Computation 2.7 · Computation: \(T_\lambda\ast\varphi_0\) on hermitian \(2\times2\) matrices

1. Formula. \((T_\lambda\ast\varphi_0)(x)=\#\{gK\subseteq Kt^\lambda K:\ g^{-1}xg^{-*}\in\mathrm{GL}_2(\mathcal O_E)\}\). The cosets in \(Kt^{(1,0)}K/K\) are represented by \(g=\left(\begin{smallmatrix}t&b\\0&1\end{smallmatrix}\right)\), \(b\in\mathbb F_{q^2}\), and \(g=\operatorname{diag}(1,t)\): \(q^2+1\) cosets.
2. Value at \(x=t\cdot1\). For the first family,

$$g^{-1}xg^{-*}=\begin{pmatrix}t^{-1}(1+N(b))&-b\\-\bar b&t\end{pmatrix}.$$

This is integral iff \(N(b)=-1\), which has \(q+1\) solutions in \(\mathbb F_{q^2}\), and its determinant is then \(1\). The coset \(\operatorname{diag}(1,t)\) gives \(\operatorname{diag}(t,t^{-1})\), which is not integral. Value: \(q+1\).

3. Value at \(x=\operatorname{diag}(t^2,1)\). The first family gives \(\left(\begin{smallmatrix}1+t^{-2}N(b)&-bt^{-1}\\-\bar bt^{-1}&1\end{smallmatrix}\right)\), which is integral iff \(b=0\), and then it is the identity. The second coset gives \(\operatorname{diag}(t^2,t^{-2})\). Value: \(1\).
4. Supports. By Theorem 2.6, step 2, \(T_{(1,0)}\ast\varphi_0\) is supported on \(\{(2,0),(1,1)\}\), so the first identity is complete. The remaining identities are supported on \(\{c\le2a\}\) for the same reason. Vanishing on odd \(c_1+c_2\) is the parity invariant of Theorem 2.6, step 6.
5. Enumeration. The script enumerates the Hermite representatives \(\left(\begin{smallmatrix}t^i&b\\0&t^j\end{smallmatrix}\right)\) of \(Kt^\lambda K/K\): \(10\) and \(90\) cosets for \(q=3\), \(26\) and \(650\) for \(q=5\). It tests integrality and unimodularity of \(g^{-1}xg^{-*}\) exactly, over \(\mathbb F_{q^2}[t,t^{-1}]\). The results are \((1,4)\), \((1,4,12)\), \((1,4,12)\) for \(q=3\) and \((1,6)\), \((1,6,30)\), \((1,6,30)\) for \(q=5\), as predicted, and \(0\) on the odd orbit \((1,0)\) for \(\lambda=(1,0),(0,0),(1,-1),(1,1)\). This checks the hand computation in steps 2–3; it is not used in the proofs.
6. Non-generation. On the lower set \(\{c\le(4,0)\}\cap\{c_2\ge0\}=\{(4,0),(3,1),(2,2)\}\), the Hecke span is spanned by \((1,q+1,q(q+1))\) and \((0,0,1)\). This is a plane in a \(3\)-dimensional space, containing neither \(\mathbf 1_{(3,1)}\) nor \(\mathbf 1_{(4,0)}\). On \(\{(2,0),(1,1)\}\) the span is the line through \((1,q+1)\). These are the cases \(d=1,2\) of Theorem 2.6(5): codimensions \(p_{\le2}(2)-p_{\le2}(1)=1\) and \(p_{\le2}(4)-p_{\le2}(2)=1\).
7. \(\mathrm{SL}_2\). \(K_{\mathrm{SL}}t^{\alpha^\vee}K_{\mathrm{SL}}/K_{\mathrm{SL}}\) is the set of lattices in relative position \((1,-1)\); it is the same set as for \(\mathrm{GL}_2(E)\), since determinants can be adjusted by \(\operatorname{diag}(u,1)\), \(u\in\mathcal O_E^\times\). For \(\det x=1\) the integrality test is unchanged. So \(T_{\alpha^\vee}\ast\varphi_0=\mathbf 1_{2\alpha^\vee}+(q+1)\mathbf 1_{\alpha^\vee}+q(q+1)\mathbf 1_0\) and \(T_0\ast\varphi_0=\mathbf 1_0\). On \(\{0,\alpha^\vee,2\alpha^\vee\}\) the span is \(2\)-dimensional and omits \(\mathbf 1_{\alpha^\vee}\).