Muro, Masakazu Invariant hyperfunctions on regular prehomogeneous vector spaces of commutative parabolic type. (English) Zbl 0751.46027 Tôhoku Math. J., II. Ser. 42, No. 2, 163-193 (1990). 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. Cited in 5 Documents MSC: 46F15 Hyperfunctions, analytic functionals 32A45 Hyperfunctions Keywords:regular irreducible prehomogeneous vector space; irreducible relatively invariant polynomial; relatively invariant hyperfunction; linear combination of the hyperfunctions Citations:Zbl 0639.14032 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] B. DATSKOVSKY AND D. J. WRIGHT, The adelic zeta functions associated to the space of binary cubic forms, Part II: Local theory, J. Reine Angew. Math. 367 (1986), 27-75. · Zbl 0575.10016 · doi:10.1515/crll.1986.367.27 [2] M. KASHIWARA, ^-functions and holonomic systems, Invent, math. 38 (1976), 33-53 192M. MURO · Zbl 0354.35082 · doi:10.1007/BF01390168 [3] M. 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