Proposition IIc and proof

Part (c) of Proposition II: spherical generation over \(\mathbb F_q(\!(t)\!)\) makes every intersection complex a summand of a Satake object, hence thick generation.

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.

Proposition IIc · functions generate ⇒ sheaves generate

Setting. \(X=G/G^\theta\) over \(\mathbb F_q\), \(q\) odd, with (R) and (C); sheaves in \(\mathcal C_{K,k}\) over \(k=\overline{\mathbb F}_q\), with Satake objects \(V_\lambda\ast\mathrm{IC}_0\) and intersection complexes \(\mathrm{IC}_\nu\). The functions live on \(X(F)\), \(F=\mathbb F_q(\!(t)\!)\):1

  • \(K=G(\mathbb F_q[\![t]\!])\), the stabilizer of the base point of \(\operatorname{Gr}\); \(LX(\mathbb F_q)=X(F)\).
  • \(\mathcal H_K=C_c(K\backslash G(F)/K)\), the spherical Hecke algebra, with basis \(T_\lambda=\mathbf 1_{Kt^\lambda K}\), \(\lambda\in\Lambda_T^+\), acting by convolution.
  • \(C_K=C_c^\infty(X(F))^K\), the basic function \(\varphi_0=\mathbf 1_{X(\mathbb F_q[\![t]\!])}\), and (SG): \(\mathcal H_K\ast\varphi_0=C_K\).
  • \(\operatorname{tr}\), the alternating trace of Frobenius on stalks: a Weil object \(M\) of \(\mathcal C_{K,k}\) gives \(\operatorname{tr}(M):X(F)\to\overline{\mathbb Q}_\ell\).

1 Only \(F=\mathbb F_q(\!(t)\!)\) is needed, for an arbitrary odd \(q\). Replacing \(q\) by \(q^n\) gives the same statements over \(\mathbb F_{q^n}(\!(t)\!)\) with the same sheaves (Definitions, other residue fields).

The set \(\mathcal M\). The labels whose intersection complex occurs, up to shift, as a direct summand of a Satake object:

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

By Proposition I, \(\mathcal C^{\mathrm{Sat}}_{K,k}=\operatorname{Thick}\{\mathrm{IC}_\nu:\nu\in\mathcal M\}\). So sheaf generation, \(\mathcal C^{\mathrm{Sat}}_{K,k}=\mathcal C_{K,k}\), is the statement \(\mathcal M=\Lambda_S^+\).

Claim. Assume D0–D3.

  1. Over \(\overline{\mathbb F}_q\). If (SG) holds over \(F=\mathbb F_q(\!(t)\!)\), then \(\mathcal M=\Lambda_S^+\), and therefore $$\boxed{\mathcal C^{\mathrm{Sat}}_{K,k}=\mathcal C_{K,k} .}$$
  2. Over \(\mathbb C\). If moreover D5 holds (\(p\) outside a finite set), the same holds for Betti sheaves: \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\).

The proof is one chain:

$$(\mathrm{SG})\ \xRightarrow{\ \text{step 2}\ }\ \text{every row of }m\text{ is nonzero}\ \xRightarrow{\ \text{step 3}\ }\ \mathcal M=\Lambda_S^+\ \xRightarrow{\ \text{step 4}\ }\ \mathcal C^{\mathrm{Sat}}_{K,k}=\mathcal C_{K,k} .$$

Uses parts (a) and (b) from IIa and IIb, and Proposition I. Step 6 is a pointwise refinement, not used for (a) or (b).

Sources

References

References checked 17 October 2026. The reader in D holds located extracts.

Proof

Step 1 collects the inputs, steps 2–4 are the chain above, step 5 transfers it to \(\mathbb C\), step 6 refines it to one orbit.

  1. The IC traces are a basis, the coordinates of a Satake trace are the column \((m_{\nu\lambda})_\nu\), and (SG) is the spanning of the columns:

    $$C_K\otimes\overline{\mathbb Q}_\ell\cong\bigoplus_{\Lambda_S^+}\overline{\mathbb Q}_\ell,\qquad\operatorname{tr}(V_\lambda\ast\mathrm{IC}_0)\mapsto\big(m_{\nu\lambda}\big)_\nu,\qquad m_{\nu\lambda}=\sum_i(-1)^i\operatorname{tr}A_{\nu,i}(\lambda).$$
    Proof

  2. Let \(\pi_\mu\) be the coordinate functional of \(\operatorname{tr}\mathrm{IC}_\mu\). A zero row would make \(\pi_\mu\) vanish on the span of the columns:

    $$(\mathrm{SG})\ \Longrightarrow\ \forall\mu\ \exists\lambda:\ \ m_{\mu\lambda}=\pi_\mu\big(\operatorname{tr}(V_\lambda\ast\mathrm{IC}_0)\big)\ne0 .$$
    Proof
    • Span. Under (SG) the columns span \(\bigoplus_{\Lambda_S^+}\overline{\mathbb Q}_\ell\) (IIb (iii), direction ⇒).
    • Functional. \(\pi_\mu(\operatorname{tr}\mathrm{IC}_\mu)=1\), so \(\pi_\mu\ne0\) on that span; it is linear, so it is nonzero on some column \(\lambda\).
    • Weaker hypothesis. Steps 3–5 use only "every row nonzero", which is weaker than (SG) (IIb, step 6).

  3. A nonzero coefficient needs a nonzero multiplicity space, and D2 turns that into a direct summand:

    $$m_{\mu\lambda}\ne0\ \Longrightarrow\ \exists i:\ E_{\mu,i}(\lambda)\ne0\ \Longrightarrow\ \mathrm{IC}_\mu[-i]\ \text{is a direct summand of}\ V_\lambda\ast\mathrm{IC}_0\ \Longrightarrow\ \mu\in\mathcal M .$$
    Proof
    • Vanishing. If every \(E_{\mu,i}(\lambda)\) were zero, \(m_{\mu\lambda}\) would be zero (IIb, step 5).
    • Summand. \(E_{\mu,i}(\lambda)\ne0\) means \(\mathrm{IC}_\mu[-i]\otimes E_{\mu,i}(\lambda)\) is a nonzero summand in D2.
    • All rows. With step 2, every \(\mu\) is in \(\mathcal M\): \(\mathcal M=\Lambda_S^+\).
    • What is used. Only "zero multiplicity space, zero coefficient": no positivity, no purity beyond D2.

  4. Proposition I turns \(\mathcal M=\Lambda_S^+\) into thick generation:

    $$\mathcal M=\Lambda_S^+\quad\Longrightarrow\quad\mathcal C^{\mathrm{Sat}}_{K,k}=\operatorname{Thick}\{\mathrm{IC}_\nu:\nu\in\Lambda_S^+\}=\mathcal C_{K,k} .$$
    Proof
    • Hypotheses. H1–H3 hold for \(\mathcal C_{K,k}\) by D0, D1 and (C), and D2 is the decomposition Proposition I assumes.
    • Last equality. The \(\mathrm{IC}_\nu\), \(\nu\in\Lambda_S^+\), are the simple perverse objects, which generate \(\mathcal C_{K,k}\) as a thick subcategory. Proposition I · proof →

  5. Whether \(\mathrm{IC}_\mu[-i]\) is a summand of \(V_\lambda\ast\mathrm{IC}_0\) has the same answer over \(\overline{\mathbb F}_q\) and over \(\mathbb C\) (D5). So \(\mathcal M\) is the same set over both fields, and Proposition I over \(\mathbb C\) concludes:

    $$\mathcal M_{\overline{\mathbb F}_q}=\mathcal M_{\mathbb C}=\Lambda_S^+\quad\Longrightarrow\quad\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\ \ \text{(Betti, over }\mathbb C).$$
    Proof
    • Where the statement lives. \(V_\lambda\ast\mathrm{IC}_0\) is the pushforward along the finite-type map \(\operatorname{Gr}_{\le\lambda}\mathbin{\widetilde\times}\overline{\mathcal O_0}\to\overline{\mathcal O_{\varphi(\lambda)}}\), \(\varphi(\lambda)=\lambda-\theta(\lambda)\) (D0), so "\(\mathrm{IC}_\mu[-i]\) is a summand" is a statement about one finite-type map.
    • Comparison. D5: such statements agree over \(\overline{\mathbb F}_q\) and \(\mathbb C\) (BBD §6.1; Bezrukavnikov–Finkelberg, Proposition 5), for \(p\) outside a finite set.
    • Scope. So (b) needs (SG) over \(\mathbb F_q(\!(t)\!)\) for one \(q\) whose characteristic is outside that finite set; for the symplectic family (SG) holds for every \(q\) (Proposition IV).

  6. Not needed for the claim. (SG) asks that every orbit indicator be generated; if one indicator is generated, its IC is already a summand:

    $$\mathbf 1_{\mathcal O_\mu(\mathbb F_q)}\in\mathcal H_K\ast\varphi_0\ \Longrightarrow\ \mu\in\mathcal M .$$
    Proof
    • Only \(\mathcal M\) appears. By step 3, \(m_{\nu\lambda}=0\) for \(\nu\notin\mathcal M\); so every \(\operatorname{tr}(V_\lambda\ast\mathrm{IC}_0)\), and by part (a) all of \(\mathcal H_K\ast\varphi_0\otimes\overline{\mathbb Q}_\ell\), lies in \(\operatorname{span}\{\operatorname{tr}\mathrm{IC}_\nu:\nu\in\mathcal M\}\).
    • Expansion. The indicator lies in the \(\overline{\mathbb Q}_\ell\)-span too (IIb, step 7), so \(\mathbf 1_{\mathcal O_\mu(\mathbb F_q)}=\sum_{\nu\in\mathcal M}c_\nu\operatorname{tr}\mathrm{IC}_\nu\), a finite sum, not all \(c_\nu\) zero.
    • Top term. Take \(\nu_0\) maximal with \(c_{\nu_0}\ne0\) and evaluate at \(x_{\nu_0}\). \(\operatorname{tr}\mathrm{IC}_\nu(x_{\nu_0})\ne0\) needs \(x_{\nu_0}\in\overline{\mathcal O_\nu}\), i.e. \(\nu_0\le\nu\); so only \(\nu=\nu_0\) contributes, with value \(c_{\nu_0}(-1)^{\delta(\nu_0)}\ne0\) (IIb (i)).
    • Conclusion. \(\mathbf 1_{\mathcal O_\mu}(x_{\nu_0})\ne0\), so \(x_{\nu_0}\in\mathcal O_\mu\) and \(\mu=\nu_0\in\mathcal M\). Source: the run's audit node A7-14.