"Research on prehomogeneous vector spaces and micro-local analysis"

(a) Singular invariant hyperfunctions on the space of n x n real symmetric matrices are discussed. We construct singular invariant hyperfunctions, i.e., invariant hyperfunctions whose supports are contained in the set S ={det(x) = O}, in terms of negative order coefficients of the Laurent expansions of the complex powers of the determinant function. In particular, we give an algorithm to determine the orders of poles of the complex powers of the determinant functions and the support of the singular hyperfunctions appearing in the principal part of the Laurent expansions of the complex powers.(b) Singular invariant hyperfuncsions on the space of n x n complex and quaternion matrices are discussed. We give an algorithm to determine the orders of poles of the complex power of the determinant function and to determine exactly the support of singular invariant hyperfunctions, i.e., invariant hyperfunctions whose supports are contained in the set S : ={det(x) = O}, obtained as negative-order-coefficients of the Laurent expansions of the complex powers.

(a) Singular invariant hyperfunctions on the space of n x n real symmetric matrices are discussed. We construct singular invariant hyperfunctions, i.e., invariant hyperfunctions whose supports are contained in the set S ={det(x) = O}, in terms of negative order coefficients of the Laurent expansions of the complex powers of the determinant function. In particular, we give an algorithm to determine the orders of poles of the complex powers of the determinant functions and the support of the singular hyperfunctions appearing in the principal part of the Laurent expansions of the complex powers.(b) Singular invariant hyperfuncsions on the space of n x n complex and quaternion matrices are discussed. We give an algorithm to determine the orders of poles of the complex power of the determinant function and to determine exactly the support of singular invariant hyperfunctions, i.e., invariant hyperfunctions whose supports are contained in the set S : ={det(x) = O}, obtained as negative-order-coefficients of the Laurent expansions of the complex powers.