CHARACTER DEGREE SUMS AND REAL REPRESENTATIONS OF FINITE CLASSICAL GROUPS OF ODD CHARACTERISTIC

Abstract

Let 𝔽q be a finite field with q elements, where q is the power of an odd prime, and let GSp (2n, 𝔽q) and GO ±(2n, 𝔽q) denote the symplectic and orthogonal groups of similitudes over 𝔽q, respectively. We prove that every real-valued irreducible character of GSp (2n, 𝔽q) or GO ±(2n, 𝔽q) is the character of a real representation, and we find the sum of the dimensions of the real representations of each of these groups. We also show that if G is a classical connected group defined over 𝔽q with connected center, with dimension d and rank r, then the sum of the degrees of the irreducible characters of G(𝔽q) is bounded above by (q + 1)(d+r)/2. Finally, we show that if G is any connected reductive group defined over 𝔽q, for any q, the sum of the degrees of the irreducible characters of G(𝔽q) is bounded below by q(d-r)/2(q - 1)r. We conjecture that this sum can always be bounded above by q(d-r)/2(q + 1)r.