Klingen $${\mathfrak {p}}^2$$ vectors for $$\mathrm{GSp}(4)$$

Dimension formulas for Siegel modular forms of level 4

“…Proof (i)For $$H=\mathrm{K}(\mathfrak {p})$$, see [27, Section A.8]. For $$H=\Gamma _0^{\prime }(\mathfrak {p}^2)$$, see [47, Table 1]. For every other H , there exists a conjugate subgroup $$\tilde{H}$$ such that $$\Gamma (\mathfrak {p})\subset \tilde{H}\subset K$$.…”

“…We thus get the desired dimensions from the $$r_K(\pi )$$ listed in Table 4. (ii)If $$H\ne \Gamma _0^{\prime }(\mathfrak {p}^2)$$, then a conjugate of H contains $$\Gamma (\mathfrak {p})$$, so that $$r_K(\pi )=V^{\Gamma (\mathfrak {p})}\supset V^H\ne 0$$. If $$H=\Gamma _0^{\prime }(\mathfrak {p}^2)$$, then [47, Lemma 4] shows that $$V^{\Gamma (\mathfrak {p})}\ne 0$$.$$\Box$$…”

“…Representatives for Γ 0 (2) follow from $$$$\begin{equation} \Gamma _0(2)\backslash {\rm Sp}(4,{\mathbb {Q}})/P({\mathbb {Q}})\cong P(\mathbb {F}_2)\backslash {\rm Sp}(4,\mathbb {F}_2)/P(\mathbb {F}_2) \end{equation}$$$$and the Bruhat decomposition; similarly, for $$\Gamma _0^{\prime }(2)$$ and B (2). For $$\Gamma _0^{\prime }(4)$$ and M (4), see [47, Lemma 1, Lemma 2]. For $$\Gamma =\Gamma _0(4)$$, see [42, Proposition 2.6].…”

“…For the paramodular groups, see [27]. For $$M(\mathfrak {p}^2)$$ and $$\Gamma _0^{\prime }(\mathfrak {p}^2)$$, see [47].…”

“…A key result which allows us to get information about $$\Gamma _0^{\prime }(4)$$ is [47, Lemma 4]. It implies that if an irreducible, admissible representation of $${\rm GSp}(4,{\mathbb {Q}}_2)$$ has nonzero $$\Gamma _0^{\prime }(\mathfrak {p}^2)$$‐invariant vectors, then it also has nonzero $$\Gamma (\mathfrak {p})$$‐invariant vectors, and hence, appears in Table 4.…”