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

Abstract

We determine an explicit form of the Igusa local zeta function of the simple prehomogeneous vector space $\left(GL\right(1{)}^4\times SL(2n+1),{\mathrm{\Lambda }}_2\oplus {\mathrm{\Lambda }}_1\oplus {\mathrm{\Lambda }}_1\oplus {\mathrm{\Lambda }}_1)$. As for the Igusa local zeta functions and the $p$-adic $\mathrm{\Gamma }$-factors of regular reducible simple prehomogeneous vector spaces with universally transitive open orbits, a table for explicit forms is completed by our result together with the result in Hosokawa [2].