We introduce a non-negative numerical invariant associated with each finite-dimensional representation of a Lie algebra. Given a representation, this invariant can be theoretically computed through a finite procedure involving the analysis of edges of certain convex polyhedra, but an explicit computation is quite involved. In this talk, I will demonstrate how to compute the invariant explicitly through several elementary examples. I will also briefly present some conjectures and related open questions. Finally, I plan to mention some connections of this numerical invariant with other areas of mathematics, such as analysis on homogeneous spaces, without going into technical details.
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© Toshiyuki Kobayashi