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Information Center for Mathematical Science

논문검색

Information Center for Mathematical Science

논문검색

Tohoku Mathematical Journal. Second Series
( Vol. 42 NO.2 / (1990))
Invariant Hyperfunctions on Regular Prehomogeneous Vector Spaces of Commutative Parabolic Type
Masakazu Muro,
Pages. 163-193
Abstract Let $(mathbf{G}^+_mathbf{R},
ho, mathbf{V})$ be a regular irreducible prehomogeneous vector space defined over the real field $mathbf{R}$. We denote by $P(x)$ its irreducible relatively invariant polynomial. Let $mathbf{V}_1 cup mathbf{V}_2 cup cdots cup mathbf{V}_l$ be the connected component decomposition of the set $mathbf{V}-{ x in mathbf{V}; P(x) =0}$. It is conjectured by [Mr4] that any relatively invariant hyperfunction on $mathbf{V}$ is written as a linear combination of the hyperfunctions $| P(x)|^s_i$, where $|P(x)|^s_i$ is the complex power of $|P(x)|^s$ supported on $ar{mathbf{V}}_i$. In this paper the author gives a proof of this conjecture when $(mathbf{G}^+_mathbf{R},
ho, mathbf{V})$ is a real prehomogeneous vector space of commutative parabolic type. Our proof is based on microlocal analysis of invariant hyperfunctions on prehomogeneous vector spaces.
Contents Introduction 1. Formulation of the main problem 1.1. Preliminary conditions and some definitions 1.2. Main problem 2. Regular prehomogeneous vector spaces of commutative parabolic type 2.1. Prehomogeneous vector spaces of parabolic type 2.2. Holonomic systems $\mathfrak{M}_s$ for prehomogeneous vector spaces of commutative parabolic type 3. Holonomic systems on the real locus and its solutions 3.1. Solutions with a holomorphic parameter $s$ 3.2. Real holonomy diagrams 3.3. Relations for microfunction solutions 4. Real forms of prehomogeneous vector spaces of commutative parabolic type 4.1. The list of real forms 4.2. Real holonomy diagrams of $\mathfrak{M}_s$ 5. Proof of the main theorem 5.1. Critical points for $P(x)^s$ 5.2. Proof of the main theorem at non-critical points 5.3. Proof of the main theorem at critical points 5.4. Conclusion and remark
Key words prehomogeneous vector space, invariant, hyperfunction, micro-local analysis
Mathmatical Subject Classification