Operational calculus on a~complex semisimple Lie group

D. P. Zhelobenko

Izvestiya. Mathematics , Tome 3 (1969) no. 5, pp. 881-916.

Résumé

For every complex semisimple Lie algebra $\mathfrak g$ we construct a so-called operational calculus, which consists in the isomorphic embedding of $\mathfrak g$ along with its associative hull $\mathfrak G$ into a certain algebra of operator polynomials. We investigate the image of $\mathfrak G$ under this embedding; the resulting theorems comprise the algebraic analog of the functional duality theorems of harmonic analysis (theorems of Paley–Wiener type).

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