Correction 10.3 · \(\GL_4/\Sp_4\): which Hecke translates reach the label \((1,1,1,1)\), and the corrected cancellation

Throughout, for a lattice \(M=h\cO^4\) write \(\operatorname{inv}(\cO^4,M)\) for the dominant \(c\) with \(h\in Kt^cK\), and \(M^\vee=\{x:J(x,M)\subseteq\cO\}\).

1. Labels. Two labels are comparable only if they differ by an element of the coroot lattice, hence only if they have the same \(\val\Pf\). For \(\nu=(b_1,b_1,2-b_1,2-b_1)\) one has \(\nu-\mu=k(1,1,-1,-1)\) with \(k=b_1-1\); its partial sums \(k,2k,k,0\) are nonnegative exactly when \(k\ge0\), so \(\nu\ge\mu\iff k\ge0\). Likewise \((2,2,0,0)-\nu=(2-b_1)(1,1,-1,-1)\ge0\iff b_1\le2\), and dominance \(b_1\ge2-b_1\) gives \(b_1\ge1\); so the lower set is \(\{(2,2,0,0),\mu\}\).

2. Value formula for every \(\lambda\). By Lemma A, \((T_\lambda\ast\varphi_0)(J')\) is the number of cosets \(gK\subseteq Kt^\lambda K\) with \(g^{-1}J'g^{-\mathsf T}\in X(\cO)\), i.e. with \(J'\) unimodular on \(M=g^{-\mathsf T}\cO^4\). Since \(g\mapsto g^{-\mathsf T}\) maps \(Kt^\lambda K\) onto \(Kt^{-w_0\lambda}K\), \(M\) runs bijectively over the lattices with \(\operatorname{inv}(\cO^4,M)=-w_0\lambda=(-\lambda_4,-\lambda_3,-\lambda_2,-\lambda_1)\). As \(T_\lambda\ast\varphi_0\) is \(K\)-invariant and \(\mathcal O_\mu=K\cdot\varpi J\), it suffices to evaluate at \(\varpi J\).

3. Which lattices are \(\varpi J\)-unimodular. \(\varpi J\) is unimodular on \(M\) iff \(\varpi^{-1}M^\vee=M\), i.e. \(M^\vee=\varpi M\). If \(M=h\cO^4\) with \(\operatorname{inv}=c\), then \(M^\vee=J^{-\mathsf T}h^{-\mathsf T}\cO^4\) has \(\operatorname{inv}=-w_0c\), while \(\varpi M\) has \(\operatorname{inv}=c+(1,1,1,1)\). So \(M^\vee=\varpi M\) forces \(c_4=-1-c_1\) and \(c_3=-1-c_2\), and dominance \(c_2\ge c_3\) gives \(c_2\ge0\): \(c=(a,b,-1-b,-1-a)\) with \(a\ge b\ge0\). Conversely, for such \(c\), take a symplectic basis \(e_1,f_1,e_2,f_2\) of \(\cO^4\) (\(J(e_i,f_i)=1\)) and

then \(M^\vee=\varpi^{1+a}\cO e_1\oplus\varpi^{-a}\cO f_1\oplus\varpi^{1+b}\cO e_2\oplus\varpi^{-b}\cO f_2=\varpi M\) and \(\operatorname{inv}(\cO^4,M)=c\). With step 2, \((T_\lambda\ast\varphi_0)(\varpi J)\ne0\) iff \(-w_0\lambda=c\) for such a \(c\), i.e. \(\lambda=-w_0c=(1+a,1+b,-b,-a)\). Nonnegative entries force \(a=0\), hence \(b=0\). The label \((2,0,0,0)\) would need \(-a=0\) and \(1+a=2\), which is impossible.

4. The constants. On \(\cO^4\) the Pfaffian of \(J''\) has valuation \(2\), and passing to a lattice \(M\supseteq\cO^4\) of index \(q^2\) lowers it by \(2\); so for such \(M\), \(J''\) is unimodular on \(M\) iff \(J''(M,M)\subseteq\cO\).
   • \(\lambda=(1,1,0,0)\): \(M=\cO^4+\varpi^{-1}\widetilde W\) for a plane \(W\subset\F_q^4\). Integrality of \(J''(\varpi^{-1}w,x)\) for all \(x\in\cO^4\) forces the \(e_2,f_2\)-coordinates of \(w\) to vanish modulo \(\varpi\), so \(W=\langle\bar e_1,\bar f_1\rangle\); conversely, for this \(W\), \(J''(w,w')\in\varpi^2\cO\) for all \(w,w'\in\cO e_1+\cO f_1+\varpi\cO^4\), so \(J''\) is integral on \(M\). Hence the value at \(J''\) is \(1\).
   • \(\lambda=(2,0,0,0)\): \(M=\cO^4+\cO\varpi^{-2}v\) with \(v\) primitive, determined by the line of \(v\) in \((\cO/\varpi^2)^4\). Integrality of \(J''(\varpi^{-2}v,\cO^4)\) means that the \(e_2,f_2\)-coordinates of \(v\) lie in \(\varpi^2\cO\), and \(J''(v,v)=0\) automatically; so the admissible \(M\) correspond to the points of \(\mathbb P^1(\cO/\varpi^2)\), and the value at \(J''\) is \(q(q+1)\).
   • At \(\varpi J\): \(\lambda=(1,1,0,0)\) gives the number of Lagrangian planes in \(\F_q^4\), \((q+1)(q^2+1)\) (Proposition 1.4), and \(\lambda=(2,0,0,0)\) gives \(0\) by step 3.
   By the support bound of Proposition 1.4, step 1, both functions are supported on the lower set \(\{(2,2,0,0),\mu\}\) of step 1, which gives the two expansions and the formula for \(\mathbf 1_{\mathcal O_\mu}\).

5. (SG). Order all labels by dominance; lower sets are finite, because comparable labels have the same \(\val\Pf\) and \(b_1\) is then bounded on both sides. For a covered label \(\nu\) take \(f_\nu=T_\lambda\) with \(\varphi(\lambda)=\nu\): by the support bound \(f_\nu\ast\varphi_0\) is supported on \(\{\nu'\le\nu\}\), and it is nonzero at \(t^\lambda x_0\in\mathcal O_\nu\) (Lemma A). For \(\nu=(a,a,a,a)\) with \(a=2k+1\) take \(f_\nu=\mathbf 1_{\varpi^kK}\ast f_\mu\), where \(f_\mu\) is the combination of step 4; then \(f_\nu\ast\varphi_0=\mathbf 1_{\varpi^kK}\ast\mathbf 1_{\mathcal O_\mu}=\mathbf 1_{\mathcal O_\nu}\), since \(\varpi^kI\cdot J'=\varpi^{2k}J'\). Proposition B gives (SG). Every count above is of configurations over \(\cO/\varpi^2\) or \(\F_q\) and does not depend on \(F\). This replaces the appeal to Corollary C in Proposition 1.4, step 1, which does not apply when uncovered labels lie below a covered one.

Check. For \(q=2\), an enumeration of all lattices \(t^3\cO^4\subseteq N\subseteq\cO^4\) (\(N=t^2M\)) over \(\F_2[\![t]\!]\) gives, at \(\varpi J\), the values \(15\), \(360\), \(960\) for \(\lambda=(1,1,0,0)\), \((2,1,0,-1)\), \((2,2,-1,-1)\) and \(0\) for every other \(\lambda\) in that range (including \((2,0,0,0)\) and \((1,1,1,-1)\)); at \(J''\) it gives \(1\) for \((1,1,0,0)\) and \(6=q(q+1)\) for \((2,0,0,0)\).

Verified by computation: gl4sp4_G1.py.