We study the complex-time Segal-Bargmann transform $\mathbf{ B}_{ s,\tau}^{ K_N}$ on a compact type Lie group $K_N$, where $K_N$ is one of the following classical matrix Lie groups: the special orthogonal group $\mathrm{ SO}(N,\mathbb{ R})$, the special unitary group $\mathrm{ SU}(N)$, or the compact symplectic group $\mathrm{ Sp}(N)$. Our work complements and extends the results of Driver, Hall, and Kemp on the Segal-Bargman transform for the unitary group $\mathrm{ U}(N)$. We provide an effective method of computing the action of the Segal-Bargmann transform on \emph{ trace polynomials}, which comprise a subspace of smooth functions on $K_N$ extending the polynomial functional calculus. Using these results, we show that as $N\to\infty$, the finite-dimensional transform $\mathbf{ B}_{ s,\tau}^{ K_N}$ has a meaningful limit $\mathscr{ G}_{ s,\tau}^{ (\beta)}$ (where $\beta$ is a parameter associated with $\mathrm{ SO}(N,\mathbb{ R})$, $\mathrm{ SU}(N)$, or $\mathrm{ Sp}(N)$), which can be identified as an operator on the space of complex Laurent polynomials.
Categories: math.FA