Let \(G\) be a quasi-split connected reductive algebraic group over a \(p\)-adic field \(k\) and let \(K\) be the splitting field of \(G\). Let \(B=TU\) be a Borel \(k\)-subgroup of \(G\) and let \(P\) be a self-associate maximal parabolic \(k\)-subgroup of \(G\), with Levi \(k\)-decomposition \(P=MN\), \(M\supset T\). We set \(U_M:=U\cap M\).
Let \(\xi\) be the \(K\)-rational character of \(M\) defined by \(\xi(m):=\det(\text{Ad}_{\mathfrak n}(m))\) for \(\mathfrak n\) the Lie algebra of \(N\). Let \(\pi\) be an irreducible generic admissible representation of \(M(k)\) and let \(C_\psi(s,\pi)\) denote the corresponding local coefficient (which is defined by the way of an intertwining operator and of Whittaker functionals).
Then, under some simplifying hypotheses, including the dimension condition \(\dim(U_M\backslash N)=2\), the authors are able to express the inverse of the local coefficient as, up to an abelian \(\gamma\)-factor, a genuine Mellin transform of a partial Bessel function of a type they have analysed in a previous work. Then they prove that \(C_\psi(s,\pi)\) is stable, that is, if \(\pi_1\) and \(\pi_2\) are two irreducible generic admissible representations of \(M(k)\) which share the same central character, then \[ C_\psi(s,\pi_1\otimes\tilde\nu)=C_\psi(s,\pi_2\otimes\tilde\nu), \] for all sufficiently highly ramified characters \(\nu\) of \(K^\times\), identified as a character \(\tilde\nu\) of \(M(k)\) by \(\tilde\nu(m):=\nu(\xi(m))\). They illustrate their results in particular in the case of quasi-split unitary groups.
This proves stability of \(\gamma\)-factors (which are defined inductively by means of local coefficients) in all the cases of interest in functoriality, and hopefully, the analysis given in the article will open the door to a proof of the general stability and the equality of \(\gamma\)-factors obtained from different methods through integration over certain quotient spaces whose generic fibres are closed.