Fundamental Hermite constants of linear algebraic groups

Abstract

Let $G$ be a connected reductive algebraic group defined over a global field $k$ and $Q$ a maximal $k$-parabolic subgroup of $G$. The constant $\gamma (G,Q,k)$ attached to $(G,Q)$ is defined as an analogue of Hermite's constant. This constant depends only on $G,$$Q$ and $k$ in contrast to the previous definition of generalized Hermite constants ([W1]). Some functorial properties of $\gamma (G,Q,k)$ are proved. In the case that $k$ is a function field of one variable over a finite field, $\gamma (G{L}_n,Q,k)$ is computed.