ScalingStacks

Proof. We follow similar arguments to [Sau16, Lemma 1].

By the explicit description of nilpotent representations, $M$ is a direct sum of modules of the form $E(i,l).$ We can hence choose a basis of $\operatorname{soc}(M)$ by a choosing non-zero vectors in each $\operatorname{soc}(E(i,l)).$ With respect to this basis, $P$ is generated by elementary matrices and the proof can be reduced to the case $M=E(i,l_{1})\oplus E(i,l_{2}).$ We denote the standard basis vectors of $E(i,l_{1})$ and $E(i,l_{2})$ by $e_{j}$ and $f_{j},$ respectively.

The socle of $\operatorname{soc}(M)=\operatorname{soc}(E(i,l_{1}))\oplus\operatorname{soc}(E(i,l_{2}))=ke_{i}\oplus kf_{i}$ is two-dimensional in degree $i.$ Assume that $l_{1}>l_{2}.$ Then the radical filtration of the socle is

$$I^{\prime}=(0\subset ke_{i}\subset ke_{i}\oplus kf_{i}).$$

Let $g\in P.$ For $j<l_{2}$ we define the action $\theta(g)$ on $ke_{i-j}\oplus kf_{i-j}$ via the natural isomorphism $ke_{i-j}\oplus kf_{i-j}\cong ke_{i}\oplus kf_{i}.$ For $j\geq l_{2},$ we define the action $\theta(g)$ on $ke_{i-j}$ via the natural isomorphism $ke_{i-j}\cong ke_{i}.$ It can easily be checked that $\theta(g)\in\operatorname{Aut}(M)$ and $\operatorname{Res}\theta(g)=g.$ The case $l_{1}=l_{2}$ is proven similarly. ∎

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. ∎