Real representations of the finite orthogonal and symplectic groups of odd characteristic

Abstract

Let q be a power of an odd prime p. Let Sp(2n, q) denote the symplectic group of degree 2n over GF(q). Let O+(n, q) denote the split orthogonal group of degree n over GF(q), corresponding to a symmetric form of maximal Witt index, and 0 (n, q) denote the non-split orthogonal group. Unless we need to be more specific, we will simply write O(n, q) for either of these two orthogonal groups. In addition, we will often abbreviate these symbols further to O(n) and Sp(2n) whenever q is understood to be fixed. It is known from conjugacy class considerations that all complex irreducible characters of the finite orthogonal groups O(n) are real-valued. The same is true of the symplectic groups Sp(2n, q) provided that q z 1 (mod 4). When q= 3 (mod 4), not all characters of Sp(2n, q) are realvalued. However, as the total number of irreducible characters is a manic polynomial in q of degree n and this is true of the number of real-valued irreducible characters, we can assert that the majority of characters is realvalued in this case. It is of interest to know whether or not the real-valued characters of these families of groups are the characters of representations that are defined over the real numbers. This paper gives a solution to this problem by proving the following theorem.