Analogies between finite-dimensional irreducible representations of SO(2n)and infinite-dimensional irreducible representations of Sp(2n,R). I. Characters and products

The analogy between the finite-dimensional spin representation Delta of SO( 2n) and the infinite-dimensional representation <(Delta)over tilde> of Sp(2 n,R) is made precise. It is then shown that this analogy can be extended so as to provide a precise link between each finite dimensional unitary irred ucible representation of SO(2n) and a corresponding infinite-dimensional un itary irreducible representation of Sp(2n,R). The analogy shows itself at t he level of the corresponding characters and difference characters, and inv olves the use of Schur function methods to express both characters and diff erence characters of SO(2n) and Sp(2n,R) in terms of characters of irreduci ble representations of their common subgroup U(n). The analogy is extended still further to cover the explicit decomposition of not only tensor produc ts of Delta and <(Delta)over tilde> with other unitary irreducible represen tations of SO(2n) and Sp(2n,R), respectively, but also arbitrary tensor pow ers of Delta and <(Delta)over tilde>. (C) 2000 American Institute of Physic s. [S0022-2488(00)00307-8].