Contragredients and a multiplicity one theorem for general spin groups

Abstract

Each orthogonal group $${\text {O}}(n)$$ has a nontrivial $${\text {GL}}(1)$$ -extension, which we call $${\text {GPin}}(n)$$ . The identity component of $${\text {GPin}}(n)$$ is the more familiar $${\text {GSpin}}(n)$$ , the general Spin group. We prove that the restriction to $${\text {GPin}}(n-1)$$ of an irreducible admissible representation of $${\text {GPin}}(n)$$ over a nonarchimedean local field of characteristic zero is multiplicity free and also prove the analogous theorem for $${\text {GSpin}}(n)$$ . Our proof uses the method of Aizenbud, Gourevitch, Rallis and Schiffman, who proved the analogous theorem for $${\text {O}}(n)$$ , and of Waldspurger, who proved that for $${\text {SO}}(n)$$ . We also give an explicit description of the contragredient of an irreducible admissible representation of $${\text {GPin}}(n)$$ and $${\text {GSpin}}(n)$$ , which is needed to apply their method to our situations.