A p-adic analog of Hasse--Davenport product relation involving ϵ-factors

In this paper we prove some generalizations of the classical Hasse–Davenport product relation for certain arithmetic factors defined on a p -adic field F , among them one finds the ϵ-factors appearing in Tate’s thesis. We then show that these generalizations are equivalent to some representation theoretic identities relating the determinant of ramified local coefficients matrices defined for coverings of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>SL</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>F</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {SL₂(F)} to Plancherel measures and γ-factors.