Lie Groups and Geometry

李群和几何

• 批准号: 0405606
• 负责人: John Millson
• 金额: $ 12.97万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-15 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0405606&HistoricalAwards=false
• 关键词: Lie; Groups; Geometry

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.

米尔森和他的合作者将继续探索在对称空间中构造三角形和欧几里得建筑的问题以及表示理论家感兴趣的代数问题之间的联系。他们对分裂p进群G的球形赫克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特别感兴趣。这两个环是同构的，并且具有由相同的半群 S（G 的主权重集）参数化的自然基。然而，两个环之间的同构（佐竹变换）并不将一个自然基础传递给另一个环。基矩阵的变化是三角形的，由Lusztig计算。因此，比较 S 中相同参数值的结构常数是一个自然的问题。 Kapovich、Leeb 和 Millson 获得的主要定理之一是，如果朗兰兹对偶群的表示环的结构常数不消失，则对应的（即对于相同参数值）赫克环的结构常数不会消失。对于 GL(n)（由 Hall、Green 和 Klein 的经典著作得出），反之亦然，但对于其他群则不然。 Millson 打算解决的主要问题之一是一般 G 的逆问题，即如果 Hecke 环的结构常数非零，那么表示环的相应结构常数（或相关结构常数）可以说是怎样的。 米尔森的工作始于高中几何的第一个定理——该定理认为三个正实数a、b、c是平面中三角形的边长当且仅当它们满足“三角形不等式”，即a、b、c中的每一个都小于或等于另外两个的和。尝试给出任何齐次几何中测地线段的三个等距类的条件是一个自然的问题，这些条件是必要且充分的，以便可以将它们组装成三角形。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，并且当且仅当三个长度满足上述三角形不等式时，三个测地线段可以组装成三角形。然而，对于许多示例（例如，秩 r 大于 1 的非紧对称空间），测地线段通过维度 r 的单纯圆锥中的元素参数化为等距。人们应该将该锥体中的一点视为“向量值长度”。值得注意的是，在这样的长度向量的三重中存在一个齐次线性不等式系统，它为将具有这些长度的三个线段组装成三角形提供了充分必要条件。米尔森和他的合作者将这些不等式称为“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理论中某些基本代数问题所必需的条件。这些条件几乎已经足够了，米尔森和他的合作者将在未来的工作中明确这一点。