Smoothness of the complete mixed truncated flag variety F_{σ̲}(λ)

Determine whether, for any simple simply connected complex Lie group G with opposite Borel subgroup B_− and Weyl group W, and for any regular dominant integral weight λ, the variety F_{σ̲}(λ) is smooth when σ̲ lists all elements of W. Here, for each σ∈W, let D(σ)⊂L(λ) be the opposite Demazure submodule generated by the extremal vector v(σ), let L_σ(λ)=L(λ)/D(σ), and let X_σ(λ)⊂P(L_σ(λ)) be the closure of the B_−-orbit of the line [v_σ(λ)]; for a tuple σ̲=(σ_1,…,σ_m), define F_{σ̲}(λ) as the closure of the B_−-orbit of ([v_{σ_1}(λ)],…,[v_{σ_m}(λ)]) in the product ∏_{i=1}^m X_{σ_i}(λ).

Background

The paper develops a framework connecting blow-ups of Schubert subvarieties in flag varieties with truncated representations and truncated flag varieties. For a simple simply connected group G, a regular dominant weight λ, and each Weyl group element σ, the truncated module L_σ(λ)=L(λ)/D(σ) yields a truncated flag variety X_σ(λ)=\overline{B_- [v_σ(λ)]}⊂P(L_σ(λ)).

For a collection σ̲=(σ1,…,σ_m), the authors define F{σ̲}(λ) inside ∏i X{σi}(λ) as the closure of the B- orbit of the product of highest-weight lines. This construction generalizes the mixed blow-up picture proved for Grassmannians, where the “complete” mixed blow-up is shown to be smooth. Motivated by that example, the authors conjecture smoothness in the general complete case when σ̲ runs over all Weyl group elements.