Group schemes with additional structures and Weyl group cosets.

*(English)*Zbl 1084.14523
Faber, Carel (ed.) et al., Moduli of abelian varieties. Proceedings of the 3rd Texel conference, Texel Island, Netherlands, April 1999. Basel: Birkhäuser (ISBN 3-7643-6517-X/hbk). Prog. Math. 195, 255-298 (2001).

The paper under review reports on the progress concerning the classification problems of finite group schemes with additional structure.

Let \(k\) be an algebraically closed field of characteristic \(p>0\). Let \(D\) be a finite-dimensional semisimple \(\mathbb{F}_p\)-algebra. The first classification problem addressed in the paper is (GE): Classify pairs (\(\mathcal G, \iota\)) where \(\mathcal G\) is a \(\text{BT}_1\) over \(k\) and \(\iota\colon D\to\text{End}_k(\mathcal G)\) defines an action of \(D\) on \(\mathcal G\) (with \(\iota(1)=\text{id}_{\mathcal G}\)). (Here \(\text{BT}_1\) stands for a truncated Barsotti-Tate group of level \(1\).)

Associated to such a group scheme \(\mathcal G\), there is the Dieudonné module \((N, F_N, V_N)\). There are two discrete invariants which play essential roles in the problem (GE). Let \(G:= G_D(N)\), and consider the structure of \(N\) as a \(D\)-module. Another discrete invariant, which is called the multiplication type of the pair \((\mathcal G,\iota)\), is the structure of \(\text{Ker}(F)\subset N\) as a \(D\)-module. The multiplication type corresponds to a conjugacy class \(X\) of parabolic subgroups of \(G\), which gives rise to a subgroup \(W_X\) of the Weyl group \(W_G\) of \(G\). The coset space \(W_X\backslash W_G\) is a finite set which can be described explicitly.

Given a pair \((\mathcal G,\iota)\) with discrete invariants \(N\) and \(X\), define an element \(\underline{w}(\mathcal G,\iota)\in W_X \backslash W_G\). The first main result on the problem (GE) follows. Theorem 1: Associating \(\underline{w}(\mathcal G,\iota)\) to the pair \((\mathcal G,\iota)\) gives a bijection between the isomorphism classes of pairs (\(\mathcal G,\iota\)) with invariants \(N\) and \(X\), and the coset \(W_X\backslash W_G\).

The second classification problem (GPE) concerns triples \((\mathcal G,\lambda,\iota)\), where \(\mathcal G\) is a \(\text{BT}_1\) over \(k\), \(\lambda: \mathcal G \buildrel\sim\over\rightarrow \mathcal G^D\) is a principal quasi-polarization and \(\iota\) is defined in the same way as (GE). Let \(G=\text{Sp}_D(N,\psi)\), where \(\psi\) is the symplectic form on the Dieudonné module \(N\) corresponding to the polarization \(\lambda\). As for (GE), consider the multiplication type. To a triple \((\mathcal G, \lambda, \iota)\), associate an element \(\underline{w}(\mathcal G, \lambda, \iota)\in W_X\backslash W_G\). The inclusion \(f: \text{Sp}_D(N,\psi)\to\text{GL}_D(N)\) induces an injective map \(W(f): W_X\backslash W_G\hookrightarrow W_X\backslash W_{\text{GL}_D(N)}\) which sends \(\underline{w}(\mathcal G,\lambda,\iota)\) to \(\underline{w}(\mathcal G,\iota)\). The main result on the problem (GPE) is the following.

Theorem 2: Assume that \(k\) is an algebraically closed field of characteristic \(>2\). (i) Let \(\mathcal G\) be a \(\text{BT}_1\) over \(k\) with an action \(\iota\) of the semisimple algebra \(D\). Then there exists a principal quasi-polarization \(\lambda\) such that \((\mathcal G,\lambda,\iota)\) is a triple as in (GPE) if and only if \(\underline{w}(\mathcal G,\iota)\in\break W_X\backslash W_{\text{GL}_D(N)}\) is in the image of the map \(W(f)\).

(ii) If a form \(\lambda\) as in (i) exists, then it is unique up to isomorphism (respecting the \(D\)-action). In other words: associating \(\underline{w}(\mathcal G,\lambda,\iota)\in W_X\backslash W_G\) to a triple \((\mathcal G,\lambda,\iota)\) gives a bijection between isomorphism classes of triples \((\mathcal G,\lambda,\iota)\) with invariants \(N\) and \(X\), and cosets

\(W_X\backslash W_G\).

For the entire collection see [Zbl 0958.00023].

Let \(k\) be an algebraically closed field of characteristic \(p>0\). Let \(D\) be a finite-dimensional semisimple \(\mathbb{F}_p\)-algebra. The first classification problem addressed in the paper is (GE): Classify pairs (\(\mathcal G, \iota\)) where \(\mathcal G\) is a \(\text{BT}_1\) over \(k\) and \(\iota\colon D\to\text{End}_k(\mathcal G)\) defines an action of \(D\) on \(\mathcal G\) (with \(\iota(1)=\text{id}_{\mathcal G}\)). (Here \(\text{BT}_1\) stands for a truncated Barsotti-Tate group of level \(1\).)

Associated to such a group scheme \(\mathcal G\), there is the Dieudonné module \((N, F_N, V_N)\). There are two discrete invariants which play essential roles in the problem (GE). Let \(G:= G_D(N)\), and consider the structure of \(N\) as a \(D\)-module. Another discrete invariant, which is called the multiplication type of the pair \((\mathcal G,\iota)\), is the structure of \(\text{Ker}(F)\subset N\) as a \(D\)-module. The multiplication type corresponds to a conjugacy class \(X\) of parabolic subgroups of \(G\), which gives rise to a subgroup \(W_X\) of the Weyl group \(W_G\) of \(G\). The coset space \(W_X\backslash W_G\) is a finite set which can be described explicitly.

Given a pair \((\mathcal G,\iota)\) with discrete invariants \(N\) and \(X\), define an element \(\underline{w}(\mathcal G,\iota)\in W_X \backslash W_G\). The first main result on the problem (GE) follows. Theorem 1: Associating \(\underline{w}(\mathcal G,\iota)\) to the pair \((\mathcal G,\iota)\) gives a bijection between the isomorphism classes of pairs (\(\mathcal G,\iota\)) with invariants \(N\) and \(X\), and the coset \(W_X\backslash W_G\).

The second classification problem (GPE) concerns triples \((\mathcal G,\lambda,\iota)\), where \(\mathcal G\) is a \(\text{BT}_1\) over \(k\), \(\lambda: \mathcal G \buildrel\sim\over\rightarrow \mathcal G^D\) is a principal quasi-polarization and \(\iota\) is defined in the same way as (GE). Let \(G=\text{Sp}_D(N,\psi)\), where \(\psi\) is the symplectic form on the Dieudonné module \(N\) corresponding to the polarization \(\lambda\). As for (GE), consider the multiplication type. To a triple \((\mathcal G, \lambda, \iota)\), associate an element \(\underline{w}(\mathcal G, \lambda, \iota)\in W_X\backslash W_G\). The inclusion \(f: \text{Sp}_D(N,\psi)\to\text{GL}_D(N)\) induces an injective map \(W(f): W_X\backslash W_G\hookrightarrow W_X\backslash W_{\text{GL}_D(N)}\) which sends \(\underline{w}(\mathcal G,\lambda,\iota)\) to \(\underline{w}(\mathcal G,\iota)\). The main result on the problem (GPE) is the following.

Theorem 2: Assume that \(k\) is an algebraically closed field of characteristic \(>2\). (i) Let \(\mathcal G\) be a \(\text{BT}_1\) over \(k\) with an action \(\iota\) of the semisimple algebra \(D\). Then there exists a principal quasi-polarization \(\lambda\) such that \((\mathcal G,\lambda,\iota)\) is a triple as in (GPE) if and only if \(\underline{w}(\mathcal G,\iota)\in\break W_X\backslash W_{\text{GL}_D(N)}\) is in the image of the map \(W(f)\).

(ii) If a form \(\lambda\) as in (i) exists, then it is unique up to isomorphism (respecting the \(D\)-action). In other words: associating \(\underline{w}(\mathcal G,\lambda,\iota)\in W_X\backslash W_G\) to a triple \((\mathcal G,\lambda,\iota)\) gives a bijection between isomorphism classes of triples \((\mathcal G,\lambda,\iota)\) with invariants \(N\) and \(X\), and cosets

\(W_X\backslash W_G\).

For the entire collection see [Zbl 0958.00023].

Reviewer: Noriko Yui (MR1827024)

##### MSC:

14L15 | Group schemes |