Identify symmetric-pair medium algebras with type -1 coinvariants

Construct, for every principal involution η of a complex reductive Lie algebra g and every dominant integral weight λ of its fixed-point Lie algebra k=g^η, a dominant integral weight λ' of g such that the symmetric-pair medium algebra M^λ(g,η) is isomorphic, possibly after taking the torsion-free quotient, to the type -1 coinvariant algebra M^{λ'}(g)_ι, compatibly with an isomorphism C[p]^k ≅ C[g]^g_ι.

Background

The paper introduces symmetric-pair medium algebras for a reductive Lie algebra g with involution η, writing g=k⊕p and defining Mλ(g,η) from k-equivariant maps p→k. For principal involutions, invariant-theoretic properties make these algebras natural candidates for fixed-point or coinvariant constructions. The conjecture would connect symmetric-pair representation theory, type -1 medium-algebra automorphisms, and quasi-split/quasi-anisotropic duality.

References

In this case, we expect a connection to the automorphisms of \ref{subsec:MA_autos} of the following form. Let $g$, $\eta$, $k$ and $p$ be as above, and assume that $\eta$ is principal. Let $\lambda$ be a dominant integral weight of $k$. Then there exist a dominant integral weight $\lambda'$ of $g$ such that the automorphism $\iota = \iota_{\eta,-1}$ of $M{\lambda'}(g)$ is defined, and we have $M\lambda(g,\eta) \cong M\lambda(g)_\iota$, possibly after taking the torsion-free quotient on the right-hand side, compatibly with an isomorphism $C[p]k \cong C[g]g_\iota$.

— Automorphisms of medium algebras and quasi-anisotropic real affine Grassmannians  (2610.18170 - Elkner, 13 Oct 2026) in Conjecture, Section 10, “Medium algebras for symmetric pairs” (label conj:principal_coinv)

Nevertheless, calculations in the real case suggest the following: With notation as above, suppose $\sigma$ is a quasi-anisotropic real structure and $Lie(\Check{G}\sigma)$ does not contain a copy of $sl_{2n+1}(R)$ for any $n\in Z_{>0}$. If $\sigma(\lambda) = \lambda$, then $(I_\lambda){\sigma*}\cong J\lambda$, and so we have the following strengthened version of eq:coinv_main_total:

— Automorphisms of medium algebras and quasi-anisotropic real affine Grassmannians  (2610.18170 - Elkner, 13 Oct 2026) in Conjectures 1 and 2, Section 6.4, subsection “Specialisation of coinvariants and torsion”