Invariant Theory Beyond Reductivity: Hilbert and Schwarz Theorems for Discrete Lorentz and Cocompact Groups

Abstract: Classical invariant theory provides a complete framework for reductive and compact Lie groups, but the non-reductive and non-compact cases are significantly more complex and less completely understood. This work investigates two distinct regimes: discrete subgroups of the Lorentz group $O(n,1)$ acting on $\mathbb{R}^{n,1}$, and cocompact actions of discrete groups on $\mathbb{R}^n$. We prove that for discrete Lorentz groups, the ring of polynomial invariants remains finitely generated, yet the classical Schwarz Theorem, stating that smooth invariants are generated by polynomial ones, fails. Conversely, for cocompact actions, the polynomial invariant ring collapses to constants, but a persistent structure of smooth invariants persists and is finitely generated over the smooth functions on the quotient. These results reveal a fourfold classification of invariant-theoretic regimes, extending the classical framework and delineating the boundaries of the Hilbert-Weyl and Schwarz theorems.