  Let C be a smooth projective complex curve of genus at least 2. For a
simply-connected complex Lie group G the vector space of global sections
H^0(M(G), L^l) of the l-th power of the ample generator L 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) 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.
