Proposition II as a paper · statement and complete proof (PDF) →

Definitions

• Group data over a finite field

  \(p\) is an odd prime and \(q\) a power of \(p\). \(G\) is a connected reductive group over \(\mathbb F_q\) and \(\theta\) an involution of \(G\) over \(\mathbb F_q\):

  $$X=G/G^\theta,\qquad x_0=eG^\theta .$$

  \(S\) is a maximal \(\theta\)-split torus and \(T\supseteq S\) a \(\theta\)-stable maximal torus. Labels and the leading-term map are those of the pages:

  $$\Lambda_T^+\subseteq X_*(T),\qquad\Lambda_S^+=X_*(S)\cap\Lambda_T^+,\qquad\varphi(\lambda)=\lambda-\theta(\lambda),$$

  with the dominance order on \(\Lambda_S^+\), in which lower sets \(\{\nu\le\mu\}\) are finite.

• Sheaves over \(k=\overline{\mathbb F}_q\)

  \(LX\) and \(L^+G\) are the loop space and the arc group over \(k\); the orbits are \(\mathcal O_\mu=L^+G\cdot x_\mu\) with \(\tau(x_\mu)=t^\mu\). The category is Chen–Yi's \(\ell\)-adic one:

  $$\mathcal C_{K,k}=\{\text{bounded }L^+G\text{-equivariant constructible }\overline{\mathbb Q}_\ell\text{-complexes on }LX\text{, supported on finitely many orbit closures}\}.$$

  \(\mathrm{IC}_\nu\) is the intersection complex of \(\overline{\mathcal O_\nu}\) with constant coefficients, \(V_\lambda\) the Satake sheaf of \(\operatorname{Gr}_{\le\lambda}\), and \(\mathrm{IC}_0\) the constant sheaf on \(L^+X\):

  $$\mathrm{IC}_\nu|_{\mathcal O_\nu}\simeq\overline{\mathbb Q}_\ell[\delta(\nu)],\qquad\mathcal C^{\mathrm{Sat}}_{K,k}=\operatorname{Thick}_{\mathcal C_{K,k}}\{V_\lambda\ast\mathrm{IC}_0:\lambda\in\Lambda_T^+\}.$$

  The Betti category \(\mathcal C_K\) of the pages is the same construction over \(\mathbb C\) with complex coefficients.

• Functions over \(F=\mathbb F_q(\!(t)\!)\)

  \(F=\mathbb F_q(\!(t)\!)\) has ring of integers \(\mathcal O=\mathbb F_q[\![t]\!]\). Its points are the \(\mathbb F_q\)-points of the loop spaces:

  $$LX(\mathbb F_q)=X(F),\qquad LG(\mathbb F_q)=G(F),\qquad L^+G(\mathbb F_q)=K=G(\mathcal O).$$

  The spherical Hecke algebra \(\mathcal H_K\) consists of the compactly supported \(K\)-bi-invariant functions on \(G(F)\); it has the basis \(T_\lambda\), \(\lambda\in\Lambda_T^+\) (Cartan decomposition), and acts by convolution on the \(K\)-invariant functions on \(X(F)\), with Haar measure normalized by \(\operatorname{vol}K=1\):

  $$\mathcal H_K=C_c(K\backslash G(F)/K),\qquad T_\lambda=\mathbf 1_{Kt^\lambda K},\qquad(f\ast\varphi)(x)=\sum_{gK\in G(F)/K}f(g)\,\varphi(g^{-1}x),$$ $$C_K=C_c^\infty(X(F))^K,\qquad\varphi_0=\mathbf 1_{X(\mathcal O)}.$$

  Spherical generation.

  $$\text{(SG)}\qquad\mathcal H_K\ast\varphi_0=C_K.$$

  For a field \(E\) of characteristic \(0\), \(\mathcal H_K\ast\varphi_0\otimes E\) is the \(E\)-span of the \(\mathbb Z\)-valued functions \(T_\lambda\ast\varphi_0\) and \(C_K\otimes E\) the \(K\)-invariant compactly supported \(E\)-valued functions; (SG) does not depend on \(E\) (IIb, step 7).

  Other residue fields. Proposition II needs only \(F\). Since \(\overline{\mathbb F}_{q^n}=\overline{\mathbb F}_q\), replacing \(q\) by \(q^n\) leaves \(\mathcal C_{K,k}\) and its objects unchanged and gives the same statements over \(F_n=\mathbb F_{q^n}(\!(t)\!)\), with \(K_n=G(\mathbb F_{q^n}[\![t]\!])\), \(\mathcal H_{K_n}\), \(C_{K_n}\), \((\mathrm{SG})_n\), the trace \(\operatorname{tr}_n\) of the \(n\)-th iterate of \(\Phi\) and the matrix \(m^{(n)}\) (trace dictionary). Several \(n\) at once are used only for the converse (II, limits; Proposition IX).

• Weil objects and trace functions

  A Weil object is a pair \((M,\Phi)\) with \(M\in\mathcal C_{K,k}\) and \(\Phi:\mathrm{Fr}^*M\xrightarrow{\sim}M\), where \(\mathrm{Fr}\) is the \(\mathbb F_q\)-linear Frobenius endomorphism of \(LX\) (coordinates raised to the \(q\)-th power), so that \(LX(\mathbb F_q)=X(F)\) is its set of fixed points. For \(x\in X(F)\) and a geometric point \(\bar x\) over it, \(\mathrm{Fr}(\bar x)=\bar x\), so \(\Phi_{\bar x}\) acts on each stalk cohomology \(\mathcal H^i(M)_{\bar x}\), and the trace function of \(M\) is

  $$\operatorname{tr}(M)(x)=\sum_i(-1)^i\operatorname{tr}\big(\Phi_{\bar x},\ \mathcal H^i(M)_{\bar x}\big)\in\overline{\mathbb Q}_\ell .$$

  The function \(\operatorname{tr}(M)\) is \(K\)-invariant and lies in \(C_K\otimes\overline{\mathbb Q}_\ell\) (IIb, steps 1–2); it is additive on Frobenius-compatible distinguished triangles, and \(\operatorname{tr}(M[1])=-\operatorname{tr}(M)\). \(V_\lambda\), \(\mathrm{IC}_0\) and \(\mathrm{IC}_\nu\) carry canonical Weil structures (eigenvalue \(1\) on the lowest stalk cohomology of the open stratum, no Tate twist), and \(V_\lambda\ast\mathrm{IC}_0\) the induced one. The trace dictionary: properties, conventions and sources →

• Multiplicity spaces and the matrix \(m\)

  By D2 each Satake object is a finite sum of shifted intersection complexes. With \(P_i={}^p\mathcal H^i(V_\lambda\ast\mathrm{IC}_0)\), the \(i\)-th perverse cohomology sheaf, the multiplicity space \(E_{\nu,i}(\lambda)\) counts the copies of \(\mathrm{IC}_\nu[-i]\):

  $$V_\lambda\ast\mathrm{IC}_0\simeq\bigoplus_{\nu,i}\mathrm{IC}_\nu[-i]\otimes E_{\nu,i}(\lambda),\qquad E_{\nu,i}(\lambda)=\operatorname{Hom}\big(\mathrm{IC}_\nu,P_i\big).$$

  The Weil structure induces the Frobenius \(A_{\nu,i}(\lambda)\in\mathrm{GL}(E_{\nu,i}(\lambda))\) (IIb, step 4). The matrix has rows indexed by orbit labels and columns by Hecke labels:

  $$m_{\nu\lambda}=\sum_i(-1)^i\operatorname{tr}A_{\nu,i}(\lambda)\qquad(\nu\in\Lambda_S^+,\ \lambda\in\Lambda_T^+).$$

  Each column \((m_{\nu\lambda})_\nu\) has finitely many nonzero entries. "Every row is nonzero" and "the columns span \(\bigoplus_{\Lambda_S^+}\overline{\mathbb Q}_\ell\)" are the two conditions compared on these pages.

• Proposition I (summand criterion)

  Assume H1–H3 and that every \(V_\lambda\ast\mathrm{IC}_0\) is a finite sum of shifted \(\mathrm{IC}_\nu\) with constant coefficients. Put

  $$\mathcal M=\{\mu\in\Lambda_S^+:\mathrm{IC}_\mu[s]\text{ is a direct summand of some }V_\lambda\ast\mathrm{IC}_0,\ s\in\mathbb Z\}.$$

  Then

  $$\mathcal C_K^{\mathrm{Sat}}=\operatorname{Thick}\{\mathrm{IC}_\nu:\nu\in\mathcal M\},\qquad\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\iff\mathcal M=\Lambda_S^+ .$$

  So \(\mathcal M\) is the set of labels whose intersection complex occurs, up to shift, as a direct summand of a Satake object, and the sheaf statement \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\) is the statement \(\mathcal M=\Lambda_S^+\). For \(\mathcal C_{K,k}\): H1 follows from D0, H2 from D1, H3 from the connectedness hypothesis. Proposition I · proof →

Hypotheses (R), (C) and inputs D0–D3, D5

• (R) · Rationality.

  \(G\), \(\theta\) and \(X\) are defined over \(\mathbb F_q\), \(q\) odd; \(T\supseteq S\) is split over \(\mathbb F_q\); and \(G\) has a \(\theta\)-stable Borel pair \((B_0,T_0)\) over \(\mathbb F_q\) with \(T_0\) split (Chen–Yi, §8.2). Then every \(t^\mu\), \(\mu\in\Lambda_S\), lies in \(S(F)\).
• (C) · Connectedness.

  $$G^\theta\ \text{ and }\ Z_{G^\theta}(S)\ \text{ are connected}.$$
• D0 · Orbit geometry over \(k\).

  For \(p\ne2\) (Chen–Yi, Prop. 17, Thm 18 and the pages' Proposition III): $$LX(k)=\bigsqcup_{\mu\in\Lambda_S^+}\mathcal O_\mu(k),\quad\tau(x_\mu)=t^\mu,\qquad\overline{\mathcal O_\mu}=\bigsqcup_{\nu\le\mu}\mathcal O_\nu,\qquad\operatorname{supp}(V_\lambda\ast\mathrm{IC}_0)\subseteq\overline{\mathcal O_{\varphi(\lambda)}},$$ $$\operatorname{Stab}_{L^+G}(x_\mu)=Z_{G^\theta}(\mu)\ltimes U_\mu,\ \ U_\mu\ \text{pro-unipotent},\qquad\pi_0Z_{G^\theta}(\mu)=1\ \text{ for all }\mu\ \text{ under (C)}.$$
• D1 · Traces and t-structure.

  Proper base change and the Grothendieck–Lefschetz trace formula on the fibres of the convolution map; Chen–Yi's sheaf theory on \(L^+G\backslash LX\) (App. A.5) with a Frobenius-compatible perverse t-structure of finite length whose simple objects are the \(\mathrm{IC}_\nu\), \(\operatorname{End}(\mathrm{IC}_\nu)=\overline{\mathbb Q}_\ell\).
• D2 · Decomposition.

  \(V_\lambda\ast\mathrm{IC}_0\) is pure, and over \(k\) $$V_\lambda\ast\mathrm{IC}_0\simeq\bigoplus_{\nu,i}\mathrm{IC}_\nu[-i]\otimes E_{\nu,i}(\lambda),\qquad\dim E_{\nu,i}(\lambda)\lt\infty$$ (Sun, Thm 1.2, on finite-type equivariant models; Chen–Yi, Prop. 70 and the proof of Thm 93).
• D3 · Lang's theorem.

  For \(L^+G=\varprojlim_mG(\mathbb F_q[t]/t^m)\) and its stabilizers: $$H^1(\mathbb F_q,L^+G)=1,\qquad H^1\big(\mathbb F_q,\operatorname{Stab}_{L^+G}(x_\mu)\big)=H^1\big(\mathbb F_q,\pi_0\operatorname{Stab}_{L^+G}(x_\mu)\big).$$
• D5 · Comparison with \(\mathbb C\).

  The statement "\(\mathrm{IC}_\mu\) is a summand of \(V_\lambda\ast\mathrm{IC}_0\)", which lives on the finite-type model \(\operatorname{Gr}_{\le\lambda}\mathbin{\widetilde\times}\overline{\mathcal O_0}\to\overline{\mathcal O_{\varphi(\lambda)}}\), has the same answer over \(\overline{\mathbb F}_q\) and over \(\mathbb C\) (BBD §6.1; Bezrukavnikov–Finkelberg, Prop. 5, for \(p\) outside a finite set). Used only for the Betti form of the conclusion.

Proof outline

Step 0 fixes the field on which the levels can be compared. Steps 1–3 are the three parts, each proved on its own page; step 4 reads off the consequence. Two limits close the page.

0. Purpose Compare the levels where a functor links them

   The sheaf statement of the pages is Betti over \(\mathbb C\), Theorem E is \(p\)-adic, and only over \(\mathbb F_q(\!(t)\!)\) does a functor relate sheaves and functions:

   $$\operatorname{tr}:\{(M,\Phi)\}\longrightarrow C_K\otimes\overline{\mathbb Q}_\ell .$$
   Proof

   • Sheaves. The inputs that decide \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\) (Chen–Yi) are \(\ell\)-adic in positive characteristic; Chen–Yi only assert that their proofs carry over to mixed Hodge modules (§1.2.6). D5 is the bridge to \(\mathbb C\).
   • Functions under (A). Theorem E assumes a local field of characteristic zero (Sakellaridis, §1.1).
   • Trace. A Frobenius trace needs a finite residue field and Weil structures; it sends Weil objects of \(\mathcal C_{K,k}\) to functions on \(X(\mathbb F_q(\!(t)\!))\).
   • Output. A cell of the table read across Betti sheaves and \(p\)-adic functions compares two statements with no map between them. The comparison is made over \(\mathbb F_q(\!(t)\!)\) against \(\mathcal C_{K,k}\).
1. Method (a) Take traces of Satake objects

   The trace of \(V_\lambda\ast\mathrm{IC}_0\) is the Hecke translate of \(\varphi_0\) by \(\operatorname{tr}V_\lambda\), and the \(\operatorname{tr}V_\lambda\) are unitriangular in the basis \(T_\mu\):

   $$\operatorname{tr}(V_\lambda\ast\mathrm{IC}_0)=\operatorname{tr}(V_\lambda)\ast\varphi_0,\qquad\operatorname{tr}V_\lambda=(-1)^{\langle2\rho,\lambda\rangle}T_\lambda+\sum_{\mu\lt\lambda}a_{\lambda\mu}T_\mu .$$
   Outline

   • Points. Lang's theorem (D3) identifies the \(\mathbb F_q\)-points of \(\operatorname{Gr}\), \(L^+X\) and the convolution space with \(G(F)/K\), \(X(\mathcal O)\) and \(G(F)\times^KX(\mathcal O)\).
   • Trace formula. Proper base change and Grothendieck–Lefschetz (D1) on the projective fibres of the convolution map give the convolution formula, as in Zhu's Lemma 5.6.1 with \(LX\) in place of \(\operatorname{Gr}\).
   • Basis. Open-cell value and finite lower sets. Full proof: IIa →
2. Method (b) Expand traces in the IC basis

   Orbits, then IC traces, are bases of \(C_K\otimes\overline{\mathbb Q}_\ell\); the coordinates of a Satake trace are the alternating traces \(m_{\nu\lambda}\), and (SG) is the spanning of the columns:

   $$\operatorname{tr}\mathrm{IC}_\nu=(-1)^{\delta(\nu)}\mathbf 1_{\mathcal O_\nu(\mathbb F_q)}+\sum_{\nu'\lt\nu}c_{\nu'\nu}\mathbf 1_{\mathcal O_{\nu'}(\mathbb F_q)},\qquad\operatorname{tr}(V_\lambda\ast\mathrm{IC}_0)=\sum_\nu m_{\nu\lambda}\operatorname{tr}\mathrm{IC}_\nu .$$
   Outline

   • Orbits. Lang's theorem and (C): one compact open \(K\)-orbit of rational points per label.
   • Frobenius on multiplicities. Frobenius acts on each multiplicity space \(E_{\nu,i}(\lambda)\) by \(A_{\nu,i}(\lambda)\); additivity gives \(m\).
   • Columns. Coordinates in the IC basis carry the span of the Satake traces onto the span of the columns, so one is everything exactly when the other is; (SG) does not depend on the coefficient field. Full proof: IIb →
3. Result (c) Functions generate ⇒ sheaves generate

   (SG) makes every row of \(m\) nonzero, a nonzero entry \(m_{\mu\lambda}\) makes \(\mathrm{IC}_\mu\) a summand (\(\mu\in\mathcal M\)), and Proposition I turns \(\mathcal M=\Lambda_S^+\) into thick generation:

   $$(\mathrm{SG})\ \Longrightarrow\ \forall\mu\ \exists\lambda:\ m_{\mu\lambda}\ne0\ \Longrightarrow\ \mathcal M=\Lambda_S^+\ \Longrightarrow\ \mathcal C^{\mathrm{Sat}}_{K,k}=\mathcal C_{K,k} .$$
   Outline

   • Rows. A zero row would make the coordinate functional of \(\operatorname{tr}\mathrm{IC}_\mu\) vanish on the span of the columns.
   • Summands. If every \(E_{\mu,i}(\lambda)\) vanished, so would \(m_{\mu\lambda}\). Method · Proposition I: summands generate
   • Over \(\mathbb C\). D5 transfers the summand statements. Full proof: IIc →
4. Conclusion No reverse divergence

   Over \(\mathbb F_q(\!(t)\!)\), functions holding and sheaves failing never happens. For \(p\)-adic \(F\) the table excludes it row by row.

   Proof

   • Over \(\mathbb F_q(\!(t)\!)\). Part (c), under (R), (C) and D0–D3.
   • Other fields. Proposition II does not apply. In every row of the verdict's table where SG holds, \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\) is proved: the spherical-generation pages (group case, spheres, \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\), rank one) and Proposition IV (symplectic rows). The rows \(\mathrm{SL}_2/\mathrm{SO}_2\) (Proposition III), \(\mathrm{GL}_n/\mathrm O_n\) and \(\mathrm{PGL}_2/\mathrm{PO}_2\) fail at both levels, and at \(E_6/F_4\) (Proposition VIII) the sheaf statement holds.
   • What remains. The opposite divergence, sheaves holding and functions failing, is not excluded here; it occurs at \(E_6/F_4\), across fields.

What the argument does not give

1. Result The converse needs spanning columns

   Here \(n\) varies: \(m^{(n)}\) is the matrix over \(\mathbb F_{q^n}(\!(t)\!)\) (footnote 1), with \(m^{(n)}_{\nu\lambda}=\sum_i(-1)^i\operatorname{tr}A_{\nu,i}(\lambda)^n\) and \(m^{(1)}=m\). Sheaf generation gives a nonzero entry in every row, each at its own \(n\); (SG) asks that the columns of \(m\) span. The converse fails (Proposition IX).

   $$\mathcal C^{\mathrm{Sat}}_{K,k}=\mathcal C_{K,k}\ \Longrightarrow\ \forall\mu\ \exists\lambda,\ n\le\dim\textstyle\bigoplus_iE_{\mu,i}(\lambda):\ \ m^{(n)}_{\mu\lambda}\ne0 .$$
   Proof

   • Summand. Proposition I (\(\Rightarrow\)) gives \(E_{\mu,i}(\lambda)\ne0\) for some \(\lambda,i\).
   • No cancellation across \(n\). By purity (D2) the eigenvalues of \(A_{\mu,i}(\lambda)\) have absolute value \(q^{(w+i-w_\mu)/2}\), distinct for distinct \(i\). So \(m^{(n)}_{\mu\lambda}=\sum_\alpha c_\alpha\alpha^n\) with distinct \(\alpha\) and all \(c_\alpha\ne0\), and a Vandermonde determinant gives the bound on \(n\). Equivalently, over \(\mathbb F_{q^n}(\!(t)\!)\) the trace of \(V_\lambda\ast\mathrm{IC}_0\) is not in the span of the traces of \(\mathrm{IC}_\nu\), \(\nu\ne\mu\).
   • The gap. SG over \(\mathbb F_q(\!(t)\!)\) says that the columns of \(m=m^{(1)}\) span. Nonzero rows, even at \(n=1\), do not give that; trace functions over all \(\mathbb F_{q^n}\) together determine a class in the Grothendieck group, and one \(n\) does not (Laumon, Theorem 1.1.2). Thick generation is closed under retracts, SG is a statement about linear spans.
   • Status. \(\mathcal C^{\mathrm{Sat}}_{K,k}=\mathcal C_{K,k}\) does not imply SG, even under (R) and (C): at \(E_6/F_4\) sheaves generate over \(\overline{\mathbb F}_q\), while over \(\mathbb F_q(\!(t)\!)\) the columns do not span, since \(C_K\) needs two generators (Proposition VII, Proposition IX).
2. Result Theorem E does not reach F_q((t))

   Theorem E assumes characteristic \(0\), so no cell "SG under (A)" can feed Proposition II; only function results proved over \(\mathbb F_q(\!(t)\!)\) can.

   Proof

   • Hypothesis. Sakellaridis works over the integers of a local non-archimedean field of characteristic zero (§1.1), which is A1.
   • Bridge. The computations of the restriction map \(r\) do not depend on the field; Theorem 8.0.2 itself reaches \(\mathbb F_q(\!(t)\!)\) for symmetric \(X\) by transfer from ramified \(p\)-adic fields, and holds there for \(E_6/F_4\) (Proposition VI).
   • What does feed it. The unconditional function results (Theorem 4 of the run, the classical page's Propositions D, G, J) hold over every non-archimedean \(F\), including \(\mathbb F_q(\!(t)\!)\).