Proof workspace · spherical level · functions

The classical analogue

Compactly supported functions on \(X(F)\): what the basic function \(\mathbf 1_{X(\mathcal O)}\) generates over the spherical Hecke algebra. A triangular criterion that is the function-level form of Proposition I, the group case, Sakellaridis's theorem with its hypotheses listed one by one, the two \(\mathrm{GL}\) families through their restriction maps, the affine space and \(M_n\), and rank one on the tree. Every variety at both levels, with the reason it holds or fails, is on the Summary tab. Explained slowly, with the spherical and Iwahori levels side by side →

Categorical statements; the dictionary between the two levels is on the Iwahori workspace's Proposition I page, and the Iwahori level of the same question, the basic function over the Iwahori–Hecke algebra, is on the Iwahori pages. Conventions: \(X=G/H\) with left actions, left \(\mathcal H_K\)-modules, \(\operatorname{vol}K=1\); Sakellaridis writes \(H\backslash G\) and right modules, exchanged with these by \(x\mapsto x^{-1}\). No implication between the two levels is proved on these pages.

1 · Setting

\(F\) a non-archimedean local field with ring of integers \(\mathcal O\), uniformizer \(\varpi\) and residue field \(\mathbb F_q\); \(\operatorname{char}F=0\) where a cited theorem needs it (4). \(G\supseteq B\supseteq T\) split connected reductive over \(\mathcal O\), \(X\) a spherical \(G\)-variety over \(\mathcal O\), a model of the paper's \(X\) (smooth, affine, connected generic \(B\)-stabilizers); 5 and 6 say where an example leaves that class. Coefficients \(\mathbb C\).

$$K:=G(\mathcal O),\qquad\mathcal S(X):=C_c^\infty(X(F)),\qquad C_K:=\mathcal S(X)^K,\qquad\mathcal H_K:=C_c(K\backslash G(F)/K),\qquad\varphi_0:=\mathbf 1_{X(\mathcal O)}\in C_K .$$

Convolution and Hecke operators. Haar measure on \(G(F)\) with \(\operatorname{vol}(K)=1\). For \(f\in\mathcal H_K\), \(\varphi\in C_K\), and a dominant cocharacter \(\lambda\in\Lambda_T^+\) with \(t^\lambda:=\lambda(\varpi)\),

$$(f\ast_K\varphi)(x):=\int_{G(F)}f(g)\,\varphi(g^{-1}x)\,dg=\sum_{g\in G(F)/K}f(g)\,\varphi(g^{-1}x),\qquad T_\lambda:=\mathbf 1_{Kt^\lambda K},\qquad G(F)=\bigsqcup_{\lambda\in\Lambda_T^+}Kt^\lambda K .$$
Well defined; \(C_K\) is a unital \(\mathcal H_K\)-module; the Satake transform
  • The summand depends only on \(gK\), and finitely many cosets contribute. \((f\ast_K\varphi)(kx)=(f\ast_K\varphi)(x)\) for \(k\in K\) by \(g\mapsto k^{-1}g\), and \(\operatorname{supp}(f\ast_K\varphi)\subseteq\operatorname{supp}f\cdot\operatorname{supp}\varphi\). The same formula makes \(\mathcal H_K\) an associative algebra with unit \(\mathbf 1_K\) and \(C_K\) a unital left module. Hecke algebras of compact open subgroups: Cartier, §1.3–1.4.
  • The Cartan decomposition makes \(\{T_\lambda\}_{\lambda\in\Lambda_T^+}\) a basis of \(\mathcal H_K\). The Satake transform $$\mathsf Sf(\chi):=\int_{T(F)}\delta_B^{1/2}(t)\,\chi(t)\int_{U(F)}f(tu)\,du\,dt,\qquad\mathsf S:\mathcal H_K\xrightarrow{\ \sim\ }\mathbb C[A^*]^W,\qquad A^*:=\operatorname{Hom}(\Lambda_T,\mathbb C^\times),$$ is an isomorphism of algebras onto the \(W\)-invariant Laurent polynomials on the dual torus \(A^*\) of unramified characters of \(T(F)\); so \(\mathcal H_K\) is commutative, and \(\mathsf S(T_\lambda)=q^{\langle\rho,\lambda\rangle}x^\lambda+\) lower monomials in the dominance order (Cartier, §4; Macdonald, Spherical functions on a group of p-adic type).
  • For \(f\in\mathcal H_K\) the operator \(\varphi\mapsto f\ast_K\varphi\) on \(\mathcal S(X)\) is \(\pi(f)=\int f(g)\pi(g)\,dg\) for \((\pi(g)\varphi)(x)=\varphi(g^{-1}x)\); so every \(G(F)\)-map \(\mathcal S(X)\to V\) restricts to an \(\mathcal H_K\)-map \(C_K\to V^K\).

Lemma A (orbits and translates).

  1. Every \(K\)-orbit in \(X(F)\) is compact, and \(\mathbf 1_{Kx}\in C_K\) when \(Kx\) is open. If \(X=G/H\) is homogeneous, \(X(F)\) is a finite disjoint union of open \(G(F)\)-orbits, every \(K\)-orbit is compact open, and \(C_K\) has the basis \(\{\mathbf 1_{Kx}\}_{Kx\in K\backslash X(F)}\).
  2. For \(\lambda\in\Lambda_T^+\), $$(T_\lambda\ast_K\varphi_0)(x)=\#\{gK\subseteq Kt^\lambda K:\ x\in g\,X(\mathcal O)\}\in\mathbb Z_{\ge0},\qquad\operatorname{supp}(T_\lambda\ast_K\varphi_0)=K\,t^\lambda X(\mathcal O),$$ the union of the \(K\)-orbits of the points \(t^\lambda x\), \(x\in X(\mathcal O)\); a finite union when \(X(\mathcal O)\) meets finitely many \(K\)-orbits, in particular for homogeneous \(X\). For \(x_0\in X(\mathcal O)\), \((T_\lambda\ast_K\varphi_0)(t^\lambda x_0)\ge1\).
Proof
  • (1) \(Kx\) is the image of the compact group \(K\). For homogeneous \(X\): \(X(F)\) is the disjoint union of the \(G(F)\)-orbits of representatives of \(\ker\big(H^1(F,H)\to H^1(F,G)\big)\), finitely many since \(\operatorname{char}F=0\) (Borel–Serre, Théorèmes de finitude en cohomologie galoisienne, 1964; Serre, Galois cohomology, I.5.4 and III.4.3); each orbit \(G(F)x\simeq G(F)/G_x(F)\) is open, since the orbit map \(G\to X\) is smooth and a submersion of \(p\)-adic manifolds is open, and \(Kx\) is the image of the open subset \(K\) under the open map \(G(F)\to G(F)x\). A \(K\)-invariant \(\varphi\in\mathcal S(X)\) is constant on each orbit, and its compact support meets finitely many of the open orbits.
  • (2) With \(\operatorname{vol}K=1\), \((T_\lambda\ast_K\varphi_0)(x)=\sum_{gK\subseteq Kt^\lambda K}\mathbf 1_{X(\mathcal O)}(g^{-1}x)\). The term for \(gK\) is \(1\) iff \(x\in gX(\mathcal O)\); so the support is \(Kt^\lambda K\cdot X(\mathcal O)=Kt^\lambda X(\mathcal O)\), as \(X(\mathcal O)\) is \(K\)-stable. \(X(\mathcal O)\) is compact, so it meets finitely many orbits when these are open. At \(x=t^\lambda x_0\) the coset \(t^\lambda K\) contributes \(1\).

2 · Spherical generation and a triangular criterion

The statement, and its reading through Lemma A(1) for homogeneous \(X\):

$$\boxed{\ \text{(SG)}\qquad\mathcal H_K\cdot\varphi_0=C_K\ }\qquad\Longleftrightarrow\qquad\text{every orbit indicator }\mathbf 1_{Kx}\text{ is a Hecke translate }f\ast_K\varphi_0 .$$

Proposition B (triangular generation). Let \(\Omega\) be a set of compact open \(K\)-orbits in \(X(F)\) whose indicators span \(C_K\) (all orbits, when \(X\) is homogeneous), with a partial order \(\le\) on \(\Omega\) in which every lower set \(\{\nu\le\mu\}\) is finite. Suppose that for every \(\mu\in\Omega\) there is \(f_\mu\in\mathcal H_K\) with

$$\operatorname{supp}(f_\mu\ast_K\varphi_0)\subseteq\bigcup_{\nu\le\mu}O_\nu\qquad\text{and}\qquad f_\mu\ast_K\varphi_0\ne0\ \text{ on }O_\mu .$$

Then (SG) holds. Conversely, under (SG) such \(f_\mu\) exist for every order, with \(f_\mu\ast_K\varphi_0=\mathbf 1_{O_\mu}\).

Proof
  • Induction on \(n(\mu):=|\{\nu\le\mu\}|\). \(f_\mu\ast_K\varphi_0\) is \(K\)-invariant, hence constant on orbits, and its support lies in the finite union above: \(f_\mu\ast_K\varphi_0=c_\mu\mathbf 1_{O_\mu}+\sum_{\nu<\mu}c_\nu\mathbf 1_{O_\nu}\) with \(c_\mu\ne0\). Each \(\nu<\mu\) has \(n(\nu)

Corollary C (dominant lifts). Let \(X\) be homogeneous, and let \(\Lambda_X^+\) index \(K\backslash X(F)\), \(\mu\mapsto O_\mu\), with a partial order with finite lower sets. If every \(\mu\in\Lambda_X^+\) has a Hecke label \(\lambda\in\Lambda_T^+\) with

$$t^\lambda x_0\in O_\mu\qquad\text{and}\qquad K\,t^\lambda X(\mathcal O)\subseteq\bigcup_{\nu\le\mu}O_\nu ,$$

then (SG) holds, with \(f_\mu=T_\lambda\).

Proof

Proof and dictionary

  • Lemma A(2): \(\operatorname{supp}(T_\lambda\ast_K\varphi_0)=Kt^\lambda X(\mathcal O)\) and \((T_\lambda\ast_K\varphi_0)(t^\lambda x_0)\ge1\). Proposition B.
  • Dictionary with Proposition I. Finite lower sets are H1(a); a label \(\lambda\) with \(t^\lambda x_0\in O_\mu\) is a dominant lift of \(\mu\) under the leading-term map, the condition \(\varphi(\Lambda_T^+)=\Lambda_S^+\) of Proposition III; the support bound and the nonzero value on the top orbit are the function-level reading of H5, whose sheaf form says that the top part of \(V_\lambda\ast\mathrm{IC}_0\) on \(\mathcal O_{\varphi(\lambda)}\) is a nonzero constant sheaf that splits off. At function level the top value is a positive integer for free, and the content is the support bound; at sheaf level the same set \(L^+G\,t^\lambda L^+X\) carries \(V_\lambda\ast\mathrm{IC}_0\). No implication between the levels is proved.

3 · The group

Proposition D (group case). \(X=H\) under \(G=H\times H\), \((h_1,h_2)\cdot x=h_1xh_2^{-1}\), \(K=K_H\times K_H\). Then \(C_K=\mathcal H_{K_H}\), \(\varphi_0=\mathbf 1_{K_H}\) is the unit of \(\mathcal H_{K_H}\), \(\mathcal H_K=\mathcal H_{K_H}\otimes\mathcal H_{K_H}^{\mathrm{op}}\) acts by \((a\otimes b)\cdot\varphi=a\ast_K\varphi\ast_Kb\), and (SG) holds:

$$\mathcal H_K\cdot\varphi_0=\mathcal H_{K_H}\ast_K\mathbf 1_{K_H}\ast_K\mathcal H_{K_H}=\mathcal H_{K_H}=C_K .$$
Proof

Proof; the triangular reading

  • \(\mathcal S(H)^{K_H\times K_H}=C_c(K_H\backslash H(F)/K_H)\), and \(\mathbf 1_{K_H}\) is the unit for \(\operatorname{vol}K_H=1\); \(a\ast_K\mathbf 1_{K_H}\ast_Kb=a\ast_Kb\) ranges over all of \(\mathcal H_{K_H}\) (take \(b=\mathbf 1_{K_H}\)).
  • In the language of Corollary C: the orbits are the double cosets \(K_Ht^\lambda K_H\), \(\lambda\in\Lambda_{T_H}^+\), each is its own lower set, the lift of \(\lambda\) is \((\lambda,0)\), and \(K\,t^{(\lambda,0)}X(\mathcal O)=K_Ht^\lambda K_H\) is a single orbit: the triangular system is diagonal. This is the function-level shadow of Proposition II.

4 · Sakellaridis's theorem, with its hypotheses

Sakellaridis, Spherical functions on spherical varieties, Amer. J. Math. 135 (2013), arXiv:0905.4244v3 (PDF), decides (SG) for affine homogeneous \(X\) under standing assumptions stated in his §1.7, §7 and §8, listed here as (A1)–(A5) because every use of his theorem on these pages assumes them; none of them is verified on these pages for any family, and what he asserts for the paper's families is recorded after the theorem.

The invariants: \(A_X:=L(X)/(L(X)\cap H)\) for the Levi \(L(X)\) of the parabolic stabilizing \(\mathring X\), \(\Lambda_X:=X_*(A_X)\), \(\Lambda_X^+\) the \(X\)-anti-dominant cocharacters (the \(G\)-invariant valuations), \(A_X^*\) the dual torus of \(A_X\), mapped to \(A^*\) by \(T\twoheadrightarrow A_X\); \(W_X\) the little Weyl group; \(\delta(X)\) the modular character of \(L(X)\cap B\), and \(\delta(X)^{1/2}A_X^*\subseteq A^*\) the translate, with the \(W_X\)-action transported from \(A_X^*\). These are the invariants of Sakellaridis–Venkatesh, §2.1, defined for any spherical variety over an algebraically closed field of characteristic \(0\). For symmetric \(X=G/G^\theta\): \(A_X\simeq S\) via \(t\mapsto t\theta(t)^{-1}\), \(\Lambda_X^+=\Lambda_S^+\), and \(W_X=N_G(S)/Z_G(S)\) is the restricted Weyl group.

Theorem E (Sakellaridis, Theorem 8.0.2, Corollary 8.0.4, Example 8.0.1). Assume (A1)–(A5).

  1. There is an isomorphism of \(\mathcal H_K\)-modules \(C_K\simeq\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X}\) sending \(\varphi_0\) to \(1\), where \(\mathcal H_K\simeq\mathbb C[A^*]^W\) acts through the restriction map $$r:\mathbb C[A^*]^W\longrightarrow\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X}.$$
  2. (SG) holds if and only if \(r\) is surjective.
  3. For \(X=\mathrm{SL}_2\times\mathrm{SL}_2\times\mathrm{SL}_2\backslash\mathrm{Sp}_4\times\mathrm{Sp}_4\), the first two factors embedded as \(\mathrm{Sp}_2\) in the two copies of \(\mathrm{Sp}_4\) and the third diagonally, \(r\) is not surjective and (SG) fails.
Proof

Proof of (2) from (1); what is and is not checked

  • Under (1) the submodule \(\mathcal H_K\cdot\varphi_0\) corresponds to \(r(\mathbb C[A^*]^W)\cdot1=\operatorname{im}r\), so (SG) is the surjectivity of \(r\); the implication \(\Leftarrow\) is Corollary 8.0.4(1) of loc. cit. (1) and (3) are Theorem 8.0.2 and Example 8.0.1 there: in (3), \(W_X\)-nonconjugate points of \(A_X^*\) become conjugate under \(W\).
  • For the table of examples on his p. 76 he writes that the groups are taken split, with convenient choices of centres, “so that the assumptions of Theorem 7.2.1 are satisfied”: rows 1, 2 and 12 are \(\mathrm{GL}_m\times\mathrm{GL}_n\backslash\mathrm{GL}_{m+n}\), \(\mathrm{Sp}_{2n}\backslash\mathrm{GL}_{2n}\) and \(H\backslash H\times H\). So (A4) is asserted there for the \(\mathrm{GL}\) families and the group; (A3) he proves only at almost all places; (A5) is the remark below. The spheres \(\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}\) and \(E_6/F_4\) are not in his table.
  • (A5) and the paper's class. If the stabilizer \(B_x\) of a point of the open \(B\)-orbit is connected, as the paper assumes, then \(H^1(F,B_x)=1\): \(B_x\) is a connected solvable subgroup of the split \(B\), its maximal torus is split (a subtorus of a \(B(F)\)-conjugate of \(T\)) and its unipotent radical is split, so Hilbert 90 and \(H^1(F,\mathbb G_a)=0\) give the vanishing; hence \(\mathring X(F)=(B/B_x)(F)\) is a single \(B(F)\)-orbit and (A5) holds. For \(\mathrm{SL}_2/\mathrm{SO}_2\) the generic stabilizer is \(\{\pm1\}\), \(\mathring X(F)/B(F)\simeq H^1(F,\mu_2)=F^\times/F^{\times2}\), and (A5) fails: the square classes of Proposition J.

Proposition F (the two \(\mathrm{GL}\) families). Assume (A1)–(A5) for \(X\). Then \(r\) is surjective, hence (SG) holds, for

  1. \(X=\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}\): \(A_X^*=\{(y_1,y_1,\dots,y_n,y_n)\}\subseteq A^*=(\mathbb C^\times)^{2n}\), \(\delta(X)^{1/2}\) moves each pair to \((q^{1/2}y_i,q^{-1/2}y_i)\), \(W_X=S_n\) permutes the pairs, and for the Laurent power sums \(p_{\pm k}:=\sum_ix_i^{\pm k}\), $$r(p_{\pm k})=\big(q^{k/2}+q^{-k/2}\big)\,p_{\pm k}(y);$$
  2. \(X=\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\), \(p\le q\): \(A_X^*=\{(y_1,\dots,y_p,1,\dots,1,y_p^{-1},\dots,y_1^{-1})\}\), \(\delta(X)^{1/2}\) moves only the middle block, \(W_X=W(C_p)\) acts by signed permutations of the \(y_i\), and $$r(p_k)=\sum_{i=1}^p\big(y_i^k+y_i^{-k}\big)+c_k,\qquad c_k\in\mathbb C .$$

Moreover (SG) for \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\) follows from (SG) for \(\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}\), with no further hypothesis.

Proof
  • Root data (Proposition III, step 7; symplectic, \(\mathrm{GL}_p\times\mathrm{GL}_q\)). (1) \(\theta(x)=(-x_2,-x_1,\dots,-x_{2n},-x_{2n-1})\) on \(\Lambda_T\), and \(T\twoheadrightarrow S=A_X\) is \((x_i)\mapsto(x_1x_2,\dots,x_{2n-1}x_{2n})\); so the unramified characters of \(T(F)\) pulled back from \(A_X\) are the \((y_1,y_1,\dots,y_n,y_n)\). \(L(X)=Z_G(S)=\mathrm{GL}_2^n\), and \(\delta(X)=\delta_{B\cap L(X)}\) is \(|a/b|\) on each block \(\operatorname{diag}(a,b)\): the shift \((q^{1/2},q^{-1/2})\) per pair. The restricted root system is \(A_{n-1}\), \(W_X=S_n\) permutes the pairs, the shift is \(W_X\)-invariant, and the transported action on \(\delta(X)^{1/2}A_X^*\) permutes the \(y_i\). (2) \(\theta\) permutes coordinates by \(w\) (\(i\leftrightarrow n+1-i\) for \(i\le p\)); the pulled-back characters are those with \(\chi\circ\theta=\chi^{-1}\), i.e. \((y_1,\dots,y_p,1,\dots,1,y_p^{-1},\dots,y_1^{-1})\). \(L(X)=\mathbb G_m^{2p}\times\mathrm{GL}_{q-p}\), so \(\delta(X)\) is trivial on the \(y\)-coordinates and moves the middle \(1\)'s to constants \(c^{(j)}=q^{(q-p+1)/2-j}\). The restricted root system is \(BC_p\) or \(C_p\), and \(W_X=W(C_p)\) acts by permutations and inversions \(y_i\mapsto y_i^{-1}\).
  • Restricted power sums. \(\mathbb C[A^*]^W=\mathbb C[x^{\pm1}]^{S_N}\) is generated by the \(p_{\pm k}\), \(k\ge1\): it is \(\mathbb C[x]^{S_N}[e_N^{-1}]\), and \(e_N^{-1}=e_N(x^{-1})\) is a polynomial in the \(p_{-k}\). Restricting: (1) \(p_{\pm k}\mapsto\sum_i\big((q^{1/2}y_i)^{\pm k}+(q^{-1/2}y_i)^{\pm k}\big)=(q^{k/2}+q^{-k/2})p_{\pm k}(y)\); (2) \(p_k\mapsto\sum_i(y_i^k+y_i^{-k})+\sum_j(c^{(j)})^k\).
  • Generation. (1) The target is \(\mathbb C[y^{\pm1}]^{S_n}\), generated by the \(p_{\pm k}(y)\), and \(q^{k/2}+q^{-k/2}\ne0\). (2) The target is \(\mathbb C[y^{\pm1}]^{W(C_p)}=\mathbb C[z_1,\dots,z_p]^{S_p}\), \(z_i:=y_i+y_i^{-1}\): a Laurent polynomial invariant under every inversion is a polynomial in the \(z_i\), since \(y^k+y^{-k}=z^k+\) lower terms; it is generated by \(P_k:=\sum_iz_i^k\), and \(r(p_k)-c_k=\sum_i(y_i^k+y_i^{-k})=P_k+(\text{polynomial in }P_1,\dots,P_{k-2})\), so by induction on \(k\) every \(P_k\) lies in \(\operatorname{im}r\).
  • \(\mathrm{SL}_{2n}\) from \(\mathrm{GL}_{2n}\). \(X_{\mathrm{GL}}(F)\) is the set of nondegenerate alternating forms on \(F^{2n}\), and \(X_{\mathrm{SL}}(F)=\{J:\operatorname{Pf}J=1\}\) is a single \(\mathrm{SL}_{2n}(F)\)-orbit, since \(\operatorname{Pf}(g^{\mathsf T}Jg)=\det g\cdot\operatorname{Pf}J\) (Kneser: \(H^1(F,\mathrm{Sp}_{2n})=1\)). The \(K\)-orbits on \(X_{\mathrm{GL}}(F)\) are the elementary-divisor classes \(a=(a_1,a_1,\dots,a_n,a_n)\), \(a_1\ge\dots\ge a_n\) (symplectic bases over \(\mathcal O\)), with \(\operatorname{val}\operatorname{Pf}=\sum_ia_i\); two forms of Pfaffian \(1\) in the same class are conjugate under \(\mathrm{SL}_{2n}(\mathcal O)\), the conjugating \(g\in\mathrm{GL}_{2n}(\mathcal O)\) having \(\det g=1\). So \(K_{\mathrm{SL}}\backslash X_{\mathrm{SL}}(F)\) is the set of classes with \(\sum a_i=0\), and \(C_K(X_{\mathrm{SL}})\simeq C_K(X_{\mathrm{GL}})_0\), the span of the classes of Pfaffian valuation \(0\), with \(\varphi_0\mapsto\varphi_0\). Grade \(\mathcal H_K(\mathrm{GL}_{2n})=\bigoplus_m\mathcal H_m\) by \(\operatorname{val}\det\); then \(\mathcal H_m\cdot C_K(X_{\mathrm{GL}})_{m'}\subseteq C_K(X_{\mathrm{GL}})_{m+m'}\), and \(\mathcal H_0\simeq\mathcal H_K(\mathrm{SL}_{2n})\) compatibly with the actions, since \(Kt^\lambda K\cap\mathrm{SL}_{2n}(F)=K_{\mathrm{SL}}t^\lambda K_{\mathrm{SL}}\) for \(\sum_i\lambda_i=0\) (move a determinant into a diagonal matrix commuting with \(t^\lambda\)). The degree-\(0\) part of \(\mathcal H_K(\mathrm{GL}_{2n})\cdot\varphi_0=C_K(X_{\mathrm{GL}})\) is \(\mathcal H_0\cdot\varphi_0=C_K(X_{\mathrm{GL}})_0\), which is (SG) for \(X_{\mathrm{SL}}\).

Across the dictionary: for \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\), \(m\ge2\), and \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\), \(n\ge3\), H5 fails at sheaf level (coverage: the label \((1,1,1,1,0,\dots)\) has no dominant lift), so Proposition I does not apply and \(\mathcal C^{\mathrm{Sat}}=\mathcal C\) is open there, while (SG) holds at function level under (A1)–(A5). An orbit without a dominant lift is reached by Hecke translates only below their top orbit; at sheaf level that would need the intersection complex of the uncovered orbit closure to be a retract of some \(V_\lambda\ast\mathrm{IC}_0\), as the paper remarks.

5 · Not homogeneous: \(\mathbb A^n\) and \(M_n\)

Proposition G (affine space). \(X=\mathbb A^n\) under \(G=\mathrm{GL}_n\), \(n\ge1\). The \(K\)-orbits on \(F^n\) are the shells \(X_m:=\{v:\operatorname{ord}v=m\}\), \(m\in\mathbb Z\), with \(\operatorname{ord}v:=\min_i\operatorname{val}v_i\), and the point \(0\); and

$$\mathbf 1_{\varpi^mK}\ast_K\varphi_0=\mathbf 1_{\varpi^m\mathcal O^n},\qquad\mathbf 1_{X_m}=\mathbf 1_{\varpi^m\mathcal O^n}-\mathbf 1_{\varpi^{m+1}\mathcal O^n}.$$

Hence (SG) holds, with the central elements \(\mathbf 1_{\varpi^mK}=T_{m\cdot\mathbf 1}\) of \(\mathcal H_K\) alone.

Proof
  • \(K\) acts transitively on the vectors of a given order (extend the primitive vector \(\varpi^{-m}v\) to a basis of \(\mathcal O^n\)). Since \(\varpi^mK=K\varpi^mK\) is a single coset, \((\mathbf 1_{\varpi^mK}\ast_K\varphi_0)(v)=\varphi_0(\varpi^{-m}v)=\mathbf 1_{\varpi^m\mathcal O^n}(v)\). A \(K\)-invariant \(\varphi\in\mathcal S(X)\) is supported in some \(\varpi^{-N}\mathcal O^n\) and constant on some \(\varpi^N\mathcal O^n\), hence \(\varphi=c\,\mathbf 1_{\varpi^N\mathcal O^n}+\sum_{-N\le m
  • Here the orbit \(\{0\}\) is not open, \(C_K\) is not spanned by orbit indicators, and every lower set \(\{X_{m'}:m'\ge m\}\cup\{0\}\) is infinite: Proposition B does not apply as stated, and differences of two central translates replace it. This is the function-level form of Proposition IV, where H1 fails and the level-wise argument uses the central Satake objects alone (IV; the same lattice computation at Iwahori level is 𝔸ⁿ pages of the Iwahori site).

Proposition H (matrices). \(X=M_n\) under \(G=\mathrm{GL}_n\times\mathrm{GL}_n\), \((g_1,g_2)\cdot x=g_1xg_2^{-1}\), \(K=K_n\times K_n\), \(\varphi_0=\mathbf 1_{M_n(\mathcal O)}\). The \(K\)-orbits are the elementary-divisor classes \(a=(a_1\le\dots\le a_n)\), \(a_i\in\mathbb Z\cup\{\infty\}\), the nonsingular ones \(Kt^aK\) being compact open.

  1. Every nonsingular orbit indicator \(\mathbf 1_{Kt^aK}\) lies in \(\mathcal H_K\cdot\varphi_0\): on \(\mathrm{GL}_n(F)\) the Satake transform of \(\varphi_0\) is the standard \(L\)-factor \(\prod_{i=1}^n(1-q^{(n-1)/2}x_i)^{-1}\), whose inverse is a symmetric polynomial.
  2. For \(n=2\), (SG) holds: the nonsingular orbit indicators and the central translates \(\mathbf 1_{\varpi^mM_2(\mathcal O)}=\mathbf 1_{\varpi^mK_2\times K_2}\ast_K\varphi_0\) span \(C_K\).
  3. For \(n\ge3\) the span near the singular locus is not settled here. The Iwahori site's F2, Proposition E, argues by induction along the dominance order, but the lower sets \(\{x:\operatorname{ed}(x)\ge a\}\) there are infinite, so that induction does not decide (SG) for \(M_n\).
Proof

Proof of (1) and (2)

  • Orbits. Smith normal form over \(\mathcal O\); a nonsingular class is compact open in \(M_n(F)\); a locally constant function on \(M_n(F)\) is determined by its restriction to the dense open \(\mathrm{GL}_n(F)\).
  • (1) \(\mathcal H_K=\mathcal H_{K_n}\otimes\mathcal H_{K_n}^{\mathrm{op}}\) acts by \((a\otimes b)\cdot\varphi=a\ast\varphi\ast b\), and on the restriction to \(\mathrm{GL}_n(F)\), which is bi-\(K_n\)-invariant, \(a\ast\varphi\ast b=(a\ast b)\ast\varphi\) by commutativity of \(\mathcal H_{K_n}\); so \((\mathcal H_K\cdot\varphi_0)|_{\mathrm{GL}_n(F)}=\mathcal H_{K_n}\ast\varphi_0|_{\mathrm{GL}_n(F)}\), a locally finite convolution. Extend \(\mathsf S\) termwise to the bi-\(K_n\)-invariant functions supported in some \(\varpi^{-N}M_n(\mathcal O)\): it stays multiplicative and injective (the \(\mathsf S(T_\lambda)\) are triangular in the monomials \(x^\lambda\), and the dominant \(\lambda\) with a given \(|\lambda|\) and \(\lambda_n\ge-N\) are finitely many). For \(t=\operatorname{diag}(\varpi^{a_1},\dots,\varpi^{a_n})\): \(tu\in M_n(\mathcal O)\) iff \(a\ge0\) and \(u_{ij}\in\varpi^{-a_i}\mathcal O\) for \(i
  • (2) A \(K\)-invariant \(\varphi\in\mathcal S(M_2)\) is supported in \(\varpi^{-N}M_2(\mathcal O)\) and constant on the cosets of \(\varpi^NM_2(\mathcal O)\). Subtracting nonsingular orbit indicators, assume \(\varphi\) vanishes on the classes with \(a_2

6 · Rank one on the tree, and \(\mathbb G_m\)

Proposition J (rank one). \(T\subseteq\mathrm{GL}_2\) the diagonal torus.

  1. \(X=\mathrm{GL}_2/T\) (the same function spaces as \(\mathrm{PGL}_2/T\)): \(K\backslash X(F)=\{d\ge0\}\), \(d\) the distance of a vertex of the tree from the apartment of \(T\), and (SG) holds.
  2. \(X=\mathrm{SL}_2/T\), \(q\) odd: (SG) fails.
  3. \(X=\mathrm{GL}_2/\mathrm{Sp}_2\simeq\mathbb G_m\), \(\mathrm{GL}_2\) acting through \(\det\): \(K\backslash X(F)=F^\times/\mathcal O^\times\simeq\mathbb Z\), and (SG) holds.
Proof
  • (1) \(X(F)=\mathrm{GL}_2(F)/T(F)\) by Hilbert 90, and \(K\backslash X(F)=T(F)\backslash\mathrm{GL}_2(F)/K\) is the set of \(T(F)\)-orbits on \(\mathrm{GL}_2(F)/K\), the lattices in \(F^2\); since \(T(F)\) contains the centre, these are the orbits of \(\overline T(F)\), \(\overline T:=T/Z\), on the vertices of the tree of \(\mathrm{PGL}_2(F)\) (Serre, Trees, II.1). Let \(A_T\) be the apartment of \(T\), with vertices \([\varpi^a\mathcal Oe_1\oplus\varpi^b\mathcal Oe_2]\), and \(d(v)\) the distance of a vertex \(v\) from \(A_T\). \(\overline T(F)\) acts on \(A_T\) by all translations, and \(T(\mathcal O)\) fixes \(A_T\) pointwise and acts transitively on the vertices at distance \(d\ge1\) above a given vertex of \(A_T\), which are the \([x:1]\), \(x\in(\mathcal O/\varpi^d)^\times\), by \(x\mapsto u_1u_2^{-1}x\). So \(K\backslash X(F)=\{d\ge0\}\) and \(\varphi_0=\mathbf 1_{\{d=0\}}\). The Hecke operator \(T_1:=T_{(1,0)}=\mathbf 1_{K\operatorname{diag}(\varpi,1)K}\) is the adjacency operator of the tree (a lattice has \(q+1\) sublattices of index \(q\)), so on the orbit indicators $$T_1\mathbf 1_{\{0\}}=2\,\mathbf 1_{\{0\}}+\mathbf 1_{\{1\}},\qquad T_1\mathbf 1_{\{1\}}=(q-1)\,\mathbf 1_{\{0\}}+\mathbf 1_{\{2\}},\qquad T_1\mathbf 1_{\{d\}}=q\,\mathbf 1_{\{d-1\}}+\mathbf 1_{\{d+1\}}\ \ (d\ge2)$$ (a vertex of \(A_T\) has two neighbours on \(A_T\) and \(q-1\) at distance \(1\); a vertex at distance \(d\ge1\) has one neighbour at distance \(d-1\) and \(q\) at distance \(d+1\)). By induction \(\mathbf 1_{\{d\}}\in\mathbb C[T_1]\cdot\varphi_0\) for every \(d\), and these span \(C_K\); the central \(T_{(1,1)}=\mathbf 1_{\varpi K}\) acts trivially. In the language of Proposition B: order by \(d\), \(f_d=T_1^d\). It is also the case \(p=q=1\) of Proposition F, where \(r(x_1+x_2)=y+y^{-1}\) and \(r(x_1x_2)=1\) generate \(\mathbb C[y^{\pm1}]^{y\leftrightarrow y^{-1}}\).
  • (2) \(X(F)=\mathrm{SL}_2(F)/T(F)\) and \(X(\mathcal O)=K\cdot x_0\), \(x_0:=eT\). Let \(u\in\mathcal O^\times\) be a non-square unit, \(s:=\operatorname{diag}(u,1)\in\mathrm{GL}_2(\mathcal O)\), and \(\sigma:gT\mapsto sgs^{-1}T\) the induced automorphism of \(X(F)\). Since \(s\) normalizes \(K\), \(T\) and every double coset \(Kt^\lambda K\) and fixes \(x_0\), \(\sigma\) fixes \(\varphi_0\) and commutes with every Hecke operator, \(\sigma(f\ast_K\varphi)=f\ast_K\sigma(\varphi)\); hence every element of \(\mathcal H_K\cdot\varphi_0\) is \(\sigma\)-invariant. But for \(x\in F\) with \(\operatorname{val}x=-1\) the \(K\)-orbits of $$y:=\begin{pmatrix}1&x\\0&1\end{pmatrix}T\qquad\text{and}\qquad\sigma(y)=\begin{pmatrix}1&ux\\0&1\end{pmatrix}T$$ are distinct: \(k\begin{pmatrix}1&x\\0&1\end{pmatrix}\operatorname{diag}(a,a^{-1})=\begin{pmatrix}1&ux\\0&1\end{pmatrix}\) forces \(k=\begin{pmatrix}a^{-1}&xa^{-1}(ua^2-1)\\0&a\end{pmatrix}\), which lies in \(K\) only if \(a\in\mathcal O^\times\) and \(ua^2\equiv1\bmod\varpi\), i.e. only if \(u\) is a square modulo \(\varpi\). So \(\mathbf 1_{Ky}\notin\mathcal H_K\cdot\varphi_0\). Here (A5) fails: the generic \(B\)-stabilizer is \(\{\pm1\}\), and \(\mathring X(F)\) has \(|F^\times/F^{\times2}|\) orbits of \(B(F)\) (4).
  • (3) \(\det K=\mathcal O^\times\), so \(K\backslash X(F)=F^\times/\mathcal O^\times\), and \(T_\lambda\ast_K\varphi_0=|Kt^\lambda K/K|\cdot\mathbf 1_{\varpi^{\lambda_1+\lambda_2}\mathcal O^\times}\), since \(\det\) is constant on \(Kt^\lambda K\). These span \(C_K=C_c(\mathbb Z)\).

Across the dictionary: over \(\mathbb C\) there are no square classes, and the sheaf-level counterpart of (2) is the failure of both conditions of Proposition III for \(\mathrm{SL}_2/\mathrm{SO}_2\) (split forms: \(Z_{G^\theta}(S)=\{\pm1\}\), and \(\varphi\) doubles); (1) is the case \(p=q=1\) of \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\), and (3) is \(\mathrm{GL}_2/\mathrm{Sp}_2\), where both conditions hold.

7 · Status at the level of functions

Table 1. · (SG) for compactly supported functions. “Under (A)” means under Sakellaridis's hypotheses (A1)–(A5), assumed and not verified. Both levels side by side: the Summary tab.
\(X\)(SG)reason
\(H\) under \(H\times H\)holds\(\varphi_0\) is the unit (D)
\(\mathbb A^n\) under \(\mathrm{GL}_n\)holdslattices, central translates (G)
\(M_n\) under \(\mathrm{GL}_n\times\mathrm{GL}_n\)\(n=2\) holds; \(n\ge3\) open here\(L\)-factor and central translates (H)
\(\mathrm{GL}_2/\mathrm{Sp}_2\simeq\mathbb G_m\)holds\(K\backslash X(F)=F^\times/\mathcal O^\times\) (J)
\(\mathrm{GL}_2/T\), \(\mathrm{PGL}_2/T\)holdsadjacency operator of the tree (J)
\(\mathrm{SL}_2/T\), \(q\) oddfailssquare classes; (A5) fails (J)
\(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\)holds under (A)\(r\) onto: power sums (F)
\(\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}\), \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\)holds under (A) for \(\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}\)\(r\) onto; degree-\(0\) part (F)
\(\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}\)\(n=2\) holds; \(n=3\) under (A); \(n\ge4\) open\(\mathrm{SL}_2\); \(\mathrm{SL}_4/\mathrm{Sp}_4\); \(r\) not computed
\(E_6/F_4\)opennot in Sakellaridis's table; \(r\) not computed
\(\mathrm{GL}_n/\mathrm O_n\), \(\mathrm{PGL}_2/\mathrm{PO}_2\)fails\(X(F)\) has several \(G(F)\)-orbits; translates of \(\varphi_0\) stay in \(G(F)X(\mathcal O)\)
\(\mathrm{SL}_2^3\backslash\mathrm{Sp}_4^2\)fails\(r\) not onto (E(3))

Pattern. Wherever (SG) is decided it holds, except where the orbit set is too large for the Hecke translates to reach it: the square classes of \(\mathrm{SL}_2/T\), the several \(G(F)\)-orbits of \(\mathrm{GL}_n/\mathrm O_n\), and the Weyl-group conjugacy of Sakellaridis's example, failures of (A5) or of the surjectivity of \(r\). The sheaf-level conditions of Proposition III fail for \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\) and \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\) in large rank while the function level holds there under (A), which points to \(\mathcal C^{\mathrm{Sat}}=\mathcal C\) beyond the reach of H5.

Open. \(r\) for the spheres with \(n\ge4\) and for \(E_6/F_4\), and whether (A1)–(A5) hold there; \(M_n\) for \(n\ge3\); any family with (A1)–(A5) checked rather than assumed.

Literature and remarksSakellaridis · Sakellaridis–Venkatesh · Ben-Zvi–Sakellaridis–Venkatesh · Hecke algebras and the tree · the removed section of the paper