Character of the oscillator representation (Q1359996)

This paper presents an algebraic calculation of the character of the oscillator representation \(\omega\) of \(\widetilde{Sp}(2n, {\mathbb R})\), using coherent continuation. Decomposing \(\omega = \omega_{even} \oplus \omega_{odd}\), where \( \omega_{even}\), \( \omega_{odd}\) are irreducible representations with characters \(\theta_{even}\), \(\theta_{odd}\), respectively, the author shows that \[ \theta_{even} - \theta_{odd} = {{D_{SO}} \over {D_{Sp}}}, \] where \(D_{Sp}\), \(D_{SO}\) are the Weyl denominators for \(Sp(2n, {\mathbb R})\), \(SO(2n+1)\), respectively. In particular, \(\theta_{even} - \theta_{odd}\) has the form of a transfer factor from characters of \(\widetilde{Sp}(2n, {\mathbb R})\) to characters of \(SO(2n+1)\).