LU-decomposition of a noncommutative linear system and Jacobi polynomials, con O. Brega, J. of Lie Theory 19 (), 463-481.

OSCAR BREGA; LEANDRO CAGLIERO

JOURNAL OF LIE THEORY

HELDERMANN VERLAG

Año: 2009 vol. 19 p. 463 - 481

0949-5932

In this paper we obtain the LU-decomposition of anon commutative linear system of equationsthat, in the rank one case, characterizes the image of the Lepowsky homomorphism$U(lieg)^{K} o U(liek)^{M}otimes U(liea)$.Although this system can not be expressed as asingle matrix equation with coefficients in $U(liek)$, it turns out that obtaining a triangular system equivelent to it can be reduced to obtaining the LU-decomposition of a matrix $widetilde M_0$ with entries in a polynomial algebra.We prove that both the L-part and U-part of $widetilde M_0$ are expressed in terms of Jacobi polynomials.Moreover, each entry of the L-part of $widetilde M_0$ and of its inverse is given by a single ultraspherical Jacobi polynomial. This fact yields a biorthogonality relation betweenthe ultraspherical Jacobi polynomials.