The Generalized Triangle Inequalities in Symmetric Spaces and Buildings with Applications to Algebra

By Michael Kapovich, B. Leeb, John Millson. 2002

74 citations recorded by the source corpus.

In this paper we apply our results on the geometry of polygons in infinitesimal symmetric spaces, symmetric spaces and buildings, [KLM1, KLM2], to four problems in algebraic group theory. Two of these problems are generalizations of the problems of finding the constraints on the eigenvalues (resp. singular values) of a sum (resp. product) when the eigenvalues (singular values) of each summand (factor) are fixed. The other two problems are related to the nonvanishing of the structure constants of the (spherical) Hecke and representation rings associated with a split reductive algebraic group over Q and its complex Langlands’ dual. We give a new proof of the “Saturation Conjecture” for GL(l) as a consequence of our solution of the corresponding “saturation problem” for the Hecke structure constants for all split reductive algebraic groups over Q.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/14b2e0277880e075caba594f10218d67a9b4c6ef

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In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Anosov representations: domains of discontinuity and applications (2011, background): “(d) The classical Cartan projection μ : G → a can be used to define a refined Weyl chamber valued distance function: if δ denotes the Weyl chamber valued distance on the symmetric space G/K, then μ(g) = δ(K, gK), see for example [49, 64] for an account on this subject and references therein.”
A path model for geodesics in Euclidean buildings and its applications to representation theory (2004, methodology): “Given a group G as above, we get a nonarchimedean Lie group G D G.K/, to which we can associate a Euclidean building (a Bruhat–Tits building) X D XG. We refer the reader to [BT], [ KLM3 ] and [Rou] for more detailed discussion of the properties ofX.”
Buildings, spiders, and geometric Satake (2011, background): “m~λ : Gr( ~λ ) = Gr(λ1)×̃Gr(λ2)×̃ · · · ×̃Gr(λn)−→ Gr, where each Gr(λ ) is a sphere of radius λ (in the sense of weight-valued distances [19]) in Gr.”