Algebraic Groups with Large Cayley Degree

Definitions and mathematical statement

Fix the ground field $k$ intended by the source. Let $G$ be a connected reductive algebraic $k$-group and let $\mathfrak g=\operatorname{Lie}(G)$. Use conjugation on $G$ and the adjoint action on $\mathfrak g$. A dominant rational map $f:G\dashrightarrow\mathfrak g$ is $G$-equivariant if \[ f(hgh^{-1})=\operatorname{Ad}(h)f(g) \] as an identity of rational maps. Since $\dim G=\dim\mathfrak g$, such a dominant map is generically finite; its degree is the function-field degree \[ \deg(f)=[k(G):f^*k(\mathfrak g)]. \] When this class of maps is nonempty, the source defines \[ \operatorname{Cay}(G)=\min_f\deg(f). \] The questions are:

1. Does there exist a connected reductive algebraic group in the intended field setting with $\operatorname{Cay}(G)>2$?
2. Does there exist such a group that is simple?

Here the first scope follows the class for which the source defines Cayley degree; the subsequent phrase "an algebraic group" does not supply a definition for arbitrary nonreductive or disconnected groups. For a finite defined Cayley degree, $\operatorname{Cay}(G)>2$ means that equivariant dominant maps exist but none has degree one or two. Empty-map classes are not assigned degree infinity without a stated convention.

The extract does not specify $k$, its characteristic, or the central-isogeny convention for simple groups. In the usual connected reductive usage, simplicity excludes nontrivial proper connected normal subgroups but can allow a finite center; no adjoint or simply connected restriction is imposed here.

Short English statement

Is the least degree of a conjugation-to-adjoint equivariant dominant rational map ever greater than two, and can this happen for a simple algebraic group? The field and precise simple-group convention are not supplied.

Sources

• Supplied report: Affine Algebraic Geometry (2007), Oberwolfach Reports, DOI 10.4171/owr/2007/01, pp. 5--82; proposed year 2007. https://doi.org/10.4171/owr/2007/01.
• Local UnsolvedMath registry, permanent ID 30000668, catalogue entry; created 25 August 2026, updated 16 September 2026. Its corrected_verified note says both existence questions and Cayley-degree notation were preserved. The supplied 21 August 2026 literature triage labels the extracted claim open and reports no located authoritative complete resolution. No additional external reference is supplied; this dated assessment is not independently updated here.