Geometric Implementation of Morse Theory Derived from Hecke Algebras in Ring Spaces

This paper presents a rigorous geometric realization of Morse theory within the framework of singular ring topological spaces, resolving the long-standing theoretical obstacles in non-smooth manifolds due to the lack of metric regularity. By embedding the Iwahori-Hecke algebra into the stratified filtration structure, we utilize the Jucys-Murphy elements and the Casimir center to construct a class of canonical "Hecke-Morse potential functions". The study shows that the non-commutative braiding relations of the algebra intrinsically encode the bifurcation dynamics behavior of the gradient flow across Whitney stratifications, effectively quantizing the classical Morse critical point set. Further, using the derived category tools, we establish a spectral sequence that converges from the local homology of the critical sub-manifolds to the global homology of the space. The core theorem confirms that under geometric purity conditions, the resulting Morse-Witten complex is isomorphic to the induced module structure of the Hecke algebra, proving that the Betti numbers of the ring space are strictly controlled by the Kazhdan-Lusztig basis. This framework not only eliminates the ambiguity in the definition of the gradient flow in the singular domain but also provides an intuitive geometric realization for the representation theory of quantum groups.