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 \(\{\pi _{\Lambda }(m, n)g_{1} \oplus \cdots \oplus \pi _{\Lambda }(m, n)g_{k}\}_{m, n \in {\mathbb {Z}}^{d}}\) is a frame for \(L^{2}({\mathbb {R}}\,^{d})\oplus \cdots \oplus L^{2}({\mathbb {R}}\,^{d})\) if and only if \(\cup _{i=1}^{k}\{\pi _{\Lambda ^{o}}(m, n)g_{i}\}_{m, n\in {\mathbb {Z}}^{d}}\) is a Riesz sequence, and \(\cup _{i=1}^{k} \{\pi _{\Lambda }(m, n)g_{i}\}_{m, n\in {\mathbb {Z}}^{d}}\) is a frame for \(L^{2}({\mathbb {R}}\,^{d})\) if and only if \(\{\pi _{\Lambda ^{o}}(m, n)g_{1} \oplus \cdots \oplus \pi _{\Lambda ^{o}}(m, n)g_{k}\}_{m, n \in {\mathbb {Z}}^{d}}\) is a Riesz sequence, where \(\pi _{\Lambda }\) and \(\pi _{\Lambda ^{o}}\) is a pair of Gabor representations restricted to a time–frequency lattice \(\Lambda \) and its adjoint lattice \(\Lambda ^{o}\) in \({\mathbb {R}}\,^{d}\times {\mathbb {R}}\,^{d}\).

在Dutkay等人中发展了群体表示的对偶原理。（J Funct Anal 257：1133–1143，2009），Han和Larson（Bull Lond Math Soc 40：685–695，2008）展示了一个事实，即Gabor分析中众所周知的对偶原理不是孤立的事件，而是更多的事件。普遍存在于群体表征理论中的现象。Gabor分析中还有两个其他众所周知的基本属性：Gabor分析的生物正交性和基本同一性。本文的主要目的是证明对于一般射影unit群表示，这两个基本属性仍然是正确的。此外，我们还提出了一个一般的对偶定理，该定理表明多帧生成器通过组表示的双交换对与超帧生成器相遇。\（\ {\ pi _ {\ Lambda}（m，n）g_ {1} \ oplus \ cdots \ oplus \ pi _ {\ Lambda}（m，n）g_ {k} \} _ {m，n \ {\ mathbb {Z}} ^ {d}} \）中的\（L ^ {2}（{\ mathbb {R}} \，^ {d}）\ oplus \ cdots \ oplus L ^ { 2}（{\ mathbb {R}} \，^ {d}）\）仅当\（\ cup _ {i = 1} ^ {k} \ {\ pi _ {\ Lambda ^ {o}} （m，n）g_ {i} \} _ {m，n \在{\ mathbb {Z}} ^ {d}} \}中是一个Riesz序列，并且\（\ cup _ {i = 1} ^ { k} \ {\ pi _ {\ Lambda}（m，n）g_ {i} \} _ {m，n \ in {\ mathbb {Z}} ^ {d}} \）是\（L ^ {2}（{\ mathbb {R}} \，^ {d}）\）当且仅当\（\ {\ pi _ {\ Lambda ^ {o}}（m，n）g_ {1} \ oplus \ cdots \ oplus \ pi _ {\ Lambda ^ {o}}（m，n）g_ {k} \} _ {m，n \ in {\ mathbb {Z}} ^ {d}} \}是Riesz序列，其中\（\ pi _ {\ Lambda} \）和\（\ pi _ {\ Lambda ^ {o}} \）在一对的Gabor表示的限制在一个时间-频率栅格\（\ LAMBDA \）和它的伴随晶格\（\ LAMBDA ^ {ö} \）在\（{\ mathbb {R}} \，^ {d} \次{\ mathbb {R}} \，^ {d} \）。