Stability of gamma factors for \(SO(2n+1)\)
The authors prove that gamma factors of generic representations of (split) \(G= SO_{2n+1}(k)\) over a non-Archimedean local field are stable under highly ramified twistings. This means that given two irreducible generic representations of \(G\), \(\pi_1\), \(\pi_2\), then \[ \gamma(\pi_1\times \mu,s,\psi)= \gamma(\pi_2\times \mu, s,\psi) \] for sufficiently ramified characters \(\mu\) of \(k^*\). The degree of high ramification depends on the representations. Here \(\psi\) is a fixed nontrivial character of \(k\). These gamma factors appear in the local theory of Rankin-Selberg convolutions for \(SO_{2n+ 1}\times GL_1\). They can also be obtained as local coefficients by Shahidi's method. This stability property is an important ingredient in the proof (by the authors, Kim and Shahidi) of the existence, via the converse theorem, of a weak functorial lift from irreducible automorphic cuspidal generic representations of \(SO_{2n+ 1}(A)\) to automorphic representations of \(GL_{2n}(A)\). The authors use the definition of \(\gamma(\pi\times \mu,s,\psi)\) as the proportionality factor in the local functional equation which results from the corresponding Rankin-Selberg integrals. The rough idea is to substitute appropriate data in the functional equation and to obtain (for \(\text{Re}(s)\) sufficiently negative) \[ \gamma(\pi\times \mu,s,\psi)= \int_{k^*} j_\pi(x) \mu^{-1}(x)|x|^{3/2-s-n} d^*x, \] where \(j_\pi\) is the Bessel function associated to the element \(\beta= \left(\begin{smallmatrix} && 1\\ &-I_{2n-1}\\ 1\end{smallmatrix}\right)\). This function is a very delicate object and requires a careful definition and a careful interpretation. For example, \(j_\pi(x)\) is defined through an integral which is not absolutely convergent, but rather stabilizes on certain increasing compact groups, depending on \(x\). The point now is to show that the Bessel function \(j_\pi(x)\) has a fixed asymptotic behaviour at infinity, which is independent of \(\pi\). The result is that the difference \(j_{\pi_1}(x)- j_{\pi_2}(x)\) equals the difference \(W_1(\widehat x)- W_2(\widehat x)\), where \(W_i\) are certain Whittaker functions in the Whittaker model of \(\pi_i\) and \(\widehat x= \left(\begin{smallmatrix} x\\ & I\\ && x^{-1}\end{smallmatrix}\right)\). Thus, for \(\mu\) sufficiently ramified and for \(\text{Re}(s)\) sufficiently negative (and hence for all \(s\)), we have \[ \gamma(\pi_1\times \mu, s,\psi)- \gamma(\pi_2\times \mu, s,\psi)= \int (W_1(x)- W_2(x)) \mu^{-1}(x)|x|^{3/2- s-n} d^*x= 0 \] due to the high ramification of \(\mu\) and the smoothness of \(W_i(\widehat x)\).
scientific article; zbMATH DE number 1158111