Singular invariant hyperfunctions on the space of real symmetric matrices (Q1961355)

Let \(Sym_n({\mathbf R})\) be the space of \(n\times n\) symmetric matrices over the real field \({\mathbf R}\) and let \({\mathbf V}_i:=\{x\in Sym_n({\mathbf R})|\text{det}(x)\neq 0\), \(sgn(x)=(i,n-i)\}\). The purpose of the paper is to study the following singular \(SL_n({\mathbf R})\)-invariant hyperfunction \(P^{a,s}(x)= \sum^n_{i=0} a_i\cdot |\text{det}(x)|^s_i\), where \(a=(a_0, \dots,a_n)\), \(a_i\in\mathbb{C}\), \(s\in\mathbb{C}\), \(|\text{det}(x)|^s_i= |\text{det}(x)|^s\) if \(x\in {\mathbf V}_i\) and \(|\text{det}(x) |^s_i=0\) if \(x\notin {\mathbf V}_i\). \(P^{a,s}(x)\) is a hyperfunction (or microfunction) with a meromorphic parameter \(s\), i.e. it is a hyperfunction (or microfunction) with a holomorphic parameter \(s\in\mathbb{C}-K\) with a discrete subset \(K\) of \(\mathbb{C}\), and, for each \(s_0\in K\), there exists \(m\in\mathbb{Z}_+\) such that \((s-s_0)^m P^{a,s}(x)\) is holomorphic with respect to \(s\) near \(s_0\). An algorithm to determine the orders of poles of the hyperfunction \(P^{a,s}(x)\) is given. The exact support of the principal part of the poles is determined.