Spectral Invariants of Complex Solvable Lie Algebras: Nilradical Weights and Hyperplane Arrangements

Abstract

A central open problem in Lie theory is the classification of finite-dimensional complex solvable Lie algebras, but classification up to isomorphism becomes increasingly difficult as the dimension grows and the number of non-isomorphic families explodes. This motivates a classification method by studying coarser spectral invariants arising from the characteristic polynomial of the adjoint representation. We prove that the characteristic polynomial is determined by the generalized weights of the adjoint action on the nilradical. This addresses the problem of expressing the spectral index in Lie-theoretic terms, gives sharp bounds in terms of the nilradical and quotient dimensions, and characterizes spectral equivalence. We then resolve a second problem concerning the higher Betti numbers of the eigenvariety complement. We endow the distinct non-$z_0$ factors of the characteristic polynomial with a natural matroid structure, which we call the spectral matroid, and use the Orlik--Solomon algebra of the associated hyperplane arrangement to determine the Betti numbers and Poincar\'{e} polynomial combinatorially and to prove log-concavity of the Betti sequence. Finally, we apply our theory to solvable Lie algebras with abelian nilradical to obtain explicit characteristic polynomials and spectral-equivalence criteria, including examples of non-isomorphic algebras with identical characteristic polynomials.