Slopes of Euclidean lattices, tensor product and group actions

Abstract

We study the behaviour of the minimal slope of Euclidean lattices under tensor product. A general conjecture predicts that ${\mu }_{min}(L\otimes M)={\mu }_{min}(L){\mu }_{min}(M)$ for all Euclidean lattices $L$ and $M$. We prove that this is the case under the additional assumptions that $L$ and $M$ are acted on multiplicity-free by their automorphism group, such that one of them has at most $2$ irreducible components.

Update of a previous version. The main result is now stated for Hermitian bundles over totally real or CM number fields. A few proofs of intermediate results have been simplified