Equivariant sheaves on some spherical varieties.

*(English)*Zbl 1231.14039This paper is an announcement of a following work, in which the category of equivariant vector bundle on certain toroidal spherical varieties is described. In this paper, the authors study an example and state the general result. Moreover, the method to generalize the proof is explained. Their results is a generalization of [S. Kato, J. Reine Angew. Math. 581, 71–116 (2005; Zbl 1074.14042)].

The authors consider the case of \(X=\mathbb{P}^1 \times \mathbb{P}^1 \) with the action of \(G=\mathrm{PGL}_2\) (over an algebraic closed field of characteristic 0). The generic stabilizer of a point in \(X\) is a maximal torus \(T\) of \(G\). They prove that the category of \(G\)-equivariant bundle is equivalent to the category are finite dimensional representation \(V\) of \(T\) with a filtration \(F^\bullet\) such that the action of \(\mathrm{Lie}(T)\) on \(V\) send \(F^i(V)\) to \(F^{i-1}(V)\).

To prove this theorem the authors construct a groupoid \(Y\) equivalent to the action groupoid \(G\ltimes X\), but whose representation are easier to describe. This groupoid will decompose as a semidirect product \(G_m\ltimes N\), for a \(G_m\)-equivariant group scheme \(N\) over \(\mathbf{A}^1\). This decomposition allows them to describe representations (i.e. equivariant vector bundle) of \(Y\) as \(G_m\)-equivariant vector bundle on \(\mathbf{A}^1\) with a compatible action of \(N\). The category of \(G_m\)-equivariant vector bundles on \(\mathbf{A}^1\) has a nice description that goes back to Rees.

representation of \(N\). They analyze such representations using a description of \(G_m\)-equivariant vector bundles on \(\mathbf{A}^1\). The groupoid \(Y\) is induced by the action groupoid and by a closed immersion \(\mathbf{A}^1\rightarrow X\); this immersion is invariant with respect to an appropriate maximal torus \(S\) of \(G\).

Let \(G\) be a connected reductive group and let \(B\) be a Borel subgroup. A normal \(G\)-variety \(X\) is spherical if it contains an open \(B\)-orbit \(B\cdot x\). Moreover \(X\) is toroidal if every \(B\)-stable prime divisor contains a \(G\)-orbit is \(G\)-stable. To study local problem on a spherical variety \(X\) is usually possible to reduce oneself to study a locally closed affine subvariety \(Z\), which is spherical with respect to the action of a Levi subgroup \(L\) of \(X\). If \(X\) is toroidal, then \(Z\) is a toric variety with respect to a quotient \(S\) of \(L\) (in particular, \(S\) is an algebraic torus). The inclusion of \(Z\) in \(X\) allows to define a groupoid \(Y\) as before. But, the authors state that in general one cannot decompose \(Y\) as a semi-direct-product \(S\ltimes N\) as before. In the case of \(\mathbb{P}^1\times \mathbb{P}^1\), \(L=S\). In general they need a splitting of \(L\rightarrow S\). This equivalent to require that the weight lattice of \(X\) is a direct summand of the character lattice of \(B\), where the weight lattice of \(X\) is the set of weights of the \(B\)-seminvariants rational functions on \(X\). They said that a toroidal spherical variety with such property is neutralizable.

With this hypothesis they reduce the description of \(G\)-equivariant vector bundles on \(X\) to the description of equivariant vector bundle on \(S\ltimes N\). The first step, namely the describition of the category of equivariant bundles over \(S\ltimes Z\), is provided by Klyachko.

The authors consider the case of \(X=\mathbb{P}^1 \times \mathbb{P}^1 \) with the action of \(G=\mathrm{PGL}_2\) (over an algebraic closed field of characteristic 0). The generic stabilizer of a point in \(X\) is a maximal torus \(T\) of \(G\). They prove that the category of \(G\)-equivariant bundle is equivalent to the category are finite dimensional representation \(V\) of \(T\) with a filtration \(F^\bullet\) such that the action of \(\mathrm{Lie}(T)\) on \(V\) send \(F^i(V)\) to \(F^{i-1}(V)\).

To prove this theorem the authors construct a groupoid \(Y\) equivalent to the action groupoid \(G\ltimes X\), but whose representation are easier to describe. This groupoid will decompose as a semidirect product \(G_m\ltimes N\), for a \(G_m\)-equivariant group scheme \(N\) over \(\mathbf{A}^1\). This decomposition allows them to describe representations (i.e. equivariant vector bundle) of \(Y\) as \(G_m\)-equivariant vector bundle on \(\mathbf{A}^1\) with a compatible action of \(N\). The category of \(G_m\)-equivariant vector bundles on \(\mathbf{A}^1\) has a nice description that goes back to Rees.

representation of \(N\). They analyze such representations using a description of \(G_m\)-equivariant vector bundles on \(\mathbf{A}^1\). The groupoid \(Y\) is induced by the action groupoid and by a closed immersion \(\mathbf{A}^1\rightarrow X\); this immersion is invariant with respect to an appropriate maximal torus \(S\) of \(G\).

Let \(G\) be a connected reductive group and let \(B\) be a Borel subgroup. A normal \(G\)-variety \(X\) is spherical if it contains an open \(B\)-orbit \(B\cdot x\). Moreover \(X\) is toroidal if every \(B\)-stable prime divisor contains a \(G\)-orbit is \(G\)-stable. To study local problem on a spherical variety \(X\) is usually possible to reduce oneself to study a locally closed affine subvariety \(Z\), which is spherical with respect to the action of a Levi subgroup \(L\) of \(X\). If \(X\) is toroidal, then \(Z\) is a toric variety with respect to a quotient \(S\) of \(L\) (in particular, \(S\) is an algebraic torus). The inclusion of \(Z\) in \(X\) allows to define a groupoid \(Y\) as before. But, the authors state that in general one cannot decompose \(Y\) as a semi-direct-product \(S\ltimes N\) as before. In the case of \(\mathbb{P}^1\times \mathbb{P}^1\), \(L=S\). In general they need a splitting of \(L\rightarrow S\). This equivalent to require that the weight lattice of \(X\) is a direct summand of the character lattice of \(B\), where the weight lattice of \(X\) is the set of weights of the \(B\)-seminvariants rational functions on \(X\). They said that a toroidal spherical variety with such property is neutralizable.

With this hypothesis they reduce the description of \(G\)-equivariant vector bundles on \(X\) to the description of equivariant vector bundle on \(S\ltimes N\). The first step, namely the describition of the category of equivariant bundles over \(S\ltimes Z\), is provided by Klyachko.

Reviewer: Alessandro Ruzzi (Roma)