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    Abstract The duality principle for group representations developed in Dutkay et al. (J Funct Anal 257:1133–1143, 2009), Han and Larson (Bull Lond Math Soc 40:685–695, 2008) exhibits a fact that the well-known duality principle in Gabor analysis is not an isolated incident but a more general phenomenon residing in the context of group representation theory. There are two other well-known fundamental properties in Gabor analysis: the biorthogonality and the fundamental identity of Gabor analysis. The main purpose of this this paper is to show that these two fundamental properties remain to be true for general projective unitary group representations. Moreover, we also present a general duality theorem which shows that that muti-frame generators meet super-frame generators through a dual commutant pair of group representations. Applying it to the Gabor representations, we obtain that $$\{\pi _{\Lambda }(m, n)g_{1} \oplus \cdots \oplus \pi _{\Lambda }(m, n)g_{k}\}_{m, n \in {\mathbb {Z}}^{d}}$$ { π Λ ( m , n ) g 1 ⊕ ⋯ ⊕ π Λ ( m , n ) g k } m , n ∈ Z d is a frame for $$L^{2}({\mathbb {R}}\,^{d})\oplus \cdots \oplus L^{2}({\mathbb {R}}\,^{d})$$ L 2 ( R d ) ⊕ ⋯ ⊕ L 2 ( R d ) if and only if $$\cup _{i=1}^{k}\{\pi _{\Lambda ^{o}}(m, n)g_{i}\}_{m, n\in {\mathbb {Z}}^{d}}$$ ∪ i = 1 k { π Λ o ( m , n ) g i } m , n ∈ Z d is a Riesz sequence, and $$\cup _{i=1}^{k} \{\pi _{\Lambda }(m, n)g_{i}\}_{m, n\in {\mathbb {Z}}^{d}}$$ ∪ i = 1 k { π Λ ( m , n ) g i } m , n ∈ Z d is a frame for $$L^{2}({\mathbb {R}}\,^{d})$$ L 2 ( R d ) if and only if $$\{\pi _{\Lambda ^{o}}(m, n)g_{1} \oplus \cdots \oplus \pi _{\Lambda ^{o}}(m, n)g_{k}\}_{m, n \in {\mathbb {Z}}^{d}}$$ { π Λ o ( m , n ) g 1 ⊕ ⋯ ⊕ π Λ o ( m , n ) g k } m , n ∈ Z d is a Riesz sequence, where $$\pi _{\Lambda }$$ π Λ and $$\pi _{\Lambda ^{o}}$$ π Λ o is a pair of Gabor representations restricted to a time–frequency lattice $$\Lambda $$ Λ and its adjoint lattice $$\Lambda ^{o}$$ Λ o in $${\mathbb {R}}\,^{d}\times {\mathbb {R}}\,^{d}$$ R d × R d . 
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  5. Discrete and continuous frames can be considered as positive operator-valued measures (POVMs) that have integral representations using rank-one operators. However, not every POVM has an integral representation. One goal of this paper is to examine the POVMs that have finite-rank integral representations. More precisely, we present a necessary and sufficient condition under which a positive operator-valued measure $F: \Omega \to B(H)$ has an integral representation of the form $$F(E) =\sum_{k=1}^{m} \int_{E}\, G_{k}(\omega)\otimes G_{k}(\omega) d\mu(\omega)$$ for some weakly measurable maps $G_{k} \ (1\leq k\leq m) $ from a measurable space $\Omega$ to a Hilbert space $\mathcal{H}$ and some positive measure $\mu$ on $\Omega$. Similar characterizations are also obtained for projection-valued measures. As special consequences of our characterization we settle negatively a problem of Ehler and Okoudjou about probability frame representations of probability POVMs, and prove that an integral representable probability POVM can be dilated to a integral representable projection-valued measure if and only if the corresponding measure is purely atomic. 
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