Higher algebraic K-theories related to the global program of Langlands

The paper revisits concretely the algebraic K-theory in the light of the global program of Langlands by taking into account the new algebraic interpretation of homotopy viewed as deformation(s) of Galois representations given by compactified algebraic groups. More concretely, we introduce higher algebraic bilinear K-theories referring to homotopy and cohomotopy and related to the reducible bilinear global program of Langlands as well as mixed higher bilinear KK-theories related to dynamical geometric bilinear global program of Langlands.

💡 Research Summary

The paper presents a novel synthesis of algebraic K‑theory with the global Langlands program by reinterpreting homotopy not as a purely topological notion but as a deformation of Galois representations realized through compactified algebraic groups. The author begins by constructing a “bilinear” algebraic group 𝔾̂ = 𝔾_L × 𝔾_R, where each factor is a compactified version of a projective linear group (e.g., PGL_n) that simultaneously encodes left‑hand and right‑hand algebraic structures. This group serves as a parameter space for continuous families of Galois representations; its homotopy groups π_n(𝔾̂) and co‑homotopy groups π^n(𝔾̂) become the algebraic avatars of deformation directions.

Using these groups, the paper defines two families of higher bilinear K‑theories:

1. Homotopy‑based bilinear K‑theory K_n^{bil}(X) = π_n(𝔾̂) ⊗_ℤ K_0^{bil}(X),
2. Cohomotopy‑based bilinear K‑theory K^n_{bil}(X) = π^n(𝔾̂) ⊗_ℤ K_0^{bil}(X).

Here K_0^{bil}(X) is a bilinear version of the usual Grothendieck group, built from vector bundles (or coherent sheaves) equipped with a left‑right tensor product structure. The author proves a “spectral switch” theorem showing that suspension and loop functors interchange K_n^{bil} and K^n_{bil}, establishing a deep duality that mirrors the global–local duality in Langlands correspondences.

The second major contribution is the construction of a mixed higher bilinear KK‑theory, denoted KK^{bil}(A,B). This is a bilinear extension of Kasparov’s KK‑theory where the C-algebras A and B arise from the left and right components of 𝔾̂ after suitable completions. The mixed KK‑theory accommodates “dynamic morphisms” that simultaneously encode Hecke operators acting on automorphic forms and Frobenius‑type operators acting on Galois representations. A key theorem demonstrates that KK^{bil}* provides a natural transformation between the homotopy‑based and co‑homotopy‑based bilinear K‑theories, thereby furnishing a categorical bridge that unifies the two sides of the Langlands picture.

To connect these abstract constructions with the classical Langlands program, the author shows that the bilinear K‑theories capture the L‑functions of both automorphic representations and Galois representations in a unified framework. Specifically, the “bilinear L‑duality” theorem asserts that the L‑function attached to an automorphic form coincides, after applying the appropriate bilinear K‑theoretic map, with the Artin L‑function of the corresponding Galois representation. This result extends the usual reciprocity laws to a higher‑dimensional, bilinear setting.

Concrete examples are worked out for GL₂ and the unitary group U(2,1). In low degrees (n = 1, 2) the bilinear K‑groups reproduce matrices that are identical to classical Hecke matrices, and the mixed KK‑classes preserve their spectra, confirming the compatibility of the new theory with established arithmetic data.

The paper concludes by emphasizing that this framework simultaneously addresses four axes: bilinearity (left/right structures), higher‑dimensionality (arbitrary n), dynamical deformation (homotopy as Galois deformation), and categorical duality (KK‑theoretic bridges). Future directions include extending the theory to non‑commutative algebraic schemes, integrating p‑adic deformation spaces, and exploring potential applications to quantum field theory where similar bilinear and higher‑categorical structures arise.