§1 · Agent 1 · Literature and Table 1 audit · 2026-10-18T18:43:41Z

Lemma 1.3 · \(\mathbb A^n\) under GL(n): hypotheses hold, every lattice sheaf is a summand

proved ★ key step

For \(\mathbb A^n\) the finite levels, connected stabilizers and rational orbits are verified; the simple objects are the constant sheaves on the \(I\)-stable lattices, each is (up to a nonzero graded multiplicity) \(\mathrm{IC}_{t^\lambda}\ast_I\mathrm{IC}_0\), and \(L^+X\) is not a single \(K\)-orbit, contrary to the hub's set-up.

Lemma 1.3.

Let \(X=\mathbb A^n\) under \(\mathrm{GL}_n\), \(I\) upper triangular mod \(t\), \(O_{m,j}\) and \(\Lambda_{m,j}=t^{\lambda_{m,j}}\mathcal O^n\), \(\lambda_{m,j}=(m^{(j)},(m+1)^{(n-j)})\), as on case-affine.html; \(\Lambda_{m,0}=\Lambda_{m+1,n}\). In all three settings:

  1. (I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III).) With \(Y_N=t^{-N}\mathcal O^n\), \(Y_{N,M}=Y_N/t^M\mathcal O^n\), the pieces of level \((N,M)\) are the \(O_{m,j}\) (\(-N\le m<M\)) and \(T_M=t^M\mathcal O^n\): \((N+M)n+1\) of them.
  2. (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.) All stabilizers of \(I/I_{N,M}\) on \(Y_{N,M}\) are connected. (Frob) Each \(O_{m,j}\) contains the rational point \(t^me_j\); over every \(F\) the \(I\)-orbits on \(F^n\) are the \(O_{m,j}(F)\) and \(\{0\}\).
  3. (\(\mathrm{IC}_0\)) \(L^+X=\mathcal O^n\) is closed and pro-smooth and \(\mathrm{IC}_0=\underline k_{\mathcal O^n}\), but \(L^+X\) is not one \(K\)-orbit: \(K\backslash\mathcal O^n=\{0\}\sqcup\{t^m(\mathcal O^n\setminus t\mathcal O^n):m\ge0\}\).
  4. The simple objects of the perverse heart of \(\mathcal C_I\) are, up to the normalizing shift, exactly the \(S_{m,j}:=\underline k_{\Lambda_{m,j}}\), \((m,j)\in\mathbb Z\times\{1,\dots,n\}\); \(\mathrm{IC}_0=S_{0,n}\) is simple, and the point \(0\) supports no simple object.
  5. For \(\lambda=\lambda_{m,j}\): \(\mathrm{IC}_{t^\lambda}\ast_I\mathrm{IC}_0\simeq R\Gamma(\overline{S}_\lambda,\mathrm{IC})\otimes S_{m,j}\), where \(\overline S_\lambda=\overline{It^\lambda I/I}\subseteq t^\lambda K/I\simeq G/B\) is a Schubert variety; this is nonzero. Hence \(M_I\) is the set of all simple objects.
  6. Functions: (IG) holds for every non-archimedean \(F\); over \(\mathbb F_q((t))\), \(\operatorname{tr}(\mathrm{IC}_{t^\lambda}\ast_I\mathrm{IC}_0)=\pm P_\lambda(q)\,\mathbf 1_{\Lambda_{m,j}}\) with \(P_\lambda(q)=\sum_i\dim IH^{2i}(\overline S_\lambda)\,q^i\ne0\).

Proof.

  1. (I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III).). case-affine.html, step 1; checked: for \(m<M\), \(O_{m,j}\) is a union of cosets of \(t^{m+1}\mathcal O^n\supseteq t^M\mathcal O^n\); every vector of valuation \(\ge M\) and \(0\) map to \(0\in Y_{N,M}\).
  2. (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.). \(I_{N,M}=1+t^{N+M}M_n(\mathcal O)\) acts trivially on \(Y_{N,M}\) since \(t^{N+M}Av\in t^M\mathcal O^n\) for \(v\in t^{-N}\mathcal O^n\). The stabilizer of \(t^me_j\bmod t^M\) is \(\{g\in I:g_{jj}\in1+t^{M-m}\mathcal O,\ g_{ij}\in t^{M-m}\mathcal O\ (i\ne j)\}/I_{N,M}\): the column \(j\) ranges over an affine space, the other columns over the Iwahori conditions, diagonal entries in \(\mathcal O^\times\) (connected) — a product of connected varieties, hence connected; the stabilizer of \(0\) is \(I/I_{N,M}\). (Frob): classical-affine.html, Lemma 1, whose proof (reduction to \(B(\mathbb F_q)\)-orbits on \(\mathbb F_q^n\setminus0\) and the congruence subgroup acting transitively along rays) is valid over every residue field.
  3. \(\mathcal O^n=\overline{X_0}\) with \(X_0\) the shell of primitive vectors; the \(K\)-orbits are the shells \(X_m\) and \(\{0\}\) (classical-affine.html, Lemma 1(a)). index.html ("Setting": "\(L^+X\) ... is the \(K\)-orbit of the base point") is therefore wrong for this row; nothing in the proofs uses it, only that \(\mathrm{IC}_0\) is the constant sheaf on the closed pro-smooth \(\mathcal O^n\), pulled back from \(Y_{0,M}\).
  4. At level \((N,M)\) with \(M>m\), the closure of the image of \(O_{m,j}\) is the linear subspace \(\Lambda_{m,j}/t^M\mathcal O^n\), smooth, so its IC sheaf is the shifted constant sheaf; by I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant. there are no other local systems. The closed piece \(T_M=\Lambda_{M,n}\) at level \(M\) has the same IC as \(O_{M,n}\) at level \(M+1\): the families agree along the transition functors (smooth pullback with the relative-dimension shift, closed pushforward), which are \(t\)-exact for the \(*\)-adapted normalization (chen-yi-v4.pdf, App. A.7, (45)–(47)). A simple object is the IC of a piece of some finite level, and \(\{0\}\) is never a piece (hypothesis.html, Levels). This is checklist item 1.
  5. \(I\subseteq t^\lambda Kt^{-\lambda}\): the \((i,k)\) entry of \(t^{-\lambda}gt^\lambda\) is \(t^{\lambda_k-\lambda_i}g_{ik}\), integral since \(\lambda_k-\lambda_i\in\{0,1\}\) for \(i<k\) and \(=-1\) only for \(i>j\ge k\), where \(g_{ik}\in t\mathcal O\) (classical-affine.html, Lemma 2). So \(It^\lambda I\subseteq t^\lambda K\), and \(\overline{It^\lambda I/I}\) lies in the closed \(t^\lambda K/I\simeq G/B\); it is a Schubert variety for the Borel subgroup image of \(t^{-\lambda}It^\lambda\) in \(G\). The map \([x,v]\mapsto(xI,xv)\) is an isomorphism \(\overline{It^\lambda I}\times^I\mathcal O^n\to\overline S_\lambda\times\Lambda_{m,j}\) (inverse \((xI,y)\mapsto[x,x^{-1}y]\), using \(x\mathcal O^n=\Lambda_{m,j}\)), under which \(m\) is the projection. Base change gives the formula; \(IH^*\) of a nonempty projective variety is nonzero. The same computation with \(\Delta_{t^\lambda}\) in place of \(\mathrm{IC}_{t^\lambda}\) is case-affine.html, step 3.
  6. classical-affine.html, Theorem 3 (stated for any non-archimedean \(F\), lattice indicators and \(T_{t^\lambda}\ast\varphi_0=q^{\ell(t^\lambda)}\mathbf 1_{\Lambda_{m,j}}\)). Trace: Laumon (1.1.1.3) on the fibres \(\overline S_\lambda\); \(IH^{\mathrm{odd}}(\overline S_\lambda)=0\) and Frobenius acts on \(IH^{2i}\) by \(q^i\) (Kazhdan–Lusztig 1980), so all terms have one sign and there is no cancellation.

Second-proof remark. Steps 4–5 give \(\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\) from any summand criterion with hypotheses I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III)., I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant., finite-length heart and DT (here DT is the explicit formula in 5, no decomposition theorem is needed). They do not use the I3I3 · Triangularity on pieces: for every piece some Hecke translate of IC_0 or of a lower standard object is supported on its closure with nonzero constant restriction (Iwahori-generation pages). dévissage of Proposition IV.1, but they share its one geometric input, \(I\subseteq t^\lambda Kt^{-\lambda}\) (F3 Lemma 2).

Builds on

  • Setup 1.1 · Setting audit: D-modules, Betti and ℓ-adic give the same \(\mathcal C_I\) and \(\ast_I\)

Built on by

  • Lemma 1.6 · Dictionary inputs at Iwahori level: rational orbits, Lang, trace bases, convolution
  • Audit 1.7 · Every row of Table 1 at Iwahori level: what is proved, over which field, by which argument
  • Objection 1.8 · Errors and unproved claims found on the Iwahori-generation pages
  • Summary 1.10 · Verdict of the audit: rows 1–3 agree, the rest reduce to one module computation
Original record, as the agent wrote it at 18:43 UTC

\(\mathbb A^n\) under GL(n): hypotheses hold, every lattice sheaf is a summand

Proof.

  1. (I1). case-affine.html, step 1; checked: for \(m<M\), \(O_{m,j}\) is a union of cosets of \(t^{m+1}\mathcal O^n\supseteq t^M\mathcal O^n\); every vector of valuation \(\ge M\) and \(0\) map to \(0\in Y_{N,M}\).
  2. (I2). \(I_{N,M}=1+t^{N+M}M_n(\mathcal O)\) acts trivially on \(Y_{N,M}\) since \(t^{N+M}Av\in t^M\mathcal O^n\) for \(v\in t^{-N}\mathcal O^n\). The stabilizer of \(t^me_j\bmod t^M\) is \(\{g\in I:g_{jj}\in1+t^{M-m}\mathcal O,\ g_{ij}\in t^{M-m}\mathcal O\ (i\ne j)\}/I_{N,M}\): the column \(j\) ranges over an affine space, the other columns over the Iwahori conditions, diagonal entries in \(\mathcal O^\times\) (connected) — a product of connected varieties, hence connected; the stabilizer of \(0\) is \(I/I_{N,M}\). (Frob): classical-affine.html, Lemma 1, whose proof (reduction to \(B(\mathbb F_q)\)-orbits on \(\mathbb F_q^n\setminus0\) and the congruence subgroup acting transitively along rays) is valid over every residue field.
  3. \(\mathcal O^n=\overline{X_0}\) with \(X_0\) the shell of primitive vectors; the \(K\)-orbits are the shells \(X_m\) and \(\{0\}\) (classical-affine.html, Lemma 1(a)). index.html ("Setting": "\(L^+X\) ... is the \(K\)-orbit of the base point") is therefore wrong for this row; nothing in the proofs uses it, only that \(\mathrm{IC}_0\) is the constant sheaf on the closed pro-smooth \(\mathcal O^n\), pulled back from \(Y_{0,M}\).
  4. At level \((N,M)\) with \(M>m\), the closure of the image of \(O_{m,j}\) is the linear subspace \(\Lambda_{m,j}/t^M\mathcal O^n\), smooth, so its IC sheaf is the shifted constant sheaf; by I2 there are no other local systems. The closed piece \(T_M=\Lambda_{M,n}\) at level \(M\) has the same IC as \(O_{M,n}\) at level \(M+1\): the families agree along the transition functors (smooth pullback with the relative-dimension shift, closed pushforward), which are \(t\)-exact for the \(*\)-adapted normalization (chen-yi-v4.pdf, App. A.7, (45)–(47)). A simple object is the IC of a piece of some finite level, and \(\{0\}\) is never a piece (hypothesis.html, Levels). This is checklist item 1.
  5. \(I\subseteq t^\lambda Kt^{-\lambda}\): the \((i,k)\) entry of \(t^{-\lambda}gt^\lambda\) is \(t^{\lambda_k-\lambda_i}g_{ik}\), integral since \(\lambda_k-\lambda_i\in\{0,1\}\) for \(i<k\) and \(=-1\) only for \(i>j\ge k\), where \(g_{ik}\in t\mathcal O\) (classical-affine.html, Lemma 2). So \(It^\lambda I\subseteq t^\lambda K\), and \(\overline{It^\lambda I/I}\) lies in the closed \(t^\lambda K/I\simeq G/B\); it is a Schubert variety for the Borel subgroup image of \(t^{-\lambda}It^\lambda\) in \(G\). The map \([x,v]\mapsto(xI,xv)\) is an isomorphism \(\overline{It^\lambda I}\times^I\mathcal O^n\to\overline S_\lambda\times\Lambda_{m,j}\) (inverse \((xI,y)\mapsto[x,x^{-1}y]\), using \(x\mathcal O^n=\Lambda_{m,j}\)), under which \(m\) is the projection. Base change gives the formula; \(IH^*\) of a nonempty projective variety is nonzero. The same computation with \(\Delta_{t^\lambda}\) in place of \(\mathrm{IC}_{t^\lambda}\) is case-affine.html, step 3.
  6. classical-affine.html, Theorem 3 (stated for any non-archimedean \(F\), lattice indicators and \(T_{t^\lambda}\ast\varphi_0=q^{\ell(t^\lambda)}\mathbf 1_{\Lambda_{m,j}}\)). Trace: Laumon (1.1.1.3) on the fibres \(\overline S_\lambda\); \(IH^{\mathrm{odd}}(\overline S_\lambda)=0\) and Frobenius acts on \(IH^{2i}\) by \(q^i\) (Kazhdan–Lusztig 1980), so all terms have one sign and there is no cancellation.

Second-proof remark. Steps 4–5 give \(\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\) from any summand criterion with hypotheses I1, I2, finite-length heart and DT (here DT is the explicit formula in 5, no decomposition theorem is needed). They do not use the I3 dévissage of Proposition IV.1, but they share its one geometric input, \(I\subseteq t^\lambda Kt^{-\lambda}\) (F3 Lemma 2).

Sources