arXiv subjects
Publications and source records attributed to Juan A. Tirao.
We describe families of polynomials arising in the study of the universal central extensions of Lie algebras introduced by Date, Jimbo, Kashiwara, and Miwa in their work on the Landau-Lifshitz equations. Two of the families of polynomials we show satisfy certain forth order linear differential equations, are orthogonal and are not of classical type.
The aim of this paper is to determine all irreducible spherical functions of the pair (G,K)=(SU(n+1),U(n)), where the highest weight of their K-types are of the form (m+l,...,m+l,m,...,m). Instead of looking at a spherical function Φof type πwe look at a matrix-valued function H defined on a section of the K-orbits in an affine subvariety of P_n(C). The function H diagonalizes, hence it can be identified with a column vector-valued function. The irreducible spherical functions of type πturn out to be parameterized by S={(w,r)\in Z x Z : 0\leq w, 0 \leq r \leq l, 0\leq m+w+r}. A key result to characterize the associated function H_{w,r} is the existence of a matrix-valued polynomial function Ψof degree l such that F_{w,r}(t)=Ψ(t)^{-1}H_{w,r}(t) becomes an eigenfunction of a matrix hypergeometric operator with eigenvalue λ(w,r), explicitly given. In the last section we assume that m\ge 0 and define the matrix polynomial P_w as the (l+1) x (l+1) matrix whose r-row is the polynomial F_{w,r}. This leads to interesting families of matrix-valued orthogonal Jacobi polynomials P_w^{α,β} for α,β>-1.