Kashiwara's bilinear-form conjecture for extremal weight modules

Establish whether the bilinear form on the extremal weight module \(V(\lambda)\) is non-degenerate and satisfies Kashiwara's stated characterization and almost-orthonormality conditions, including \((u_\lambda,G(b))=\delta_{b,u_\lambda}\), \((xu,v)=(u,\rho(x)v)\), \((G(b),G(b'))\in\delta_{b,b'}+q^{-1}\mathbb Z[[q^{-1}]]\), and orthonormality on the weight space \(V(\lambda)_\lambda\).

Background

The paper proves the existence of a nonzero symmetric bilinear form on the extremal weight module satisfying contravariance with respect to the anti-automorphism ρ\rho. It then compares this result with a stronger conjecture attributed to Kashiwara, which additionally requires non-degeneracy, a prescribed pairing with the global basis, almost-orthonormality, and orthonormality on the extremal weight space. These stronger properties are known in several special cases, including affine type and basic weights, but are not established in general.

References

In [Kas05, Conj. 2.12], Kashiwara conjectured that there is a non-degenerate symmetric bilinear form on V (\lambda), characterized by (u\lambda, G(b)) = \delta_{b,u\lambda} (b ∈ B(λ)) and (xu, v) = (u, ρ(x)v) (x ∈ eU (g)), which satisfies (G(b), G(b′)) ∈ \delta_{b,b′} + q−1Z[[q−1]] for b, b′ ∈ B(λ) and (G(b), G(b′)) = \delta_{b,b′} for b, b′ ∈ B(λ)λ.

— The crystal of the categorified quantum group  (2609.30073 - Lee, 24 Sep 2026) in Remark 8.1.7, page 42

We do not know whether (8.2) holds in general.

— The crystal of the categorified quantum group  (2609.30073 - Lee, 24 Sep 2026) in Remark 8.1.7, equation (8.2), page 42