A Real-global equivariant Segal--Becker splitting, explicit Brauer induction, and global Adams operations

We prove a splitting result in global equivariant homotopy theory that is a simultaneous refinement of the Segal--Becker splitting and its `Real' and equivariant generalizations, and of the explicit Brauer induction of Boltje and Symonds. We show that the morphism of ultra-commutative Real-global ring spectra from $\Sigma^\infty_+ B_{\text{gl}}U(1)$ to the Real-global K-theory spectrum that classifies the tautological Real $U(1)$-representation admits a section on underlying Real-global infinite loop spaces. We prove that this global Segal--Becker splitting induces the classical Segal--Becker splittings on equivariant cohomology theories, and that it induces the Boltje--Symonds explicit Brauer induction on equivariant homotopy groups. As an application we rigidify the unstable Adams operations in Real-equivariant K-theory to global self-maps of the Real-global space $\mathbf{BUP}$.