Epstein vector zeta functions related to the ADE Lie algebras

TL;DR

This paper introduces a vector-valued generalization of Epstein zeta functions linked to ADE Lie algebra root lattices, establishing their functional equations and classifying cases where these hold.

Contribution

It constructs vector-valued Epstein zeta functions associated with ADE root lattices and classifies when their matrix functional equations are satisfied.

Findings

Derived a matrix functional equation of Riemann type for the zeta functions.

Classified lattices and invariant subspaces satisfying the functional equation.

Connected the functional equations to the Weil representation of the metaplectic group.

Abstract

We introduce a vector-valued generalization of the Epstein zeta functions associated with the root lattices of ADE-type Lie algebras. The quadratic forms defining these lattices correspond to the Gram matrices of the simple roots. Using the discriminant group D = P/Q, we construct vector-valued theta series that realize the Weil representation of the metaplectic group Mp(2,Z). The proposed Epstein vector zeta functions are obtained as the Mellin transform of these theta series. By exploiting the equivariance properties of the theta vectors, we derive a matrix functional equation of the Riemann type. We show that the existence of this functional equation is governed by a selection rule: it holds specifically for the subspace of C-invariant vectors, where C is the central element of Mp(2,Z). Finally, we provide a complete classification of the lattices and invariant subspaces for which this matrix functional equation is satisfied.