Symmetries for Siegel Theta Functions, Borcherds Lifts and Automorphic Green Functions

Let q be an integral quadratic form of signature (2,m+2). We will show that the Siegel theta functions attached to q satisfies certain symmetries. As an application, we prove the symmetries for automorphic forms on the orthogonal group of q closely related to Heegener divisors (Borcherds lifts and automorphic Green functions).