The local period integrals and essential vectors (2106.08234v5)

Abstract: By applying the formula for essential Whittaker functions established by Matringe and Miyauchi, we study five integral representations for irreducible admissible generic representations of ${\rm GL}_n$ over $p$-adic fields. In each case, we show that the integrals achieve local formal $L$-functions defined by Langlands parameters, when the test vector is associated to the newform. We give the relation between local periods involving essential Whittaker functions and special values of formal $L$-factors at $s=1$ for certain distinguished or unitary representations. The period integrals are also served as standard non-zero distinguished forms.