Audit 1.7 · Every row of Table 1 at Iwahori level: what is proved, over which field, by which argument

Every row of Table 1 at Iwahori level: what is proved, over which field, by which argument

The deciding statement for every open row (over \(\overline{\mathbb F}_q\)). Let \(M\) be Chen–Yi's affine Lusztig–Vogan module (chen-yi-v4.pdf, Theorem 1, §8.2.1): free over \(\mathbb Z[q^{\pm1/2}]\) on \(\mathcal D=\{(\xi,v)\}\), with \(\mathcal H\) acting through the classes of \(\mathcal C_I\), and IC basis \(c_{\xi,v}=[\mathrm{IC}_{\xi,v}]\) (Theorems 3, 82). Put \(m_0=[\mathrm{IC}_0]\) and \(C'_w=[\mathrm{IC}_w]\).

1. Sheaves. By Proposition 70(ii) and Theorem 81, \(\mathrm{IC}_w\ast_I\mathrm{IC}_0\simeq\bigoplus_{(\xi,v),i}\mathrm{IC}_{\xi,v}[-i]\otimes E_{\xi,v,i}\) with \(E_{\xi,v,i}\) pure of a weight that is an affine function of \(i\); hence in \(M\) (coefficients in \(\mathbb Z[A]\), \(A=\overline{\mathbb Q}_\ell^\times/\mu\), graded by weight) the coefficient of \(c_{\xi,v}\) in \(C'_w\,m_0\) is \(\sum_i(-1)^i[E_{\xi,v,i}]\) with distinct \(i\) in distinct weights: it vanishes iff all \(E_{\xi,v,i}=0\). So

and, given the summand criterion, S \(\iff M_I=\mathcal D\).

2. Functions over \(\mathbb F_q((t))\), where (D3)–(D4) of Lemma 1.6 hold. Traces identify \(M\otimes_{q^{1/2}\mapsto\sqrt q}\overline{\mathbb Q}_\ell\) with \(C_I\otimes\overline{\mathbb Q}_\ell\), \(m_0\mapsto\pm\varphi_0\), \(C'_w\mapsto\operatorname{tr}\mathrm{IC}_w\), so (IG) \(\iff\) the specializations of \(\{C'_w\,m_0\}_w\) span \(M\otimes\overline{\mathbb Q}_\ell\).
3. So both levels are statements about one module and one cyclic vector: S is a support statement, (IG) a span statement after specialization. Span \(\Rightarrow\) support (a basis vector in the span must occur in some \(C'_wm_0\)): this is the bridge. Support \(\not\Rightarrow\) span in general (two basis vectors occurring only through \(c_1+c_2\)).
4. Row data. \(\mathrm{GL}_2/T\): the \(I\)-orbits and formula (29) are explicit (Mars–Springer types I–IV, Chen–Yi Lemma 28), so \(M_I\) is a finite computation per level; it has not been done. For the other symmetric rows the same computation is needed, plus Frobenius-stability (D6) to use the bridge, plus a transfer of the characteristic-\(0\) witnesses to \(\mathbb F_q((t))\) (an isomorphism of \(\mathcal H_I\)-modules \(C_c^\infty(X(F))^I\simeq C_c^\infty(X(F'))^I\), \(\varphi_0\mapsto\varphi_0\), for \(F\), \(F'\) with the same \(q\); Shtotland's presentation, arXiv:2406.16070, would give it if it is uniform in \(F\) — not checked).

Sources of Table A. summary.html §2; index.html §4; classical.html (Prop. B–G, Table 2); classical-families.html (Prop. B, Thm C, Prop. D); case-*.html; group.html. The (IG) failures for \(\mathrm{GL}_{2p}/\mathrm{GL}_p^2\) and \(\mathrm{GL}_3/\mathrm{GL}_1\times\mathrm{GL}_2\) use Shtotland's Theorem 1.5 (\(\operatorname{char}F=0\) by summary.html, Table 2) or Matringe ("\(G=\mathrm{GL}_n\) over a \(p\)-adic field"); only \(\mathrm{GL}_2/T\) (classical F: an explicit \(T\)-invariant functional on \(\mathrm{St}=C^\infty(\mathbb P^1)/\mathbb C\), Frobenius reciprocity, \(\mathrm{St}^I\ne0=\mathrm{St}^K\)) is characteristic-free.