Lie Groups and Geometry

米尔森和他的合作者将继续探索在对称空间和欧几里得建筑中构造三角形的问题与表示理论家感兴趣的代数问题之间的联系。他们特别感兴趣的是分裂p-ady群G的球面Hecke环的结构常数与朗兰兹对偶群的表示环的结构常数之间的关系。这两个环是同构的，并且有由相同的半群S(G的占优牛群)参数化为自然基。然而，两个环之间的同构(萨塔克变换)并不带有一个自然的基到另一个。基矩阵的变化是三角形的，用Lusztig算法计算。因此，比较S中相同参数值的结构常数是很自然的问题。Kapovich，Leeb和Millson得到的主要定理之一是，如果朗兰兹对偶群的表示环的结构常数不为零，则相应的(即对于相同参数值)Hecke环的结构常数也不为零。对于GL(N)(根据Hall，Green和Klein的经典工作)，相反是正确的，但对于其他组是错误的。Millson打算研究的主要问题之一是一般G的逆问题，即如果Hecke环的结构常数为非零，那么关于表示环的相应结构常数(或相关结构常数)可以说是什么。米尔森的工作始于高中几何中最早的一个定理--三个正实数a，b，c是平面上三角形的边长的定理，当且仅当它们满足“三角形不等式”，即a，b，c中的每一个都小于或等于其他两个的和。在任何齐次几何中，试图给出三类测地线段的等距条件是一个自然问题，这些条件是人们将它们组装成一个三角形的充要条件。在欧几里得几何和双曲几何的情况下，测地线段由其长度确定为等距，当且仅当三条测地线段的长度满足上述三角形不等式时，才能将三条测地线段组装成一个三角形。然而，对于许多例子(例如，秩为r的非紧对称空间大于1)，测地线段可由r维单圆锥中的元素参数化至等距。人们应将该圆锥体中的一点视为“向量值长度”。一个值得注意的事实是，在这样的长度向量的三倍的情况下，存在一个齐次线性不等式系统，这给出了将具有这些长度的三个线段组装成一个三角形的充要条件。Millson和他的合作者称这些不等式为“广义三角形不等式”。更值得注意的是，广义三角不等式给出了解决代数群论中某些基本代数问题所必需的条件。这些条件几乎是充分的，在米尔森和他的合作者未来的工作中也是如此。

Millson and his collaborators will continue to explore the connections between the problems of constructing triangles in symmetric spaces and Euclidean buildings and problems in algebra of interest to representation theorists. They are especially interested in the relations between the structure constants of the spherical Hecke ring of a split p-adic group G and those of the representation ring of the Langlands dual group. The two rings are isomorphic and have natural bases parametrized by the same semigroup S (the set of dominant coweights of G). However the isomorphism between the two rings (the Satake transform) does not carry one natural basis to the other. The change of basis matrix is triangular and was computed by Lusztig. Thus it is a natural problem to compare the structure constants for the same parameter values in S. One of the main theorems obtained by Kapovich, Leeb and Millson is that if a structure constant for the representation ring of the Langlands' dual group does not vanish then the corresponding (ie for the same parameter values) structure constant of the Hecke ring does not vanish. The converse is true for GL(n) (by classical work of Hall, Green and Klein) but is false for other groups. One of the main problems Millson intends to work on is the converse problem for general G i.e if the structure constant for the Hecke ring is nonzero what can be said about the corresponding structure constant (or related structure constants) for the representation ring. Millson's work began with one of the first theorems of high-schoolgeometry - the theorem that three positive real numbers a,b,c are the side-lengths of a triangle in the plane if and only if they satisfy the"triangle inequalities", that is, each of a,b,c is less than or equalthe sum of the other two. It is a natural problem to try to give conditions on three isometry classes of geodesic segments in any homogeneous geometry that are necessary and sufficient in order that one can assemble them intoa triangle. In the case of Euclidean and hyperbolic geometries a geodesicsegment is determined up to isometry by its length and three geodesicsegments can be assembled into a triangle if and only if the three lengthssatisfy the above triangle inequalities. However for many examples (e.g the noncompact symmetric spaces of rank r larger than 1) geodesic segments are parametrized up to isometry by elements in a simplicial cone of dimension r. One should think of a point in this cone as a "vector-valued length". It is a remarkable fact that there is a system of homogeneous linear inequalities ina triple of such length vectors that give necessary and sufficient conditions for assembling three segments with these lengths into a triangle.Millson and his collaborators call these inequalities the "generalizedtriangle inequalities". It is even more remarkable that the generalizedtriangle inequalities give conditions that are necessary in order thatcertain fundamental algebra problems in the theory of algebraic groupscan be solved. These conditions are almost sufficient as will be madeclear in future work of Millson and his collaborators.