Newforms of Half-integral Weight: The Minus Space Counterpart

We study genuine local Hecke algebras of the Iwahori type of the double cover of SL₂(ℚ_p) and translate the generators and relations to classical operators on the space S_k+1/2(γ₀(4M)), M odd and square-free. In [9] Manickam, Ramakrishnan, and Vasudevan defined the new space of S_k+1/2(γ₀(4M)) that maps Hecke isomorphically onto the space of newforms of S_2k(γ₀(2M)). We characterize this newspace as a common -1-eigenspace of a certain pair of conjugate operators that come from local Hecke algebras. We use the classical Hecke operators and relations that we obtain to give a new proof of the results in [9] and to prove our characterization result.