We consider the natural embedding for SO(r) into SL(r) and study the corresponding map between the moduli spaces of
principal bundles on smooth projective curves. We compare the spaces of global sections of natural line bundles
(non-abelian theta functions) for these moduli spaces and their twisted analogues with the space of theta functions.
We will discuss how these results can be applied to obtain an alternate proof of a result of Pauly-Ramanan. If time permits,
we will also discuss some applications to the monodromy of the Hitchin/WZW connections. This is a joint work with H. Zelaci.