Birational Asymmetry Between Representations and Their Duals

Definitions and mathematical statement

Problem 2: birational dual asymmetry. Fix an algebraically closed field $k$ of characteristic zero. Do there exist a connected semisimple algebraic group $G$ and a finite-dimensional algebraic $G$-module $V$, with trivial stabilizers in general position, such that $V$ and its dual $V^*$ are not birationally $G$-isomorphic? Trivial stabilizers in general position means that there is a nonempty $G$-invariant Zariski-open subset on which every stabilizer is trivial, the usual generically free condition in this setting. It is not merely faithfulness of the representation.

The dual has the contragredient action $(g\cdot\ell)(v)=\ell(g^{-1}\cdot v)$. A birational $G$-isomorphism would be a rational map $\phi:V\dashrightarrow V^*$ with rational inverse and $\phi(g\cdot v)=g\cdot\phi(v)$ wherever both sides are defined, equivalently an equivariant isomorphism after restricting to suitable dense invariant open subsets. The desired example excludes such nonlinear birational maps, not just linear intertwining maps. Ordinary birationality of the underlying affine spaces is not the question, nor is stable birationality after adding auxiliary representations. The acting group is fixed in the equivariance condition; a group automorphism is not silently allowed as part of $\phi$.

The original record also asks about an automorphism $\sigma\in\operatorname{Aut}(G)$ and $V$ versus its $\sigma$-twist, for example the action $g\cdot_\sigma v=\sigma(g)\cdot v$ under a stated convention. The verification field identifies that as a distinct generalization recorded in OWR-1452-030. It is preserved here as a source cross-reference, not merged into this item's existential dual-representation question. The exact twist convention belongs to that separate formulation.

The supplied triage checked 2026-08-21 labels this extracted problem open and reports no located complete resolution, citing only the report. No precise later example or theorem is reproduced. The dated status assertion is retained without independent current-literature verification.

Short English statement

Find a generically free representation of a connected semisimple group that is not equivariantly birational to its contragredient dual. The obstruction must exclude nonlinear birational maps, not just linear isomorphisms. The automorphism-twist generalization belongs to the separately identified next source item.

Sources

• Local UnsolvedMath registry, permanent ID 30000665, code catalogue entry; Birational Asymmetry Between Representations and Their Duals, Oberwolfach Reports Open Problems; verification identifies source Problem 2.
• Affine Algebraic Geometry (2007), Oberwolfach Reports, DOI 10.4171/owr/2007/01, pp. 5--82. https://doi.org/10.4171/owr/2007/01. Proposed year 2007; no proposer named.
• Supplied cross-reference OWR-1452-030 for the distinct automorphism-twist generalization included in the raw original passage.
• The full report is the sole supplied literature source; no separate authoritative birational-asymmetry theorem is supplied. Literature triage checked 2026-08-21, status open; record updated 2026-09-16. The source-verification and dated status claims were not independently checked here.