Algebras with Involution and Classical Groups Jean-Pierre Tignol∗ Universit´e catholique de Louvain B–1348 Louvain-la-Neuve, Belgium Deep relations between linear algebraic groups over an arbitrary field and central simple algebras with involution can be traced back to two main sources. In [40], Weil1 shows that the connected component of the identity in the au- tomorphism group of a separable algebra with involution is almost2 always a semisimple linear algebraic group of adjoint type and that, conversely, almost3 every semisimple linear algebraic group of adjoint type can be obtained in this way. The groups derived from algebras with involution are called classical. On the other hand, investigating representations of a semisimple linear algebraic group G over an arbitrary field F , Tits proves in [38] that some irreducible representations ρ over a separable closure can be descended to rep- resentations of G in the group GL(Aρ) of invertible elements in some uniquely determined central simple F -algebra Aρ. This algebra carries a canonical invo- lution in certain cases. Although these results have been known and used for more than twenty years, investigations from different sources unexpectedly converged these last few years to renew interest in this topic. The primary reason for this conver- gence may be that the theory of linear algebraic groups provides the proper level of generality to make explicit the relationship between various results on central simple algebras and quadratic forms. Index reduction formulas are a case in point. It is usually difficult to control the behaviour of the Schur index of a central simple algebra under scalar extension. However, Schofield and Van den Bergh [32] found a way to use Quillen’s computation of K-groups of Severi-Brauer varieties to determine by how much the index of a central simple algebra is reduced when scalars are extended to the function field of the Severi-Brauer variety of another central simple algebra. This idea of Schofield and Van den Bergh was soon used to de- rive index reduction formulas for scalar extension to function fields of various ∗Supported in part by the National Fund for Scientific Research (Belgium). 1Weil attributes the idea to Siegel and to “a famous winner of many cocycle races”. 2Factors of degree 2 with involution of orthogonal type are excluded, since their auto- morphism group is commutative. There are restrictions on the characteristic of the base field also: see the end of section 1. 3 The exceptional groups and the groups of trialitarian type D4 are excluded. 1 classes of varieties X from K-theory information on X: Blanchet [6] considered the case of twisted Grassmannians and Merkurjev [24] the case of quadrics. Tao [35] then considered varieties arising from central simple algebras with involution, which he called involution varieties. In a sense, these varieties are hybrids of Severi-Brauer varieties and quadrics, making clear that orthogonal4 involutions on central simple algebras may be thought of as twisted forms of projective quadrics: the involution variety of a central simple algebra A with involution of orthogonal type may be regarded as a quadric hypersurface on the Severi-Brauer variety of A. It was soon realized thereafter that all the va- rieties thus considered, including the transfers of Severi-Brauer varieties and of twisted Grassmannians treated by Saltman [30] by different K-theory tech- niques, were special cases of varieties of parabolic subgroups in a semisimple linear algebraic group. A general index reduction formula was then set up by Merkurjev, Panin and Wadsworth [26], subsuming all the particular cases found so far and enlightening the occurrence of auxiliary central simple algebras in these formulas: these are the Tits algebras associated with representations of the group. Simultaneously, a similar generalization was achieved by Kersten and Rehmann [11], who showed that various types of generic splitting fields designed for central simple algebras and for quadratic forms are special cases of functions fields of varieties of parabolic subgroups. Another impetus for the study of algebras with involution was given by Knus, Ojanguren, Parimala and Sridharan in a series of papers5 investigating quadratic forms of low rank. In [12], [17], quadratic forms of dimension 6 and discriminant 1 are shown to be (generalized) pfaffians of central simple algebras of degree 4 with involution. Generalized pfaffians are used in [18] and [19] to define a discriminant for involutions, a notion which was already implicit (at least for involutions on central simple algebras) in Jacobson’s and Tits’ construction of Clifford algebras [9], [37]. The case of algebras of degree 4 has been specially fascinating since the time when Albert proved that every central simple algebra of degree 4 with an involution leaving the center elementwise invariant decomposes into a tensor product of two quaternion subalgebras [2] (see also [29]). For a given involution there may not be any decomposition into quaternion subalgebras stable under the involution, however, as was shown by Amitsur, Rowen and Tignol [3]. Knus, Parimala and Sridharan [18], [20] showed that a stable decomposition exists if and only if the discriminant of the involution is trivial. After [18], it became apparent that seemingly unrelated investigations by Lam, Leep and Tignol [21] on the discriminants of subfields of biquaternion algebras could yield significant information on the discriminants of involutions on these algebras; this was made explicit by Knus, Lam, Shapiro and Tignol [15]. With hindsight, the relationship between 6-dimensional quadratic forms 4See section 1 for the definition of an orthogonal involution. 5A comprehensive account of these investigations can be found in [13], [14]. 2 and biquaternion algebras, first observed by Albert [1], as well as various re- sults relating orthogonal involutions on biquaternion algebras with quaternion algebras over quadratic extensions, and symplectic involutions on biquaternion algebras with 5-dimensional quadratic forms, can all be seen to derive from the exceptional isomorphisms of classical groups suggested by the equalities of Dynkin diagrams A3 = D3, D2 = A1 × A1 and B2 = C2. This variety of approaches to biquaternion algebras was put to remarkable use by Rost in his computation of the reduced Whitehead group of these algebras: see [25], [16]. The rest of this survey purposes to outline the properties of biquaternion algebras which derive from the exceptional isomorphisms of classical groups. General properties of central simple algebras with involution, their invariants and their automorphism groups, are discussed in the first three sections. A detailed exposition with complete proofs will be found in [16]. 1 Algebras with involution The involutions considered in this paper are ring anti-automorphisms of pe- riod 2. The most important examples in relation with algebraic groups are obtained as follows: if b is a nonsingular symmetric or skew-symmetric bilin- ear form on a finite-dimensional vector space V over a field F of arbitrary characteristic, there is a unique involution σb on EndF (V ) such that b f(x),y = b x, σb(f)(y) for all x, y ∈ V and all f ∈ EndF (V ). The map σb is called the adjoint involution with respect to the bilinear form b.Forf ∈ F = F · IdV , we clearly have σb(f)=f. It is easily seen that every involution on EndF (V )whichleavesF elementwise invariant is the adjoint involution with respect to some nonsingular symmetric or skew-symmetric bilinear form on V , uniquely determined up to afactorinF × (see [31, §8.7], [16, Ch. I]). Similarly, if W is a finite-dimensional vector space over a field K and ι is an automorphism of period 2 of K, every involution on EndK (W ) which restricts to ι on K is adjoint to a nonsingular hermitian form on W with respect to ι, uniquely determined up to a factor in the subfield of K elementwise invariant under ι. This example can be further generalized: every finite-dimensional simple algebra may be represented as EndD(U) for some finite-dimensional division al- gebra D and some finite-dimensional right vector space U over D,byatheorem of Wedderburn, and every involution on EndD(U) is adjoint to a nonsingular hermitian or skew-hermitian form on U with respect to an involution on D. This form is uniquely determined up to a central factor invariant under the involution [31, §8.7]. Therefore, it is equivalent to study involutions on central 3 simple algebras or hermitian and skew-hermitian forms up to a scalar factor over division algebras. The point of view of hermitian spaces affords greater flexibil- ity because the orthogonal sum defines a group structure on Witt-equivalence classes; on the other hand, involutions are better behaved regarding scalar ex- tension: if σ is an involution on an F -algebra A and L/F is any field extension, the involution σL = σ⊗IdL on AL = A⊗F L is clearly related to σ,whereasthe relationship between the hermitian forms associated with σ and σL is unclear, if only because the division algebras associated with A and AL by the Wed- derburn theorem are different. Moreover, the point of view of hermitian forms does not shed much light on involutions on division algebras, which correspond to hermitian or skew-hermitian forms of rank 1. It may be more suggestive to think of involutions leaving F elementwise invariant on a central simple F -algebra A as twisted forms of nonsingular sym- metric or skew-symmetric bilinear forms up to scalar factors, since under scalar ⊗ ' extension to a separable closure Fs the algebra splits as A Fs EndFs (V ) for some vector space V over Fs, and every involution on EndFs (V )isadjoint to a nonsingular symmetric or skew-symmetric bilinear form. In the case where the center K is not elementwise invariant under the involution, the algebra structure should be considered over the subfield F of invariant elements (of codimension 2 in K), so that the involution extends to scalar extensions.