n-Dimensional global correspondences of Langlands over singular schemes (II)

TL;DR

This paper develops a comprehensive algebraic framework for understanding singularities within Langlands global correspondences, involving singularizations, deformations, and monodromies on sheaves over algebraic groups, to construct holomorphic and cuspidal representations.

Contribution

It introduces a new algebraic approach to singularities in Langlands correspondences, focusing on resolutions, deformations, and dynamics of singularities in n-dimensional representations.

Findings

Develops a phenomenology of singularities in Langlands correspondences.

Analyzes the geometry of versal deformations and blowups.

Studies dynamics involving singular hyperbolic and strange attractors.

Abstract

A rather complete phenomenology of the singularities is developed according to a new algebraic point of view in the frame of Langlands global correspondences. That is to say,a process of: -singularizations and versal deformations of these, -singularizations and monodromies of these, is envisaged on all the sections of sheaves of differentiable (bi)functions on (bi)linear algebraic (semi)groups constituting the n-dimensional representations of the global Weil groups. To get the searched holomorphic and cuspidal representations,it is necessary to consider: -the resolutions of the singularities and the blowups of the versal deformations; -the resolutions of the singularities in the monodromy cases. Furthermore, the geometry of the versal deformations and of their blowups is studied, as well as the associated dynamics leading to the consideration of singular hyperbolic attractors and of singular strange attractors.