1. Introduction. Let K be a commutative separable cubic algebra over a nonarchimedean local field k. The aim of this paper is to study conditions under which irreducible, admissible representations of GL2 (K) have GL2 (k)-invariant linear forms. This work is an extension of author's previous work [P] in which he studied this question for the case K= k E k E k. In this paper we will only consider the case when K is either of the form KE k where K is a quadratic field extension of k, or is a cubic field extension of k. Let Dk be the unique quaternion division algebra over k, and let D, K= Dk? k 1K-We recall that to a discrete series representation of GL2 (k)(by which we will always mean an irreducible representation which has a twist whose matrix coefficients are square integrable modulo centre) there is a finite dimensional irreducible representation V'of D, associated by Jacquet-Langlands which satisfies the character iden-tity ch (V)(x)=-ch (V')(x) at all the regular elliptic elements x of GL2 (k). We extend this correspondence to one between representations of GL2 (K) and representations of D, as follows. If K= K) k, where K is a quadratic field extension of k, then GL2 (K)= GL2 (K) x GL2 (k) and a representation V of GL2 (K) is the tensor product V1 0 V2 of a representation V1 of GL2 (K) and a representation V2 of GL2 (k). In this case D,= GL2 (K) x D,. We define the representation V'of D, to be V1 0 V2 if the representation V2 of GL2 (k) is a discrete series rep-resentation, and to be the zero representation if V2 is not a discrete series representation. If K is a cubic field extension of k then D, K is the unique quaternion division algebra over the field K and we let V'be the representation of D, associated to the representation V of GL2 (1K)(by the Jacquet-Langlands correspondence) if it is a discrete series rep-resentation, and zero otherwise.