Local gamma factor - Knowlpedia

For a Weil–Deligne representation $V$ of a local field $F$ and a nontrivial additive character $\psi$ , the local gamma factor is

\gamma(s,V,\psi)= \varepsilon(s,V,\psi) \frac{L(1-s,V^\vee)}{L(s,V)},

where $\varepsilon(s,V,\psi)$ is the local epsilon factor and $V^\vee$ is the dual representation. This convention places the functional equation around $s=\tfrac12$ ; shifts occur when an L-function is normalized differently.

Analytic characterization

In Tate's thesis the factor is characterized by the local functional equation for Fourier transforms. Higher-rank Rankin–Selberg integrals and the Langlands–Shahidi method similarly compare a zeta integral with its dual and produce a gamma factor.

Gamma factors are especially useful because local converse theorems can recognize a representation from families of twisted gamma factors. Under local Langlands for $\operatorname{GL}_n$ , the analytic factors agree with the corresponding Artin factors on the Weil–Deligne side.

Convention warning

Some authors write the quotient of L-factors in the reciprocal order or use $\psi^{-1}$ on the dual side. A bare symbol $\gamma(s,\cdots)$ is therefore not portable without its functional equation.

References

John Tate, “Fourier analysis in number fields and Hecke's zeta-functions,” in Algebraic Number Theory, Academic Press, 1967, 305–347.
Hervé Jacquet, Ilya Piatetski-Shapiro, and Joseph Shalika, “Rankin–Selberg convolutions,” American Journal of Mathematics 105 (1983), 367–464. JSTOR.