Singular invariant tempered distributions on regular prehomogeneous vector spaces
The author proves an excellent result: It is established (with necessary and sufficient conditions) that a \(G^ 1\)-invariant tempered distribution supported in S (defined below) is obtained as a finite linear combination of negative Laurent coefficients of the tempered distributions \(| P(x)|\) \(s_ i.\) Let P(x) be an irreducible homogeneous polynomial with real coefficients on real vector space V. Let \(S=\{x\in V;P(x)=0\}\) decompose into finite number of G 1-orbits with \(G^ 1=\{g\in GL(V);P(g\cdot x)=P(x)\}\). Let \(P(x)\) also denote an irreducible homogeneous polynomial of degree d on a f.d. complex vector space \(V_ C\) satisfying that det. of the Hessian \(\partial P(x)/\partial x_ i\partial x_ j\) does not identically vanish on \(V_ C\), and let \(G\) \(1_ C=\{g\in GL(V_ C);P(g\cdot x)=P(x)\}\). Suppose that (A) (i) The set \(S_ C=\{x\in V_ C;P(x)=0\}\) decomposes into a finite number of \(G^ 1_ C\)-orbits. (ii) The open set \(V_ C-S_ C\) is a single \(G^ 1_ C\times GL_ 1(C)\)-orbit. and (iii) The restriction of \(P(x)\) on V is a polynomial with real coefficients. The main result proved is stated as follows: Theorem: Under the condition (A), following conditions (B) and (C) are equivalent: (B) Any \(G^ 1\)-invariant singular tempered distribution is obtained as a finite linear combination of \(T^ j_{ik}(x)\) (details omitted). (C) For \(\lambda\in C\), any relatively invariant tempered distribution \(T(x)\) corresponding to the character \(\chi(g)^{\lambda}\) is obtained as a linear combination: \(T(x)=\sum^{\ell}_{i=1}a_ i(s)\cdot | P(x)|\) \(s_ i|_{s=\lambda}\), with holomorphic functions \(a_ i(s)\) at \(s=\lambda\); \(| P(x)|\) \(s_ i=\sum_{j_ k\leq j<\infty}T^ j_{i_ k}(x)\cdot (s-\lambda_ k)^ j\).
scientific article; zbMATH DE number 4060100