla aJoint work with F. A. Gr¨unbaum, University of California at Berkeley 1 Outline of the talk The theory of matrix valued orthogonal polynomials (MVOP) was introduced by M. G. Krein in 1949. Systematically studied in the last 15 years. In 1997, Dur´an raised the problem of characterizing MVOP satisfying second order differential equations. Scalar case: Bochner (1929): Hermite-Laguerre-Jacobi ⇒New (non-trivial) matrix examples (2003): Dur´an-Gr¨unbaum-Pacharoni-Tirao. New phenomena: •MVOP satisfying first order differential operators. •Richer behavior of the algebra of differential operators. 2 Background I Given a self adjoint positive definite matrix weight function W we define a skew symmetric bilinear form: (P, Q) = Zb a P(t)W(t)Q∗(t)dt. This leads to a family of MVOP {Pn}n≥0with deg Pn=n, non–singular leading coefficient and (Pn, Pm) = Θ, n 6=m. Examples considered here satisfy DPn≡P′′ n(t)A2(t) + P′ n(t)A1(t) + Pn(t)A0= ΛnPn(t). Dis symmetric if (DP, Q) = (P, DQ). 3 Background II Generating examples: Solving symmetry equations. [Dur´an–Gr¨unbaum]: A2W=WA∗ 2 2(A2W)′=WA∗ 1+A1W, (A2W)′′ −(A1W)′+A0W=WA∗ 0. with boundary conditions: l´ım t→xW(t)A2(t) = Θ = l´ım t→x(W(t)A1(t)−A∗ 1(t)W(t)),for x=a, b. Matrix spherical functions associated with PN(C) = SU(N+ 1)/U(N). [Gr¨unbaum–Pacharoni–Tirao]. 4 Goal of this talk: the algebra Given a fixed family of MVOP {Pn}n≥0we study the algebra over C: D={D:DPn(t) = Λn(D)Pn(t), n = 0,1,2,...} where D= r X j=0 ∂j tFj(t) and Fj(t) = j X i=0 Bj iti. The map Λn:D −→ M(N, C), n = 0,1,2,... is a faithful representation, i. e., Λn(D1D2) = Λn(D1)Λn(D2) with D1, D2∈ D and if Λn(D) = Θ with D∈ D for all n, then D= Θ. 5 Scalar case If His the corresponding second order differential operator of the classical orthogonal polynomials (Hermite, Laguerre or Jacobi): Hn(t)′′ −2tHn(t)′=−2nHn(t) tLα n(t)′′ + (α+ 1 −t)Lα n(t)′=−nLα n(t) t(1 −t)P(α,β) n(t)′′ + (α+ 1 −(α+β+ 2)t)P(α,β) n(t)′= −n(n+α+β+ 1)P(α,β) n(t) then U= k X i=0 ciHi, where ci∈Cand Uis an even order differential operator. Then D=<H> 6 Matrix case Dcan be non–commutative, have more than one generator and have relations among the generators: Castro-Gr¨unbaum, The algebra of differential operators associated to a given family of matrix valued orthogonal polynomials: five instructive examples, IMRN, (2006). Gr¨unbaum-Pacharoni-Tirao, Matrix valued orthogonal polynomials of the Jacobi type: The role of group representation theory, Ann. Inst. Fourier, (2005). 7 The one step example: the initial differential operator W(t) = tα(1 −t)βT(t)T∗(t), α, β > −1,H1=A2(t)d2 dt2+A1(t)d dt +A0(t), A2(t) = t(1 −t)I A1(t) = 0 B B @ α+ 3 0 0 −1α+ 2 0 0−2α+ 1 1 C C A −t 0 B B @ (α+β+ 4) 0 0 0 (α+β+ 5) 0 0 0 (α+β+ 6) 1 C C A A0(t) = 0 B B @ 0 2(β−k+ 1) 0 0−(α+β−k+ 2) β−k+ 2 0 0 −2(α+β−k+ 3) 1 C C A ,1≤k≤β. 8 The one step example: polynomial eigenfunctions Given H1, there are two ways to generate a sequence of MVOP: In terms of the matrix hypergeometric function: 2F1(C, A, B;t) = X i≥0 (C, A, B)i ti i!, where (C, A, B)i+1 = (C+iI)−1(A+iI)(B+iI)(C, A, B)i, i≥0 and (C, A, B)0=I, introduced by Tirao in The matrix valued hypergeometric equation, Proc. Nat. Acad. Sci. U.S.A., (2003). Solving directly the algebraic matrix equations from the differential equation. 9 Closing remarks and future directions We hope to find theoretical proofs of these new phenomena. The matrix case is much richer than the scalar one. Possible applications: •Quantum mechanics: [Dur´an-Gr¨unbaum] P A M Dirac meets M G Krein: matrix orthogonal polynomials and Dirac´s equation, J. Phys. A: Math. Gen. (2006). •Time-and-band limiting: [Dur´an-Gr¨unbaum]A survey on orthogonal matrix polynomials satisfying second order differential equations, J. Comput. Appl. Math. (2005). 16