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

表示论中的一个基本问题是将群G的不可约表示分解为群G的子群L的不可约表示。对于复约化李群G的有限维表示G，这个问题的解原则上由Kostant分支定理给出，该分支定理给出了不可约L-表示的重数的组合表达式。然而，多重性的公式是由发生抵消的交替和给出的。期望具有由先验正项给出的多重性的公式，例如通过计算Rn的某些凸多面体或紧凸集的整数点的数目。我们建议在Rn中构造凸体Δ，使得Δ的积分点至少在渐近意义上计数Π的不可约L-表示的重数。这些物体是通过将表示X的几何实现构造成旗簇X上的线丛的全纯截面的空间。这个想法是Δ应该编码沿着完整标志X（0）<.的全纯截面的行为。

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)<...