Investigating the Concept of Mirror Symmetry and the K-theory of P-adic Groups

The relationship between mirror symmetry and the K-theory associated with p-adic groups, seems to be neglected and distant. This paper explores a deep connection between mirror symmetry and the k-theory related to p-adic groups. Originating from string theory, mirror symmetry provides significant revelations about the geometry of complex manifolds. In both algebraic topology and algebraic geometry, k-theory is presented as an influential tool, where the study investigates how mirror symmetry, primarily used in the context of complex varieties, can be modified for application to p-adic groups. By delving into the essential structures and relationships, this research presented a novel perspective on how arithmetic geometry is related to theoretical physics. The results of this investigation deepen the understanding of the algebraic and geometric properties of p-adic groups, with ramifications for number theory, p-adic approximations of vertex functions, representation theory, and the area of geometric engineering.