Further results on the paramodular Hecke algebra

In this paper we further examine the structure of the graded paramodular Hecke algebra $\mathcal{H}$. We define a commutative subring ${\mathcal{H}}^{\alpha }$ of $\mathcal{H}$ as the elements fixed by a certain automorphism $\alpha$, and we show that the elements $T(q^k)$, defined as the sum of the double cosets of degree $k$, are contained in ${\mathcal{H}}^{\alpha }$. We prove that the formal power series in the indeterminate $t$ with coefficients $T(q^k)$ is an explicit rational function in $t$ with coefficients from ${\mathcal{H}}^{\alpha }$. We also determine the structure of ${\mathcal{H}}^{\alpha }$ and its relation to $\mathcal{H}$.