Let C be a smooth projective complex curve of genus at least 2. For a simplyconnected complex Lie group G the vector space of global sections H 0 (M(G), L ⊗l G ) of the l-th power of the ample generator L G of the Picard group of the moduli stack of principal G-bundles over C is commonly called the space of generalized G-theta functions or Verlinde space of level l. In the case G = G 2 , the exceptional Lie group of automorphisms of the complex Cayley algebra, we study natural linear maps between the Verlinde space H 0 (M(G 2 ), L G2 ) of level one and some Verlinde spaces for SL 2 and SL 3 . We deduce that the image of the monodromy representation of the WZW-connection for G = G 2 and l = 1 is infinite.

Let G 2 be the exceptional Lie group of automorphisms of the complex Cayley algebra and C be a smooth, connected, projective curve of genus at least 2. Using the map obtained from extension of structure groups, we prove explicit links between the space of generalized G 2 -theta functions over C and spaces of generalized theta functions associated to the classical Lie groups SL 2 and SL 3 .