Newforms for odd orthogonal groups

Symplectic Langlands parameters of dimension 2n naturally arise from l-adic realization of symplectic motives like H_1(X), X curve of genus n. It corresponds to automorphic representations of odd special orthogonal group SO(2n+1) under the Langlands conjecture. It has been known that when n=1 the Langlands correspondence associates a specific automorphic form, the newform, to a 2-dimensional Galois representation attached to an elliptic curve. Such a newform has the property that it is fixed under an arithmetic group of prescribed level which is equal to the Artin conductor of the Galois representation. We will generalize the notion of newforms associated to symplectic motives of higher ranks by determining its local components, the new vectors, associated to the symplectic Langlands parameters.