Representation of algebraic groups preserving quaternion skew-hermitian forms

Abstract

Introduction. Let K be an infinite perfect field of characteristic different from 2, let e3 be a quaternion division algebra over K, and let tt-denote the canonical involution of the first kind on Z. Let V be a finite-dimensional right vector space over Z. A quaternion skew-hermitian form H over Z is a sesquilinear form on VX V, i.e., H is a map from VX V to Z such that (i) H(x, Yl+Y2) =H(x, yi) +H(x, Y2) and H(x, yax) =H(x, y)a for all x, y, Y1, y2 in V and ai in ; (ii) H(x, y) = -H(y, x) for all x, y in V. Let {xl, . . *, x.) be a basis for V over Q. We say that H is nondegenerate if the reduced norm in Mn(Z) of the matrix (H(xi, xj)) is not zero. Associated to such a nondegenerate form H are 3 invariants, the dimension of V over 0, dime(V), the discriminant of H, b(H), and the Clifford algebra of H, G. Let G be a simply connected semisimple algebraic group (in some GL(m, X)) which is defined over K and let p:G-?GL(V/Z) be an absolutely irreducible representation of G defined over K into the group of all nonsingular Z-linear endomorphisms of V. We shall assume that there is a nondegenerate quaternion skew-hermitian form H on V which is invariant with respect to p (G). The purpose of this paper is to describe the Clifford algebra of the invariant form H in terms of p, G, and the Steinberg group associated to G. In a previous paper, we have described dimsz(V) and b(H) in such a way and have indicated how representations such as p arise [2, Theorem I.2 ]. The invariant -y(G) plays an important role in our study and so we recall some of its properties in ?1. In 2, we define the invariant Y using the representation p. Jacobson first constructed the Clifford algebra of a quaternion skew-hermitian form [4]. In this paper, however, we shall follow a method due to Satake [6]. We give some examples in ?3 with special emphasis on the case where G is absolutely simple.