Idempotency of the twisted locally analytic distribution algebra

Determine whether the twisted locally analytic distribution algebra D^{la}(G,B)′ is idempotent under derived tensor product over B[G]′; specifically, prove or disprove that D^{la}(G,B)′ ⊗^L_{B[G]′} D^{la}(G,B)′ ≅ D^{la}(G,B)′.

Background

The authors prove idempotency for the h-analytic and h+-analytic twisted distribution algebras, showing D_{h-an}(G,B)′ ⊗L_{B[G]′} D_{h-an}(G,B)′ ≅ D_{h-an}(G,B)′ and similarly for D{h+-an}(G,B)′. They also note that in the non-twisted case one can show D{la}(G,B) ⊗L_{B[G]} D{la}(G,B) ≅ D{la}(G,B).

However, they do not know if the analogous idempotency statement holds for the twisted locally analytic distribution algebra D{la}(G,B)′. Establishing this would extend full faithfulness and structural results to the locally analytic (la) level in the semilinear setting.

References

One can also show (similarly to Corollary 5.11) that {D}la(G,B)\otimesL_{B_{}[G]}{D}la(G,B) = {D}la(G,B). We omit the details since this identity will not be used anywhere in the article. We do not know if a similar identity holds for {D}la(G,B)' - this seems to be an interesting problem.

— Solid locally analytic representations in mixed characteristic  (2510.13673 - Porat, 15 Oct 2025) in Remark after Theorem “idempotent_dists”, Section 6.2 (Idempotency of distribution algebras)