Non-Archimedean Spectral Invariants of Hecke Operators on Metaplectic Covers: Motivic Satake Formalisms and Symplectic Stability

Mathematical Applications of Science Fiction We construct a rigorous framework for spectral invariants associated with the metaplectic double cover of the symplectic group over a non-Archimedean local field. Using the geometric Satake correspondence for twisted affine Grassmannians, we define a motivic Hecke algebra and establish an isomorphism with the Grothendieck ring of representations of the Langlands dual group. We prove that the spectral action of Hecke operators on the L2 spectrum of automorphic forms satisfies a strict symplectic stability condition, bounded by the motivic volume of the associated infinite-level Shimura variety. Furthermore, we demonstrate that the local character identities lift to the derived category of l-adic sheaves on the moduli stack of G-bundles, resolving the cocycle obstruction inherent in the Weil representation. These results have direct applications to the nonperturbative stability of BPS states in supersymmetric gauge theories defined on rigid analytic spaces.