| Summary: | This is a review paper on the generalization of Euclidean as well as pseudo-Euclidean groups of interest in quantum mechanics. The Weyl–Heisenberg groups, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>H</mi><mi>n</mi></msub></semantics></math></inline-formula>, together with the Euclidean, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mi>n</mi></msub></semantics></math></inline-formula>, and pseudo-Euclidean <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub></semantics></math></inline-formula>, groups are two families of groups with a particular interest due to their applications in quantum physics. In the present manuscript, we show that, together, they give rise to a more general family of groups, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>K</mi><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub></semantics></math></inline-formula>, that contain <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>H</mi><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub></semantics></math></inline-formula> as subgroups. It is noteworthy that properties such as self-similarity and invariance with respect to the orientation of the axes are properly included in the structure of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>K</mi><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub></semantics></math></inline-formula>. We construct generalized Hermite functions on multidimensional spaces, which serve as orthogonal bases of Hilbert spaces supporting unitary irreducible representations of groups of the type <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>K</mi><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub></semantics></math></inline-formula>. By extending these Hilbert spaces, we obtain representations of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>K</mi><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub></semantics></math></inline-formula> on rigged Hilbert spaces (Gelfand triplets). We study the transformation laws of these generalized Hermite functions under Fourier transform.
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