Averages over classical Lie groups, twisted by characters

We compute E G ( ∏ i tr ( g λ i ) ) , where g ∈ G = Sp ( 2 n ) or SO ( m ) ( m = 2 n , 2 n + 1 ) with Haar measure. This was first obtained by Diaconis and Shahshahani [Persi Diaconis, Mehrdad Shahshahani, On the eigenvalues of random matrices, J. Appl. Probab. 31A (1994) 49–62. Studies in applied probability], but our proof is more self-contained and gives a combinatorial description for the answer. We also consider how averages of general symmetric functions E G Φ n are affected when we introduce a character χ λ G into the integrand. We show that the value of E G χ λ G Φ n / E G Φ n approaches a constant for large n. More surprisingly, the ratio we obtain only changes with Φ n and λ and is independent of the Cartan type of G. Even in the unitary case, Bump and Diaconis [Daniel Bump, Persi Diaconis, Toeplitz minors, J. Combin. Theory Ser. A 97 (2) (2002) 252–271. Erratum for the proof of Theorem 4 available at http://sporadic.stanford.edu/bump/correction.ps and in a third reference in the abstract] have obtained the same ratio. Finally, those ratios can be combined with asymptotics for E G Φ n due to Johansson [Kurt Johansson, On random matrices from the compact classical groups, Ann. of Math. (2) 145 (3) (1997) 519–545] and provide asymptotics for E G χ λ G Φ n .