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.

中文翻译：

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经典矩阵李群上的 Segal-Bargmann 变换

我们研究了紧凑型李群 $K_N$ 上的复时间 Segal-Bargmann 变换 $\mathbf{B}_{s,\tau}^{K_N}$，其中 $K_N$ 是以下经典矩阵 Lie 之一群：特殊正交群$\mathrm{SO}(N,\mathbb{R})$、特殊酉群$\mathrm{SU}(N)$，或紧致辛群$\mathrm{Sp}( N)$。我们的工作补充并扩展了 Driver、Hall 和 Kemp 对酉群 $\mathrm{U}(N)$ 的 Segal-Bargman 变换的结果。我们提供了一种计算 Segal-Bargmann 变换对 \emph {trace polynomials} 的作用的有效方法，它包括扩展多项式泛函演算的 $K_N$ 上的平滑函数子空间。使用这些结果，我们表明作为 $N\to\infty$，有限维变换 $\mathbf{B}_{s,\tau}^{K_N}$ 有一个有意义的极限 $\mathscr{G}_ {s,