Test vectors for local periods

Abstract

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>E</m:mi> <m:mo>/</m:mo> <m:mi>F</m:mi> </m:mrow> </m:math> {E/F} be a quadratic extension of non-Archimedean local fields of characteristic zero. An irreducible admissible representation π of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>GL</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:mi>E</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathrm{GL}(n,E)} is said to be distinguished with respect to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>GL</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:mi>F</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathrm{GL}(n,F)} if it admits a non-trivial linear form that is invariant under the action of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>GL</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:mi>F</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathrm{GL}(n,F)} . It is known that there is exactly one such invariant linear form up to multiplication by scalars, and an explicit linear form is given by integrating Whittaker functions over the F -points of the mirabolic subgroup when π is unitary and generic. In this paper, we prove that the essential vector of [14] is a test vector for this standard distinguishing linear form and that the value of this form at the essential vector is a local L -value. As an application we determine the value of a certain proportionality constant between two explicit distinguishing linear forms. We then extend all our results to the non-unitary generic case.