Earlier paper version (PDF)

Definition 1 · The norm action

Fix an odd prime power \(q\), \(k=\overline{\mathbb F}_q\), \(F=\mathbb F_q((t))\), \(E=\mathbb F_{q^2}((t))\), and \(\mathcal O=\mathbb F_q[[t]]\). Let

$$G=\operatorname{Res}_{\mathbb F_{q^2}/\mathbb F_q}\mathbb G_m,\quad H=\ker(N:G\to\mathbb G_m),\quad X=G/H\simeq\mathbb G_m.$$

The action is \(g\cdot x=N(g)x\); the involution is \(\theta(g)=\sigma(g)^{-1}\), so \(H=G^\theta\). Put \(K=\mathbb F_{q^2}[[t]]^\times\), \(\varphi_0=\mathbf1_{\mathcal O^\times}\), and use Haar measure \(\operatorname{vol}(K)=1\).

Definition 2 · The two generation questions

For \(\ell\ne\operatorname{char}k\), let \(\mathcal C_{K,k}\) be the bounded orbit-constructible \(L^+G_k\)-equivariant sheaf category on \(LX_k\), with finite orbit support, and let \(\mathrm{IC}_0\) be its constant sheaf on \(L^+X_k\). Define

$$\mathcal C_{K,k}^{\mathrm{Sat}}=\operatorname{Thick}\{V_\lambda*\mathrm{IC}_0:\lambda\in X_*(G_k)\},\quad C_K=C_c^\infty(X(F))^K,\quad\mathcal H_K=C_c(K\backslash G(F)/K).$$

Thick closure includes shifts, cones and direct summands; \(V_\lambda\) is the point Satake sheaf of the torus \(G_k\).

Definitions and hypotheses for the split case
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 IX · Sheaf generation does not imply function generation

For the norm action above, the geometric category is generated, whereas the basic function misses every odd valuation class:

$$\mathcal C_{K,k}^{\mathrm{Sat}}=\mathcal C_{K,k},\qquad\mathcal H_K*\varphi_0=\bigoplus_{a\in\mathbb Z}\mathbb C\,\mathbf1_{t^{2a}\mathcal O^\times}\ \subsetneq\ C_K=\bigoplus_{j\in\mathbb Z}\mathbb C\,\mathbf1_{t^j\mathcal O^\times}.$$

This symmetric example has connected \(H\), but \(G\) is nonsplit, so it lies outside (R). The split \(E_6/F_4\) implication is stated separately with its inputs in Step 7.

Steps 1–2 · Geometric generation · Steps 3–5 · Rational functions · Steps 6–7 · The converse

  1. Geometric generation

    Every geometric valuation class is reached by a Satake object.

  2. Split the norm action

    Over \(k\), the norm becomes multiplication.

    $$G_k=\mathbb G_m^2,\qquad H_k=\{(a,a^{-1}):a\in\mathbb G_m\},\qquad(a,b)\cdot x=abx.$$
    Proof

    The two embeddings of \(\mathbb F_{q^2}\) into \(k\) split restriction of scalars; the norm is the product of the two factors. The kernel and quotient follow directly.

  3. Reach every simple object

    For each \(j\in\mathbb Z\), use the geometric cocharacter \((j,0)\).

    $$\mathcal O_j=t^jk[[t]]^\times,\qquad V_{(a,b)}*\mathrm{IC}_0\simeq\mathrm{IC}_{a+b},\qquad V_{(j,0)}*\mathrm{IC}_0\simeq\mathrm{IC}_j.$$
    Proof
    1. The valuation identifies the orbits with \(\mathbb Z\). Multiplication by \(t^{a+b}\) sends \(\mathcal O_0\) isomorphically to \(\mathcal O_{a+b}\), and the Satake sheaf of a torus is supported on a point, so convolution is this translation.
    2. The stabilizer is \(L^+H_k\), whose finite jet groups are connected (a torus extended by unipotent groups). Hence its component group is trivial and each orbit has only the constant irreducible equivariant local system.
    3. A bounded constructible object with finite orbit support is built from these simple perverse objects by its perverse truncations and a finite Jordan–Hölder filtration. Thus reaching every \(\mathrm{IC}_j\) proves the displayed categorical equality. Equivariant self-extensions are retained by this argument.
    Sources

  4. Rational functions

    Rational Hecke labels translate valuation by twice an integer.

  5. Identify the rational orbits

    The norm on units is onto, and a uniformizer has norm \(t^2\).

    $$N(\mathbb F_{q^2}[[t]]^\times)=\mathbb F_q[[t]]^\times,\qquad N(t^a u)=t^{2a}N(u),\qquad K\backslash X(F)\simeq\mathbb Z.$$
    Proof

    The residue-field norm is onto. On the successive principal-unit quotients the norm is the nonzero field trace \(\mathbb F_{q^2}\to\mathbb F_q\); lift recursively and use completeness. Thus multiplication by \(N(K)=\mathcal O^\times\) is transitive on each set \(t^j\mathcal O^\times\), and preserves its valuation.

  6. Compute the Hecke action

    Write \(T_a=\mathbf1_{t^aK}\) and \(\delta_j=\mathbf1_{t^j\mathcal O^\times}\).

    $$\mathcal H_K=\bigoplus_{a\in\mathbb Z}\mathbb CT_a,\qquad T_a*T_b=T_{a+b},\qquad T_a*\delta_j=\delta_{j+2a}.$$
    Proof

    The group \(E^\times\) is abelian and \(E^\times/K=\mathbb Z\), so \(t^aK/K\) has one coset. In the convolution sum, \(N(t^a)^{-1}x=t^{-2a}x\) lies in \(t^j\mathcal O^\times\) exactly when \(v_F(x)=j+2a\).

  7. Separate the two parities

    The odd valuation functions form a second nonzero summand.

    $$C_K\simeq\mathbb C[u,u^{-1}],\qquad\mathcal H_K\simeq\mathbb C[z,z^{-1}],\qquad z\mapsto u^2,\qquad C_K=\mathcal H_K\delta_0\oplus\mathcal H_K\delta_1.$$
    Proof

    Identify \(u^j\) with \(\delta_j\). Step 4 identifies the image of the cyclic module with the even Laurent monomials, so \(\delta_1\) is outside it. Every Laurent monomial has exactly one parity.

    For \(F_n=\mathbb F_{q^n}((t))\), the quadratic algebra stays a field when \(n\) is odd and splits when \(n\) is even; in the split case the action is multiplication by two independent factors.

    Table 2. Generation after a residue-field extension.
    \(n\)rational norm actionfunction generation
    oddvaluation shift \(2a\)fails
    evenvaluation shift \(a+b\)holds

  8. The converse

    Geometric generation and spanning by rational trace functions impose different conditions.

  9. Track Frobenius on labels

    Geometric labels \((a,b)\) are unrestricted; rational torus Hecke labels have \(a=b\).

    $$\mathrm{Fr}(a,b)=(b,a),\qquad (j,0)+(0,j)\text{ is a Frobenius pair for }j\ne0.$$
    Proof

    Restriction of scalars exchanges the two geometric factors. A point of the affine Grassmannian descends exactly when its cocharacter is fixed, giving \((a,a)\) and hence valuation \(2a\). The geometric generator \(V_{(1,0)}*\mathrm{IC}_0\simeq\mathrm{IC}_1\) exists, but its source label is not fixed. A direct sum of the two exchanged labels has swap Frobenius and trace zero in degree one.

    This proves the failure without assuming a categorical trace–function identification for the full loop-space category. The trace formalism distinguishes the trace of a category from the image of the action functor.

  10. State the split implication

    For split \(E_6/F_4\), retain every geometric and arithmetic input of VI–VIII.

    Assume the equal-characteristic module identification of VI, the restriction-ring computation of VII, and orbit geometry and decomposition D0–D2 as used in VIII. Then

    $$\mathcal C_{K,k}^{\mathrm{Sat}}=\mathcal C_{K,k},\qquad\mathcal H_K*\varphi_0\ne C_K.$$
    Proof
    1. VIII proves the first equality under its stated D0–D2 inputs.
    2. VI identifies \(C_K\) with \(S=\mathbb C[\Lambda_S]^{W_X}\), sends \(\varphi_0\) to \(1\), and identifies \(\mathcal H_K\varphi_0\) with the image of restriction \(r\).
    3. VII exhibits \(y_1\) and \(\sigma y_1\) with equal evaluations for every function in \(\operatorname{im}r\), but unequal evaluations for an element of \(S\). Thus \(r\) is not onto. No identification of this complementary element with a single orbit indicator is required.

    The assertion is conditional on the named inputs; the complex orbit-parametrization citation alone does not establish D0 in characteristic \(p\). The elementary norm-torus proof above has no such dependency. The run record keeps the other examples and their precise hypotheses.