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We study the complex-time Segal-Bargmann transform$\mathbf{B}_{s,\tau}^{K_N}$ on a compact type Lie group $K_N$, where $K_N$ isone of the following classical matrix Lie groups: the special orthogonal group$\mathrm{SO}(N,\mathbb{R})$, the special unitary group $\mathrm{SU}(N)$, or thecompact symplectic group $\mathrm{Sp}(N)$. Our work complements and extends theresults of Driver, Hall, and Kemp on the Segal-Bargman transform for theunitary group $\mathrm{U}(N)$. We provide an effective method of computing theaction of the Segal-Bargmann transform on \emph{trace polynomials}, whichcomprise a subspace of smooth functions on $K_N$ extending the polynomialfunctional calculus. Using these results, we show that as $N\to\infty$, thefinite-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 canbe identified as an operator on the space of complex Laurent polynomials.

The Segal-Bargmann Transform on Classical Matrix Lie Groups