Branching laws for 1-parameter families of representations of Lie groups and their asymptotic behavior

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The determination of branching laws, i.e., the decomposition of representations of a group G into irreducible ones upon restriction to a subgroup H is a fundamental problem of representation theory. In physical applications it describes the breaking of symmetries for quantum mechanical systems. Many branching laws are known in principle in the form of complicated combinatorial or topological formulas expressing multiplicities as alternating sums. Exceptions to this rule are results of Littelmann and Knutson, which so far are available only for commutative H. If a representation can be realized by holomorphic sections (as in the case of compact Lie groups via the Borel-Weil Theorem) it admits a reproducing kernel which carries the entire information of the representation. Thus decompositions of this kernel into suitable pieces invariant under the action of H yield decompositions of the restricted representations. In specific examples such decompositions have been obtained by Taylor expansions transversal to an H-orbit, an idea going back to S. Martens in the 1970s. We propose a systematic study of reproducing kernels associated with representations of compact Lie groups G with the goal to describe cancelation free branching laws and their asymptotic behavior.