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An analysis of the multiplicity spaces in classical symplectic branching

Abstract

The purpose of this dissertation is to develop a new approach to Gelfand-Zeitlin theory for the rank n symplectic group Sp(n,C). Classical Gelfand-Zeitlin theory rests on the fact that branching from GL(n,C) to GL(n-1,C) is multiplicity-free. Since branching from Sp(n,C) to Sp(n -1,C) is not multiplicity-free, the theory cannot be directly applied to this case. Let L be the n-fold product of SL(2,C). Our main theorem asserts that each multiplicity space that arises in the restriction of an irreducible representation of Sp(n,C) to Sp(n-1,C), has a unique irreducible L-action satisfying certain naturality conditions. We also give an explicit description of the L- module structure of each multiplicity space. As an application we obtain a Gelfand-Zeitlin type basis for all irreducible finite dimensional representations of Sp(n,C).

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