§1 · Agent 1 · Trace formalism · 2026-10-17T21:11:10Z

Remark 1.10 · Spectral picture: generation is affineness, (SG) a closed immersion of loop spaces

heuristic

Proved for tori: \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\iff\check T_X\to\check T\) is a closed immersion, while \((\mathrm{SG})\iff\) the induced map of \(\sigma\)-coinvariants is one; for the norm example it is \(z\mapsto z^2\). Heuristic in general (relative Langlands duality is unproved): Frobenius trace turns pullback along the moment map into restriction of functions along twisted loop spaces, i.e. Sakellaridis's \(r\); failure is fusion of twisted classes, the reductive form of (E2).

Remark 1.10.

(Proved, tori.) For \(X=T/H\) as in Counterexample 1.5, let \(\check T\), \(\check T_X\) be the dual tori (\(X^*(\check T)=X_*(T)\), \(X^*(\check T_X)=X_*(X)\)), \(\check T_X\to\check T\) dual to \(X_*(T)\to X_*(X)\), and \(\check T_\sigma=\check T/(1-\sigma)\check T\) the \(\sigma\)-coinvariants (twisted conjugacy classes). Then \(\mathcal H_K\otimes\mathbb C=\mathcal O(\check T_\sigma)\), \(C_K\otimes\mathbb C=\mathcal O(\check T_{X,\sigma})\) as a module by pullback, and $$\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\ \text{and}\ \check T_X\to\check T\ \text{is a closed immersion (both always hold here)},\qquad(\mathrm{SG})\iff\check T_{X,\sigma}\to\check T_\sigma\ \text{is a closed immersion}.$$ For \(R_{\mathbb F_{q^2}/\mathbb F_q}\mathbb G_m/U_1\): \(\check T_X=\mathbb G_m\to\check T=\mathbb G_m^2\), \(z\mapsto(z,z)\), \(\check T_\sigma\cong\mathbb G_m\) by \((a,b)\mapsto ab\), and the induced map is \(z\mapsto z^2\).

(Heuristic.) By Theorem 1.3(d), \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\) says that \(\mathcal C_k\) is affine over \(\mathcal H\): \(\operatorname{Ind}\mathcal C_k\simeq A\text{-mod}\). If, as conjectured by Ben-Zvi–Sakellaridis–Venkatesh, \(\mathcal H\) and \(\mathcal C_k\) are (shearings of) \(\mathrm{QCoh}(\check{\mathfrak g}^*/\check G)\) and \(\mathrm{QCoh}(\check M/\check G)\) with \(A\mapsto\mathcal O(\check M)\), then \(\operatorname{Tr}(\mathrm{Fr})\) turns \(\mathrm{act}\) into pullback of functions along the Frobenius-twisted loop spaces \(\mathcal L_{\mathrm{Fr}}(\check M/\check G)\to\mathcal L_{\mathrm{Fr}}(\mathrm{pt}/\check G)\), the twist being the Tate shearing, and (SG) becomes surjectivity of this pullback. This matches Sakellaridis's Theorem 8.0.2 (char 0, under (A)): \(C_K=\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X}\) with \(\mathcal H_K\) acting through restriction \(r\). Affine morphisms do not induce closed immersions of twisted loop spaces; the typical obstruction is fusion: two points not conjugate under \(W_X\) whose \(q^{\rho_L}\)-translates are conjugate under \(W\).

Proof of the torus statement (proved).

  1. \(X^*(\check T_\sigma)=\{\chi\in X^*(\check T):\chi\circ(1-\sigma)=0\}=X_*(T)^\sigma\), so \(\mathcal O(\check T_\sigma)=\mathbb C[X_*(T)^\sigma]=\mathcal H_K\) (for a torus, \(K\backslash T(F)/K=T(F)/T(\mathcal O)=X_*(T)^\sigma\)). Likewise \(\mathcal O(\check T_{X,\sigma})=\mathbb C[X_*(X)^\sigma]=C_K\), and by Counterexample 1.5(b) \(\mathcal H_K\) acts through the group homomorphism \(X_*(T)^\sigma\to X_*(X)^\sigma\), i.e. by pullback along \(\check T_{X,\sigma}\to\check T_\sigma\).
  1. A homomorphism of abelian groups \(A\to B\) induces a surjection \(\mathbb C[A]\to\mathbb C[B]\) iff it is onto; by Cartier duality this is the statement that the dual map of diagonalizable groups is a closed immersion. Apply it to \(X_*(T)\to X_*(X)\) (Counterexample 1.5(a)) and to \(X_*(T)^\sigma\to X_*(X)^\sigma\) (Counterexample 1.5(b)).
  1. Example: \(X_*(T)\to X_*(X)\) is \((a,b)\mapsto a+b\), whose dual is the diagonal \(z\mapsto(z,z)\); \((1-\sigma)\check T=\{(c/d,d/c)\}\) and \(\check T_\sigma\cong\mathbb G_m\) by \((a,b)\mapsto ab\); the composite is \(z\mapsto z^2\), of degree 2, not a closed immersion, matching \(\mathcal H_K\ast\varphi_0=\mathbb C[T_1^{\pm1}]\cdot\delta_0\) = even part.

Heuristic outline (not proved; relies on unproved conjectures).

  1. Traces of \(\mathrm{QCoh}\) of perfect stacks are functions on (twisted) loop spaces (Ben-Zvi–Francis–Nadler; not in the local sources). For \(\mathrm{Rep}\,\check G\) with Frobenius acting through the Tate twist, the trace is the space of class functions, which matches \(\mathcal H_K\) (Satake; Zhu Lemma 5.6.3). If \(\mathcal C_k\simeq A\text{-mod}\) with \(A\leftrightarrow\mathcal O(\check M)\), the trace map becomes restriction of functions along \(\mathcal L_{\mathrm{Fr}}(\check M/\check G)\to\check G/\!/\check G\).
  1. In the semisimple situation of Chen–Yi's Theorem 87 (even codimensions, connected stabilizers), \(\mathrm{Perv}(L^+G\backslash LX)\simeq\mathrm{Rep}(\check G_X)\). If the abelian Hecke action were restriction along \(\check G_X\times\mathrm{SL}_2\to\check G\) with the \(\mathrm{SL}_2\) evaluated at \(\mathrm{diag}(q^{1/2},q^{-1/2})\) under Frobenius (an expectation, not proved here), then thick generation would be the dominance of restriction (Example 1.4(E2), always true for an injective map), and the trace would be \(f\mapsto(y\mapsto f(y\,q^{\rho_L}))\) on class functions, i.e. \(r\). Its surjectivity fails in particular when \(W_X\)-inequivalent \(y_1,y_2\) have \(W\)-conjugate translates (it also fails if the finite map of quotients is injective but not a closed immersion) — the gluing that the previous run found for \(E_6/F_4\) under (A1)–(A5), and Sakellaridis's Example 8.0.1 with \(\delta(X)\) trivial.
  1. Dictionary of failure modes: the label mechanism (Counterexample 1.5) is fusion created by the Frobenius action on \(\check T\) itself (\(z\mapsto z^2\)); under (R) Frobenius acts on the dual side only through the Tate twist, and fusion can only come from the \(q^{\rho_L}\)-shift relative to \(W\) — consistent with Proposition 1.8(g). \(\square\)

Builds on

  • Theorem 1.3 · Theorem T: trace of Frobenius on \(\mathcal C_k\) is \(C_K\); the missing retract classes
  • Example 1.4 · Four minimal examples: generation by an image, trace map not onto
  • Counterexample 1.5 · Counterexample outside (R): the norm torus quotient, and the torus criterion

Built on by

  • Summary 1.12 · Verdict on part (2), with consequences for (1) and (3)

Sources