Lie Groups and Geometry

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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.