| Summary: | When solving problems involving quantum mechanical systems, it is frequently desirable to find the matrix elements of a unitary representation $T$ of a real algebraic Lie group $G$. This requires defining an inner product on the Hilbert space $\mathbb{H}$ that carries the representation $T$. In the case where the representation is determined by a representation of a subgroup containing the lowest weight vector of $T$, this can be achieved through the coherent state construction. In both the scalar and vector coherent state methods, the process of finding the overlaps can be simplified by introducing the coherent state triplet ($\mathfrak{F}_{\mathbb{H}_D}$, $\mathbb{H}_D$, $\mathfrak{F}^{\mathfrak{H}_D}$) of Bargmann spaces. Coherent state wave functions -- the elements of $\mathfrak{F}_{\mathbb{H}_D}$ and of $\mathfrak{F}^{\mathbb{H}_D}$ -- are used to define the inner product on $\mathbb{H}_D$ in a way that simplifies the calculation of the overlaps. This inner product and the group action $\Gamma$ of $G$ on $\mathfrak{F}^{\mathbb{H}_D}$ are used to formulate expressions for the matrix elements of $T$ with coefficients from the given subrepresentation.
The process of finding an explicit expression for $\Gamma$ relies on matrix factorizations in the complexification of $G$ even though the representation $T$ does not extend to the complexification. It will be shown that these factorizations are, in fact, justified, that the overlaps and $\Gamma$ action can be expressed in terms of the given subrepresentation, and that it is possible to find numerical values for the inner product in $\mathbb{H}$. The scalar and vector coherent state methods will both be applied to Sp($n$) and Sp($n,\mathbb{R}$).
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