Equivariant vector bundles on certain affine \(G\)-varieties.

*(English)*Zbl 1110.14042From 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.

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.