Stability of Rankin–Selberg local <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γ</mml:mi> </mml:math> -factors for split classical groups: The symplectic case

We establish the stability of Rankin–Selberg local <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γ</mml:mi> </mml:math> -factors attached to generic representations of symplectic groups and general linear groups over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> -adic fields. Our approach uses the Langlands–Shahidi method and provides a new proof that does not rely on previously known stability results. The key innovations involve handling the geometry of orbit spaces with non-trivial stabilizers and developing new techniques for analyzing partial Bessel integrals in this setting. The results have important implications for the Langlands program, particularly for local converse theorems and the functoriality conjecture.