In this talk I will discuss some recent progress on the construction of

Stable Transfer Operators Sρ associated to an L-embedding ρ : LGLn → LG.

While many such explicit constructions remain out of reach, I will discuss a

manner in which this general case can be largely reduced to an understanding

of the case of an embedding ρ : LS → LG for a maximal torus S of GLn.

This method relies on building sections for the transfer maps associated to the

embeddings LS → LGLn for each maximal torus S of GLn, in addition to a

related family of related maps. As a guiding example, I will give a construction

of the transfer associated to the diagonal embedding LGLn → LGLn × GLn

which ought to be considered as a type of non-abelian convolution on the space

of orbital integrals on the Steinberg-Hitchin base of GLn.