ScalingStacks

Proof of Theorem 6.3. Choose any $V^{\prime}\in\operatorname{Gr}(\operatorname{soc}(M),\underline{\mathbf{d}}_{1})$ that is stabilized by $B.$ Denote the stabilizer of $V^{\prime}$ by $Q\subset\operatorname{GL}(\operatorname{soc}(M))$ and the respective Weyl groups by $W_{Q}\subset W.$ Denote by $W^{Q}$ the set of shortest coset representatives in $W/W_{Q}.$ Let $U\subset B$ be the unipotent radical, $U^{-}\subset G$ its opposite and $U_{x}=U\cap xU^{-}x^{-1}$ for $x\in W.$ Then $U_{x}\cong\mathbb{A}^{l(x)}$ is an affine space.

The Grassmannian $\operatorname{Gr}(\operatorname{soc}(M),\underline{\mathbf{d}}_{1})$ admits an affine paving by $B$-orbits

$$\operatorname{Gr}(\operatorname{soc}(M),\underline{\mathbf{d}}^{1})=\biguplus_{w\in W^{Q}}\operatorname{Gr}(\operatorname{soc}(M),\underline{\mathbf{d}}^{1})_{w}$$

such that for $w\in W^{Q}$ there is an isomorphism

$$t_{w}:U_{w}\to\operatorname{Gr}(\operatorname{soc}(M),\underline{\mathbf{d}}^{1})_{w},u\mapsto uwV^{\prime}.$$

Taking preimages under the map $p$ from (6.2) yields a decomposition

$$\operatorname{Fl}(M,\underline{d})=\biguplus_{w\in W^{Q}}\operatorname{Fl}(M,\underline{d})_{w}.$$

Denote by $\underline{\mathbf{d}}^{\star}=(\underline{\mathbf{d}}^{2},\dots)\in\operatorname{Comp}(\mathbf{d}-\underline{\mathbf{d}}^{1})$ the composition obtained by removing the first entry $\underline{\mathbf{d}}^{1}$ from $\underline{\mathbf{d}}.$ For $w\in W^{Q}$ consider the map

$$\alpha:U_{w}\times\operatorname{Fl}(M/wV^{\prime},\underline{\mathbf{d}}^{\star})\to\operatorname{Fl}(M,\underline{d})_{w},\,(u,\underline{W})\mapsto\theta(u)(\underline{W}+wV^{\prime})$$

where for a flag $\underline{W}=(0\subset W^{1}\subset\dots)$ of $M/wV^{\prime}$ we denote the lift to a flag of $M$ by $\underline{W}+wV=(0\subset wV\subset W^{1}+wV\subset\dots).$ The map $\alpha$ is an isomorphism with inverse

$$\beta:\operatorname{Fl}(M,\underline{d})_{w}\to U_{w}\times\operatorname{Fl}(M/wV^{\prime},\underline{\mathbf{d}}^{\star}),\,\underline{V}\mapsto(u(V^{1}),\theta(u(V^{1})^{-1})(\underline{V})/wV^{\prime})$$

where we write $u(V^{1})=t_{w}^{-1}(V^{1})$ and for a flag $\underline{V}=(0\subset V^{1}\subset\dots)$ of $M$ we denote by $\underline{V}/V^{1}=(0\subset V^{2}/V^{1}\subset\dots)$ the projection to a flag of $M/V^{1}.$

By induction, each $\operatorname{Fl}(M/wV^{\prime},\underline{\mathbf{d}}^{\star})$ admits an affine paving which implies that $\operatorname{Fl}(M,\underline{\mathbf{d}})$ does. ∎