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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.
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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