Abstract: Matrix special functions of symmetric pairs (G,H) are a fruitful source of special functions. One usually studies their restrictions to an appropriate torus. They diagonalise a commutative algebra of differential operators that arises from invariant differential operators on G via the radial part decomposition.
I first recap the well-known theory for Riemannian symmetric pairs (by Cartan, Harish-Chandra, and others) and then sketch some of the obstacles that the pseudo-Riemannian case presents. These obstacles are addressed by a decomposition by Matsuki, which I use to construct a radial part decomposition for the pseudo-Riemannian case.
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