Dirac Type Operators for Arithmetic Subgroups of Generalized Modular Groups

TL;DR

This paper introduces Dirac type operators on conformally flat manifolds formed by quotienting the upper half-space by arithmetic subgroups of generalized modular groups, exploring their fundamental solutions and related series.

Contribution

It develops fundamental solutions for Dirac operators on these manifolds and analyzes associated Eisenstein and Poincaré series, advancing the understanding of their spectral properties.

Findings

Fundamental solutions for Dirac operators are constructed.

Eisenstein and Poincaré series are analyzed on these manifolds.

Properties of these solutions and series are established.

Abstract

Fundamental solutions of Dirac type operators are introduced for a class of conformally flat manifolds. This class consists of manifolds obtained by factoring out the upper half-space of $\mathbb{R}^n$ by arithmetic subgroups of generalized modular groups. Basic properties of these fundamental solutions are presented together with associated Eisenstein and Poincar\'e type series.