Classification and representation theoretic study of the functional equation-spaces

让Cp(分别为Cq)是p(？)中正定实二次型的Clifford代数。Q)变量。对于Cp和Cq的张量积的表示ρ，我们可以将ρ的表示空间上的4次齐次多项式P联系起来，该多项式具有“复幂的傅里叶变换}”|P|^S与|P*|^{-m/4-S}(m=ρ的次数)，并且具有显式的伽马因子.由于准齐次向量空间的理论，不可约正则准齐次向量空间的基本相对不变量满足上述性质。然而，多项式P不一定是任何非齐次向量空间的相对不变量。多项式P是非齐次向量空间的相对不变量，仅在极少数例外情况下。在这项研究中，我们讨论了保持P不变的线性变换群的结构和作用。

Let Cp(resp. Cq) be the Clifford algebra of a positive definite real quadratic form in p(resp. q) variables. For a representationρof the tensor product of Cp and Cq, we can associate a homogeneous polynomial P of degree 4 on the representation space ofρhaving the property "The Fourier transform of the complex power}}| P|^s coincides with|P*|^{-m/4-s}(m=the degree ofρ) with an explicit gamma factor. Owing to the theory of prehomogeneous vector spaces, the basic relative invariant of an irreducible regular prehomogeneous vector space satisfies the property above. However the polynomials P are not necessarily relative invariants of any prehomogeneous vector spaces. The polynomials P are relative invariants of prehomogeneous vector spaces only for quite few exceptional cases. In this study, we discuss the structure and the action of the group of linear transformations that leave P invariant.