Embedding the spherical Hecke algebra into a completed Iwahori–Hecke algebra as functions on G^+

Determine whether there exists an embedding of the spherical Hecke algebra H_K into a completed Iwahori–Hecke algebra—realized as Iwahori-biinvariant functions on the positive Cartan semigroup G^+—that is compatible with the convolution relations of functions on G^+.

Background

For Kac–Moody groups over non-Archimedean local fields, the spherical Hecke algebra H_K requires infinite sums, and a direct embedding into the Iwahori–Hecke algebra H (or a completion thereof) as functions on G^+ is nontrivial.

The authors explain that at the level of vector spaces an expansion of spherical elements into Iwahori double cosets is possible, but multiplying such expansions leads to infinite coefficients, obstructing a straightforward embedding that respects convolution. They later introduce a bimodule approach to relate H and H_K, but an embedding that preserves functional relations remains unsettled.