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Newforms of Half-integral Weight: The Minus Space Counterpart
- Part of
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- Journal:
- Canadian Journal of Mathematics / Volume 72 / Issue 2 / April 2020
- Published online by Cambridge University Press:
- 31 October 2019, pp. 326-372
- Print publication:
- April 2020
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- Article
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We study genuine local Hecke algebras of the Iwahori type of the double cover of
$\operatorname{SL}_{2}(\mathbb{Q}_{p})$ and translate the generators and relations to classical operators on the space 
$S_{k+1/2}(\unicode[STIX]{x1D6E4}_{0}(4M))$, 
$M$ odd and square-free. In [9] Manickam, Ramakrishnan, and Vasudevan defined the new space of 
$S_{k+1/2}(\unicode[STIX]{x1D6E4}_{0}(4M))$ that maps Hecke isomorphically onto the space of newforms of 
$S_{2k}(\unicode[STIX]{x1D6E4}_{0}(2M))$. 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.
