On the collapsing of homogeneous bundles in arbitrary characteristic

The author would like to express his gratitude to Ryan Kinser for his valuable comments and suggestions on this work.

Let $G$ be a connected reductive group over $\Bbbk$, $B$ a Borel subgroup and $U$ its unipotent radical. We fix a maximal torus $T\subset B$, and denote by $X(T)$ its group of characters. We denote by $\langle\cdot,\cdot\rangle$ the standard pairing between $X(T)$ and the group of cocharacters. Let $\Phi\subset X(T)$ denote the set of roots and $\Phi_{+}\subset\Phi$ the set of positive roots with respect to the choice of $B$. We denote by $\rho$ the half sum of all the positive roots. The set of simple roots in $\Phi_{+}$ is denoted by $S$. We let $\mathcal{W}=N(T)/T$ be the Weyl group of $G$, and $w_{0}\in W$ its longest element.

For $I\subset S$, consider the standard parabolic subgroup $P:=P_{I}\subset G$. We have a Levi decomposition $P_{I}=L_{I}\ltimes U_{I}$, where $U_{I}$ is the unipotent radical of $P$ and $L:=L_{I}$ is reductive. Let $\mathcal{W}_{I}$ be the subgroup generated by the reflections $s_{\alpha}$ with $\alpha\in I$, and $w_{I}$ the longest element in $\mathcal{W}_{I}$. We choose the set $\mathcal{W}^{I}$ of representatives of the cosets of $\mathcal{W}/\mathcal{W}_{I}$ as

We have the Bruhat decomposition of $G$ into $B\times P$-orbits (see [Jan03, Section II.13]):

For $w\in\mathcal{W}^{I}$, we put $U(w):=U\cap wU^{-}w^{-1}$, where $U^{-}$ is the opposite unipotent radical. The multiplication map induces an isomorphism of $U(w)$-varieties (see [Jan03, Section II.13.8])

We denote by $X(w)_{P}$ the Schubert variety that is the image of $\overline{BwP}$ under the locally trivial projection $G\to G/P$. For $P=B$, we write $X(w):=X(w)_{B}$.

For any representation $M$ of $P$, we denote by $\mathcal{V}(M)$ the sheaf of sections of the homogeneous vector bundle $G\times_{P}M$. For $\lambda\in X(T)$, we put $\mathcal{L}(\lambda):=\mathcal{V}(\Bbbk_{-\lambda})$, where $\Bbbk_{-\lambda}$ is the $1$-dimensional representation of $B$.

A weight $\lambda\in X(T)$ is dominant if $\langle\lambda,\alpha^{\vee}\rangle\geq 0$, for all simple roots $\alpha\in S$. The set of dominant weights is denoted by $X(T)_{+}$. For $\lambda\in X(T)_{+}$, we call the space of sections

a dual Weyl module. It has lowest weight $-\lambda$ and highest weight $-w_{0}\cdot\lambda$. The module $\Delta_{G}(\lambda)=\nabla_{G}(\lambda)^{*}$ is called a Weyl module, that has a non-zero highest weight vector of weight $\lambda$, and this generates $\Delta_{G}(\lambda)$ as a $G$-module. It is known that $\nabla_{G}(\lambda)$ has a unique simple submodule, of highest weight $-w_{0}\cdot\lambda$.

When $\operatorname{char}\Bbbk=p>0$ and $e\geq 1$ is an integer such that $(p^{e}-1)\rho$ is a weight of $G$, we denote by $\operatorname{St}_{e}=\nabla_{G}((p^{e}-1)\rho)$ the $e^{\operatorname{th}}$ Steinberg module, and put $\operatorname{St}:=\operatorname{St}_{1}$. The assumption is superficial as we can always replace $G$ by $\operatorname{rad}G\times\tilde{G}$, where $\operatorname{rad}G$ denotes the radical of $G$ and $\tilde{G}$ the universal cover of $[G,G]$, and $(p^{e}-1)\rho$ is a weight of $\operatorname{rad}G\times\tilde{G}$ for all $e\geq 1$.

Let $P=P_{I}$ be a parabolic subgroup. For $\lambda\in X(T)_{+}$, put $\mathcal{V}(\lambda):=\mathcal{V}(\nabla_{L}(\lambda))$ (here $U_{I}$ acts trivially on $\nabla_{L}(\lambda)$). The quotient map $\pi:G/B\to G/P$ induces a quasi-isomorphism

By abuse of notation, we use the same notation for the respective bundles on Schubert varieties that are obtained by restriction. We record the following result.

When $\operatorname{char}\Bbbk=p>0$, for a $\Bbbk$-space $V$ and $e\in\mathbb{Z}_{\geq 0}$ we denote by $V^{(e)}$ the abelian group $V$ with the new $\Bbbk$-space structure $c\cdot v:=c^{1/p^{e}}\cdot v$. When $V$ is a module over an algebraic group $G$, then $V^{(e)}$ also has a $G$-module structure [Jan03, Section I.9.10]. If $A$ is a $\Bbbk$-algebra, then so is $A^{(e)}$ by using the same multiplicative structure.

We call a domain $A$ strongly $F$-regular if for every non-zero $c\in A$ there exists $e>0$ such that the $A^{(e)}$-map $cF^{e}:A^{(e)}\to A$ given by $x\mapsto cx^{p^{e}}$ is $A^{(e)}$-split.

As we do not need it for our purposes, we refer the reader to [HH94a] for the definition of $F$-rational rings (see (2.5) below for some of its important properties).

When $\operatorname{char}\Bbbk=0$, an algebraic variety $X$ has rational singularities, if for some (hence, any) resolution of singularities $f:Z\to X$ (i.e. $Z$ is smooth, and $f$ proper and birational), the natural map $\mathcal{O}_{X}\to\mathbf{R}f_{*}\mathcal{O}_{Z}$ is a (quasi-)isomorphism. Further, we say a ring $A$ is of strongly $F$-regular type if there exist some subring $R$ of $\Bbbk$ which is of finite type over $\mathbb{Z}$, and some $R$-algebra $A_{R}$ which is flat of finite type over $R$, such that $A_{R}\otimes_{R}\Bbbk\cong A$ and for the closed points $\mathfrak{m}$ in a dense open subset of $\operatorname{Spec}R$, the ring $A_{R}\otimes_{R}R/\mathfrak{m}$ is strongly $F$-regular.

An affine variety $X$ is $F$-rational (resp. strongly $F$-regular or of strongly $F$-regular type) if $\Bbbk[X]$ is so. We have the following implications (where CM stands for Cohen–Macaulay):

Furthermore, $F$-rationality implies pseudo-rationality [Smi97] and rational singularities in positive characteristic as defined in [Kov20]. When $\operatorname{char}\Bbbk=0$, a ring has log terminal singularities if and only if it is of strongly $F$-regular type and $\mathbb{Q}$-Gorenstein (see [HW02]).

Now let $A$ be a $G$-algebra and $\operatorname{char}\Bbbk=p>0$. We can assume that $(p^{e}-1)\rho$ is a weight of $G$ for $e\geq 1$ (otherwise replace $G$ by $\operatorname{rad}G\times\tilde{G}$). Following [Has12, Section 4], we say that $A$ is $G$-$F$-pure if there exists some $e\geq 1$ such that the map $\operatorname{id}\otimes F^{e}:\operatorname{St}_{e}\otimes A^{(e)}\to\operatorname{St}_{e}\otimes A$ splits as a $(G,A^{(e)})$-linear map.

Now we study the coordinate ring of $\overline{BwB}\subset G$, where $w\in\mathcal{W}$. Consider the rational $B\times T$-subalgebra of $\Bbbk[\overline{BwB}]^{U}$ consisting of dominant $T$-weight spaces

Consider the section ring $C(X(w)):=\bigoplus_{\lambda\in X(T)_{+}}\!H^{0}(X(w),\mathcal{L}(\lambda))$.

For the remainder of the subsection, we assume that $\operatorname{char}\Bbbk>0$.

When $\Gamma$ is saturated, the algebra $A$ as above is strongly $F$-regular [Has03, Lemma 5.6].

Take a (possibly infinite-dimensional) $G$-module $V$. Following Donkin [Don85], an ascending exhaustive filtration

of $G$-submodules of $V$ is a good filtration (resp. Weyl filtration) of $V$, if each $V_{i}/V_{i-1}$ is isomorphic to a dual Weyl module (resp. to a Weyl module). If $V$ has both good and Weyl filtrations, then we call $V$ tilting.

Now let $w\in\mathcal{W}$. We say that a $B$-module $V$ has a $w$-excellent filtration, if it has a $B$-module filtration with successive quotients isomorphic to some $H^{0}(X(w),\mathcal{L}(\lambda))$, with $\lambda\in X(T)_{+}$. This is a special type of excellent filtration, as defined in [vdK93, Definition 2.3.6]. Note that a good filtration of a $G$-module is a $w_{0}$-excellent filtration.

A finite-dimensional $G$-module $W$ good if $\operatorname{Sym}_{d}W^{*}$ has a good filtration for all $d\geq 0$. In particular, in this case $W$ must have a Weyl filtration. Similarly, we call an affine $G$-variety (resp. $B$-variety) $X$ good (resp $w$-excellent) if $\Bbbk[X]$ has a good (resp. $w$-excellent) filtration.

If $X\subset Y$ is a closed $G$-stable subvariety, then we say that $(Y,X)$ is a good pair whenever $Y$ is good and the defining ideal $I_{X}\subset\Bbbk[Y]$ has a good filtration (see [Don90, Section 1.3]). In this case $X$ is automatically good.

If $\operatorname{char}\Bbbk=0$, then all (pairs of) affine $G$-varieties are good. An important feature of good filtrations is the following result of Donkin [Don85] and Mathieu [Mat90, Theorem 1].

We list some cases that imply the existence of good filtrations (see [AJ84, Section 4]).

We further need some basic results.

We introduce a notion for generators of ideals, that is again relevant only in positive characteristic.

Let us record some useful results regarding this notion. We continue with the notation in Definition 2.13.

The proof above shows assumption (2) in Definition 2.13 can be replaced with the equivalent assumption that $\mathcal{P}$ generates $I_{X}$ and $\ker m_{\mathcal{P}}$ has a good filtration. In particular, the notion does not depend on the choice of the Borel subgroup (see [Jan03, Remark II.4.16 (2)]). We record another convenient fact.

Although we do not need it in this article, the assumption on $\bigwedge M$ in the lemma above can be weakened by requiring only that the good filtration dimension of $\bigwedge^{i}M$ is at most $i-1$, for all $i\geq 1$ (see [Don90, Section 1.3]).

We recall a filtration of algebras considered in [Pop86] and [Gro92]. There exists a homomorphism $h:X(T)\to\mathbb{Z}$ satisfying the following properties:

1. (1)

   $h(\lambda)$ is a non-negative integer for all $\lambda\in X(T)_{+}$;

2. (2)

   if $\chi^{\prime},\chi\in X(T)$ with $\chi^{\prime}>\chi$, then $h(\chi^{\prime})>h(\chi)$.

For a commutative $G$-algebra $A$ over $k$, we define the $\mathbb{Z}_{\geq 0}$-filtration

Denote by $\operatorname{gr}A$ the associated graded algebra. Then there is an injective map of $G$-algebras

which is onto if and only if $A$ has a good filtration [Gro92, Theorem 16].

Consider $L$ a linear algebraic group, and $H\subset L$ a closed subgroup. Let $N:=N_{L}(H)$ be the normalizer of $H$ in $L$. Let $R$ be an $L$-algebra. The group $N$ acts naturally on $R^{H}$ and on $H$-invariants $\Bbbk[L]^{H}=\Bbbk[L/H]$ (by right multiplication). The following is a consequence of [Pop86, Theorem 4] (see also [Gro97, Theorem 9.1]).