Let G be a connected semisimple group, let P be a parabolic subgroup of G, and let U_P be its unipotent radical. Let B be a Borel subgroup and set E=G×_B mathfraku_P^perp as a homogeneous vector bundle over G/B. The projectivization PE^ is a projective bundle. Parabolic partial-implosion conjecture. The projective bundle PE^ is of Fano type. By the preceding theorem, this is equivalent to the symplecticity of overlineT^(G/U_P). The conjecture is supported by the evidence in the cited proposition; it has been confirmed for G=operatornameSL_n and G=operatornameGL_n.

Symplecticity conjecture for parabolic partial implosions

The conjecture asserts that a certain projective bundle is of Fano type, equivalently that the affine closure of the cotangent bundle of a horospherical space is symplectic. The current paper labels this as Conjecture 1.7 and does not claim a general proof.

Known results

The conjecture is confirmed for $G=\operatorname{SL}_n$ and $G=\operatorname{GL}_n$ by T. Gannon and B. Webster; the source gives no exact publication date or separate reference.

Current arXiv version, retrieved August 27, 2026

Version $2$ of The affine closure of cotangent bundles of horospherical spaces presents the statement as an open conjecture and reports only the $\operatorname{SL}_n$ and $\operatorname{GL}_n$ cases. A catalogue entry for the same work records a February 2025 claim that Gannon proved the conjecture, but the current version does not substantiate a general proof.

Current status (as of August 2026): The general conjecture is open; the $\operatorname{SL}_n$ and $\operatorname{GL}_n$ cases are reported as confirmed, while the earlier general-proof claim remains unsupported by the current paper version.

Symplecticity conjecture for parabolic partial implosions

Primary source

Progress summary

Solutions 0