Fermionic dual pairs of representations of loop groups (Q1200386)

The theory of spinor representations of pairs of loop groups with automorphism of conjugation \(\tau_ i\) (twisted loop groups) and embedded in \(\widetilde O_ \tau(N,C)\) as commuting subgroups is developed. More precisely: pairs of groups \(O(n,C)\) and \(O(m,C)\), \(N = mn\); \(GL(n,C)\) and \(SL(m,C)\), \(N = 2mn\); \(Sp(n,C)\) and \(Sp(m,C)\), \(N = 4mn\) and the corresponding loop groups \(\widetilde G_{\tau_ i}^{(i)}\) \((i=1,2)\) are considered. A theorem is proved about the representations of the groups \(\widetilde G^{(i)}_{\tau_ i}\) in the space of spinor representations of \(\widetilde O_ \tau(n,C)\) which are completely reducible and form a dual pair. Possible application of this result is discussed.