Rational cohomology of the algebraic double loop space equals that of U(n)

Establish that for every strictly monotonic class β = (d_n, …, d_1) in A_1(Fl) (i.e., 0 < d_n < d_{n-1} < … < d_1), the inclusion map from the algebraic double loop space of the flag variety to the topological double loop space, Ω^2_β(Fl) → Ω^2_{β,top}(Fl), induces an isomorphism of rational cohomology rings H^*(Ω^2_β(Fl), Q) ≅ H^*(U(n), Q).

Background

The paper studies the algebraic double loop spaces Ω^2_β(Fl) of the complete flag variety Fl = GL_{n+1}/B, focusing on based genus-0 maps P^1 → Fl of fixed degree class β. A central topological comparison is with Ω^2_{top}(U(n+1)), whose rational homotopy type is that of U(n). Motivated by this, the authors investigate to what extent Ω^2_β(Fl) captures similar cohomological properties.

Their main theorem determines the motivic class [Ω^2_β(Fl)] in the Grothendieck ring K_0(Var), which matches [GL_n × A^{D-n^2}] under a strict monotonicity condition on β. This equality suggests (but does not prove) a deeper cohomological equivalence, leading the authors to formulate the conjecture that the rational cohomology ring of Ω^2_β(Fl) agrees with that of U(n).