T-series expansions for elements of \widetilde{H}

Ascertain whether every element of the Abdellatif–Hébert completed Iwahori–Hecke algebra \widetilde{H} admits a T-series expansion of the form \sum_{\lambda\in Y^+} h_\lambda T_\lambda with Weyl almost finite support, and, if not, characterize precisely which elements do.

Background

The paper compares the new completion \widehat{H} with \widetilde{H} (defined via a Bernstein–Lusztig style completion with W_0-almost finite support). While \widehat{H} admits natural T-series expansions by construction, the relationship for \widetilde{H} is subtler.

The authors can analyze T-expansions on \widetilde{H} \cap \widehat{H}, but whether all of \widetilde{H} admits such expansions remains unsettled.