The Igusa local zeta function of the simple prehomogeneous vector space \((\text{GL}(1)^4\times \text{SL}(2n+1),\Lambda_2\oplus \Lambda_1\oplus\Lambda_1\oplus\Lambda_1)\) (Q1775414)

In Theorem 2.1 the author determines explicitly the Igusa local zeta function of the simple prehomogeneous vector space \( \left( \text{GL}(1)\times \text{SL}(2n+1),\Lambda _{2}\oplus \Lambda _{1}\oplus \Lambda _{1}\oplus \Lambda _{1}\right) ,n\geq 1. \) The Igusa zeta function as a distribution on the space of Schwartz-Bruhat functions satisfies a functional equation of the type \(Z(s-\kappa ,\widehat{ \Phi ^{\ast }})=\gamma \left( s\right) Z^{\ast }(-s,\Phi ^{\ast })\), where \( \widehat{\left( \cdot \right) }\) denotes the Fourier transform, \(\Phi ^{\ast }\) is a Schwartz-Bruhat function, and \(Z^{\ast }(s,\Phi ^{\ast })\) is the ''dual'' of \ \(Z(s,\Phi )\). In Theorem 4.1 the author determines explicitly the \(\Gamma \)-factor \(\gamma \left( s\right) \). Furthermore, the author expresses \(\gamma \left( s\right) \) in terms of the Tate local factor and the \(b\)-function of the prehomogeneous space.