Singular invariant hyperfunctions on the space of real symmetric matrices

Singular invariant hyperfunctions on the space of real symmetric matrices o f size n are discussed in this paper. We construct singular invariant hyper functions, i.e., invariant hyperfunctions whose supports are contained in t he set of the points of rank strictly less than n, in terms of negative ord er coefficients of the Laurent expansions of the complex powers of the dete rminant function. In particular, we give an algorithm to determine the orde rs of poles of the complex powers of the determinant functions and the supp ort of the singular hyperfunctions appearing in the principal part of the L aurent expansions of the complex powers.