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Equivariant vector bundles on certain affine $$G$$-varieties. (English) Zbl 1110.14042
From the introduction: A $$G$$-variety $$X$$ is called prehomogeneous if $$G$$ acts on $$X$$ with a Zariski dense orbit. In this note, we will study the problem of concretely describing the category $$\text{Vec}^G(X)$$ of $$G$$-equivariant vector bundles on prehomogeneous $$G$$-varieties $$X$$. The simplest instance of this problem is the case when $$X$$ is a homogeneous $$G$$-variety. In this situation, the choice of a $$k$$-point $$x\in X(k)$$ provides an equivalence of categories $x^*: \text{Vec}^G(X)\widetilde\rightarrow \text{Vec}^{G_x}(\text{Spec\,}k)= \text{Rep}(G_x),$ where $$G_x$$ is the stabilizer group of $$x$$ in $$G$$.
We study the general problem as follows. For a prehomogeneous $$G$$-variety $$X$$, let $$U\subset X$$ denote the dense $$G$$-orbit in $$X$$. Restriction to $$U$$ defines a faithful functor $$\text{Vec}^G(X)\to\text{Vec\,}^G(U)$$. Choice of a point $$x\in U(k)$$ determines an equivalence of categories $$x^*: \text{Vec}^G(U)\to \text{Rep}(G_x)$$, where $$G_x$$ denotes the stabilizer group in $$G$$ of the point $$x$$. We will therefore try to describe $$\text{Vec}^G(X)$$ in terms of representations of $$G_x$$ equipped with additional structure.
The guiding example is the case $$X= \mathbb{A}^1$$ and $$G= \mathbb{G}_{{\mathbf m}}$$. In this case, the stabilizer group of any point in the open $$\mathbb{G}_{{\mathbf m}}$$-orbit is trivial. For any $$\mathbb{G}_{{\mathbf m}}$$-equivariant vector bundle $${\mathcal V}$$ on $$X$$, consider the vector space $$V=\Gamma(\mathbb{G}_{{\mathbf m}},{\mathcal V}|_{\mathbb{G}_{{\mathbf m}}})^{\mathbb{G}_{{\mathbf m}}}$$, i.e. the $$\mathbb{G}_{{\mathbf m}}$$-invariant sections over the open subset $$\mathbb{G}_{{\mathbf m}}\subset \mathbb{A}^1$$. Choose a coordinate $$x$$ on $$\mathbb{A}^1$$. We define a decreasing filtration on $$V$$ by setting $$F^p(V)$$ to be the subspace of sections $$v\in V$$ such that $$x^{-p}v$$ extends to a section of $${\mathcal V}$$ over $$X$$, i.e. by “vanishing order of sections.” This construction defines a functor $$\Phi: \text{Vec}^{\mathbb{G}}_{{\mathbf m}}(\mathbb{A}^1)\to \text{Filt}(\text{Vect}_k)$$. The following result is due to many authors.
Theorem. The functor $$\Phi: \text{Vec}^{\mathbb{G}_{{\mathbf m}}}(\mathbb{A}^1)\to \text{Filt}(\text{Vect}_k)$$ is an equivalence of categories.
If $$G$$ is a (possibly disconnected) reductive linear algebraic group, a spherical $$G$$-variety is a normal $$G$$-variety $$X$$ for which some Borel subgroup $$B\subset G$$ acts on $$X$$ with a Zariski dense orbit.
Suppose now $$X$$ is a spherical $$G$$-variety. Let $$U\subset X$$ denote the open $$G$$-orbit, let $$r\in U(k)$$ and let $$G_x$$ be the stabilizer in $$G$$ of $$x$$. In Theorem 5.2 we construct a fully-faithful functor from $$\text{Vec}^G(X)$$ into a category of multi-filtered representations of $$G_x$$, where the number of filtrations is precisely the number of codimension 1 orbits of $$G^0$$ (the identity connected component of $$G$$). When $$X$$ is an affine, spherical $$G$$-variety, the essential image of the aforementioned functor can be explicitly determined.
##### MSC:
 14L30 Group actions on varieties or schemes (quotients) 14M17 Homogeneous spaces and generalizations 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 55R91 Equivariant fiber spaces and bundles in algebraic topology
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