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

Abstract

Abstract. Fundamental calculations on singular invariant hyperfunctions on the n × n square matrix space and on the 2n × 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 hyperfunctions as 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 quasirelatively 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. §1. Introduction Let V := Mat n (R) be the real vector space of n × n real matrices. The real connected algebraic group G := GL n (R) + × SL n (R) acts on Mat n (R) algebraically by. Then the pair (G, V ) is a regular prehomogeneous vector space with an irreducible relative invariant P (x) := det(x) and V decomposes into a finite number of G-orbits. This is the first object of this paper. Next, let V := Alt 2n (R) be the real vector space of 2n × 2n real alternating matrices. The real connected algebraic group G := GL 2n (R) + acts on Alt 2n (R) algebraically by (2) (g, x) −→ g · x := gx t g with g ∈ G and x ∈ V . Then the pair (G, V ) is a regular prehomogeneous vector space with an irreducible relative invariant P (x) := Pf(x) and V decomposes into a finite number of G-orbits. Here Pf(x) stands for the Pfaffian of the alternating matrix x which is a polynomial on Alt 2n (R) given as a square root of det(x) taking the value Pf(J n ) = (−1) n(n−1)/2 for J n := 0n In −In 0n . This is the second object in this paper. These two prehomogeneous vector spaces have some common properties. Both of two have two open G-orbits, one is V + := {x ∈ V | P (x) > 0} and the other is V − := {x ∈ V | P (x) < 0}. We define the complex power of the relative invariant P (x) byfor s ∈ C. Then it is well known that the integral |P (x)| s ± f (x) dx is absolutely convergent for all the rapidly decreasing function f (x) provided that the real part (s) of s ∈ C is non-negative. In addition, |P (x)| s ± can be regarded as a tempered distribution -and hence a hyperfunctionon V with a holomorphic parameter s ∈ C and it is extended to the whole complex plane as a meromorphic function in s ∈ C. The poles of P [ a,s] (x) belong to the set Z <0 := {−1, −2, . . . }. We define the linear combination offor a := (a + , a − ) ∈ C 2 and s ∈ C, which is a relatively invariant hyperfunction under the action of G. Namely we havefor all g ∈ G with χ(g) = det(g 1 ) when V = Mat n (R) or with χ(g) = det(g) when V = Alt 2n (R). For a fixed complex number λ ∈ C, we call a...

Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians

“…The special case in which P is a determinant of a symmetric or hermitian matrix (and Ω is e.g. the cone of positive-definite matrices) has been studied by several authors [10,11,36,[72][73][74]80,81]; it plays a central role in the theory of Riesz distributions on Euclidean Jordan algebras (or equivalently on symmetric cones) [36,Chapter VII]. This case is also useful in quantum field theory in studying the analytic continuation of Feynman integrals to "complex space-time dimension" [10,35,92].…”