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Invariant hyperfunctions on regular prehomogeneous vector spaces of commutative parabolic type. (English) Zbl 0751.46027

Summary: Let \((G,\mathbb{R}^ +,\rho,V)\) be a regular irreducible prehomogeneous vector space defined over the real field \(\mathbb{R}\). We denote by \(P(x)\) its irreducible relatively invariant polynomial. Let \(V_ 1\cup V_ 2\cup\dots\cup V_ \ell\) be the connected component decomposition of the set \(V-\{x\in V;\;P(x)=0\}\). It is conjectured by the author in Proc. Jap. Acad., Ser. A 63, 66-68 (1987; Zbl 0639.14032) that any relatively invariant hyperfunction on \(V\) is written as a llinear combination of the hyperfunctions \(| P(x)|^ s_ i\), where \(| P(x)|^ s_ i\) is the complex power of \(| P(x)|^ s\) supposed on \(\overline {V}_ i\). In this paper the author gives a proof of this conjecture when \((G_ \mathbb{R}^ +,\rho,V)\) is a real prehomogeneous vector space of commutative parabolic type. Our proof is based on microlocal analysis of invariant hyperfunctions on prehomogeneous vector spaces.

MSC:

46F15 Hyperfunctions, analytic functionals
32A45 Hyperfunctions

Citations:

Zbl 0639.14032
Full Text: DOI

References:

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