The Hecke algebra of \(\mathrm{GL}_2(\mathbb{Q}_p)\) with respect to Iwahori subgroup is well-known. We extend this to study genuine Hecke algebra of the double cover \(\mathrm{DSL}_2(\mathbb{Q}_p)\) with respect to certain open compact subgroups. We translate certain elements of this Hecke algebra to obtain classical operators on the space of cusp forms of half-integral integral weight. We use these operators to indentify the ``minus space'' of level \(4M\), \(M\) odd and square-free. Our minus space is the counterpart of Kohnen's plus space in the sense that it is Hecke isomorphic to \(S_{2k}^{new}(2M)\). In an analogy with our work in the integral weight setting we show that the minus space is characterized as a common eigenspace of certain finite set of operators with given eigenvalues. This is a joint work with Prof. E. M. Baruch.

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