Asymptotic branching laws for finite dimensional representations of complex reductive Lie groups by geometric methods

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A fundamental problem in representation theory is that of decomposing an irreducible representation Π of a group G into irreducible representations of a subgroup, L, of G. For a finite dimensional representation Π of a complex reductive Lie group G the solution of this problem is given in principle by Kostant’s branching theorem, which gives a combinatorial expression for the multiplicities of the irreducible L-representations. However, the formula for the multiplicities is given by an alternating sum where cancellation occurs. It is desirable to have formulas for multiplicities given by a priori positive terms, such as for instance by counting the number of integral points of some convex polytope, or compact convex set, of Rn. We propose to construct convex bodies Δ in Rn, such that the integral points of Δ count multiplicities, at least in an asymptotic sense, of irreducible L- representations of Π. The bodies are to be constructed by the geometric realization of the representation Π as the space of holomorphic sections of a line bundle over a flag variety X. The idea is that Δ should encode the behaviour of holomorphic sections along a complete flag X(0)<...