Journal of Lie Theory 19 (2009), No. 3, 463--481
Copyright Heldermann Verlag 2009
LU-Decomposition of a Noncommutative Linear System and Jacobi Polynomials
Alfredo O. Brega
CIEM-FaMAF, Universidad Nacional de Córdoba, Ciudad Universitaria, 5000 Córdoba, Argentina
brega@famaf.unc.edu.ar
Leandro R. Cagliero
CIEM-FaMAF, Universidad Nacional de Córdoba, Ciudad Universitaria, 5000 Córdoba, Argentina
cagliero@famaf.unc.edu.ar
[Abstract-pdf]
\def\a{{\frak a}} \def\g{{\frak g}} \def\k{{\frak k}} We obtain the LU-decomposition of a non commutative linear system of equations that, in the rank one case, characterizes the image of the Lepowsky homomorphism $U(\g)^{K}\to U(\k)^{M} \otimes U(\a)$. Although this system can not be expressed as a single matrix equation with coefficients in $U(\k)$, it turns out that obtaining a triangular system equivalent 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 between the ultraspherical Jacobi polynomials.
Keywords: Noncommutative LU-factorization, Jacobi polynomials, K-invariants in the enveloping algebra of g, Lepowsky homomorphism.
MSC: 33C45, 22E46; 33C05, 16S30
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