# Local Zeta Functions Attached to the Minimal Spherical Series for a Class of Symmetric Spaces

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The aim of this paper is to prove a functional equation for a local zeta function attached to the minimal spherical series for a class of real reductive symmetric spaces. These symmetric spaces are obtained as follows. We consider a graded simple real Lie algebra \$\widetilde{\mathfrak g}\$ of the form \$\widetilde{\mathfrak g}=V^-\oplus \mathfrak g\oplus V^+\$, where \$[\mathfrak g,V^+]\subset V^+\$, \$[\mathfrak g,V^-]\subset V^-\$ and \$[V^-,V^+]\subset \mathfrak g\$. If the graded algebra is regular, then a suitable group \$G\$ with Lie algebra \$\mathfrak g\$ has a finite number of open orbits in \$V^+\$, each of them is a realization of a symmetric space \$G\slash H_p\$.The functional equation gives a matrix relation between the local zeta functions associated to \$H_p\$-invariant distributions vectors for the same minimal spherical representation of \$G\$. This is a generalization of the functional equation obtained by Godement} and Jacquet for the local zeta function attached to a coefficient of a representation of \$GL(n,\mathbb R)\$.