##### What's Hot

$Sp(2n,\mathbb{R})$ and its extension to some semigroups, called the Olshanski
semigroup for $Sp(2n,\mathbb{R})$ or Howe’s oscillator semigroup, (2)
$\mathbb{R}^{2m}\cong\mathbb{C}^{m}$, (3) path integral quantizations where the
paths are on the phase space $\mathbb{R}^{2m}\cong\mathbb{C}^{m}$. In the Main
Theorem, the metaplectic representation $\rho(e^{X})$
($X\in\mathfrak{sp}(2n,\mathbb{R})$) is expressed in terms of generalized