Singular invariant hyperfunctions on the square matrix space and the alternating matrix space

Abstract

Fundamental calculations on singular invariant hyperfunctions on the $n \times n$ square matrix space and on the $2n \times 2n$ alternating matrix space are considered in this paper.By expanding the complex powers of the determinant function or the Pfaffian function into the Laurent series with respect to the complex parameter, we can construct singular invariant hyperfunctionsas their Laurent expansion coefficients. The author presents here the exact orders of the poles of the complex powers and determines the exact supports of the Laurent expansion coefficients. By applying these results, we prove that every quasi-relatively invariant hyperfunction can be expressed as a linear combination of the Laurent expansion coefficients of the complex powers and that every singular quasi-relatively invariant hyperfunction is in fact relatively invariant on the generic points of its support. In the last section, we give the formula of the Fourier transforms of singular invariant tempered distributions.