An algebraic cell decomposition of the nonnegative part of a flag variety (Q1283666)

Let \(G\) be a reductive linear algebraic group split over \(\mathbb{R}\) with fixed pinning \((T,B^+,B^-,x_i,y_i,i\in I)\) [see \textit{G. Lusztig}, Lie theory and geometry: in honor of Bertram Kostant, Prog. Math. 123, 531-568 (1994; Zbl 0845.20034) and Positivity in Lie theory: open problems, de Gruyter Expo. Math. 26, 133-145 (1998)]. Let \(\mathcal B\) be the variety of all Borel subgroups of \(G\). Let \(W=N_G(T)/T\) be the Weyl group of \(G\) with its simple reflection set corresponding to the pinning. Write \(\ell(w)\) for the length of \(w\in W\) and \(\dot w\) for a representative of \(w\) in \(N_G(T)\). Let \(U^+\) be the unipotent radical of \(B^+\). Denote by \(U^+_{\geq 0}\) the semigroup generated by \(\{x_i(a)\mid a\in\mathbb{R}_{\geq 0},\;i\in I\}\subset U^+\). Let \(U^+_{>0}=B^-\dot w_0B^-\cap U^+_{\geq 0}\) for the longest element \(w_0\) of \(W\). For \(B\in{\mathcal B}\) and \(g\in W\), write \(g\bullet B\) for \(gBg^{-1}\). Then the totally nonnegative part \({\mathcal B}_{\geq 0}\) of the flag variety \(\mathcal B\) is defined to be the closure of \(U^+_{>0}\bullet B^-\) in \(\mathcal B\). Let \({\mathcal C}^-_w=B^-\dot w\bullet B^-\) and \({\mathcal C}^+_w=B^+\dot w\bullet B^-\). Write \({\mathcal R}_{x,y}={\mathcal C}^-_y\cap{\mathcal C}^+_x\). The main result of the paper is to show that for any \(x\leq y\) in \(W\), the intersection \({\mathcal R}_{x,y}\cap{\mathcal B}_{\geq 0}\) is homeomorphic to \(\mathbb{R}^{\ell(y)-\ell(x)}_{>0}\) by a homeomorphism which extends to a real algebraic morphism \((\mathbb{R}\setminus\{0\})^{\ell(y)-\ell(x)}\to{\mathcal R}_{x,y}\). This result was conjectured by \textit{G. Lusztig} [see loc. cit.].