Hermite constants of algebraic groups and their applications

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On the cone of positive definite n by n real symmetric matrices, Hermite's function is defined as a quotient of the arithmetical minimum function and the reduced determinant. The determination of the actual value of the maximum of Hermite's function is equivalent to the determination of the densest lattice sphere packing in an n-dimensional Euclidean space. The local maxima of Hermite's function is characterized by Voronoi's theory. In this research project, we investigated a geometric and arithmetic generalization of Voronoi's theory. We determined exact values of a generalized Hermite constant of a symplectic group and some Rankin -Hermite's constants.