Introduction

The first version of this package was released in V2.29 (October 2025). It has been developed further in each subsequent release, and is still under development. We encourage users to send feedback regarding the package, and desirable features or improvements.

The package contains implementations of certain reductive groups. The primary focus is on orthogonal and unitary groups.

The main purposes of the current functionality are to serve as a basis for representations of reductive groups and algebraic modular forms.

Definitions and Background

Reductive groups are group schemes with a trivial unipotent radical (maximal connected unipotent subgroup). Alternatively, these are the groups whose category of representations is semisimple.

If \(G\) is a reductive group, the connected component of the identity \(G^{\circ}\) is a normal subgroup, and the quotient \(G / G^{\circ}\) is the component group of \(G\). At the moment, only split extensions \(G \simeq G^{\circ} \times (G / G^{\circ})\) are supported.

If \(G\) is connected, then \(R(G)\), the radical of \(G\) (maximal connected solvable subgroup), is central and \(G / R(G)\) is semisimple. If \(G\) is connected and semisimple, it is isomorphic to a direct product product \(G_1 \times \cdots \times G_n\) of almost-simple groups \(G_1, \ldots, G_n\), where almost-simple means that they are isogenous to a simple group.

Assume that \(G\) is defined over a field \(F\), and let \(G_{\overline{F}} := G \times_{F} \overline{F}\) be its base change to \(\overline{F}\). When \(G_{\overline{F}}\) is connected and almost-simple, it is completely determined by its root datum. In order to determine \(G\), one needs to specify an element of \(H^1({\operatorname{Gal}}(\overline{F}/ F), G)\), or equivalently a Witt index and a semisimple anisotropic kernel. [Tits, 1966, Theorem 2] For the classical groups, this can be done by specifying a division algebra \(H\) over \(F\) and a sesquilinear bilinear form over \(H\). At the moment, the package only supports the case of \(H\) a field, and the bilinear form either symmetric or Hermitian with respect to complex conjugation.

Every reductive group can be embedded into \({\rm GL}_n\). In Magma, a GrpRed object consists of the group \(G\) together with its embedding \(G \to {\rm GL}_n\).

Categories

The Magma category for reductive groups is GrpRed, while elements in these spaces have type GrpRedElt.