Definitions

• Local field, groups, measures

  \(F\) is a non-archimedean local field with ring of integers \(\mathcal O\), uniformizer \(\varpi\) and residue field \(\mathbb F_q\). \(G\supseteq B\supseteq T\) is a split connected reductive group over \(\mathcal O\) with a Borel and a maximal torus, \(W\) its Weyl group and \(\widetilde W\) its extended affine Weyl group. The hyperspecial and Iwahori subgroups and their decompositions:

  $$K=G(\mathcal O),\qquad I=\{k\in K:\ k\bmod\varpi\in B(\mathbb F_q)\},\qquad G(F)=\bigsqcup_{w\in\widetilde W}IwI,\qquad K=\bigsqcup_{w\in W}IwI,$$ $$[K:I]=|G(\mathbb F_q)/B(\mathbb F_q)|=\sum_{w\in W}q^{\ell(w)}.$$

  \(G(F)/K\) and \(G(F)/I\) are discrete; sums over them are the integrals for the Haar measures with \(\operatorname{vol}K=1\), resp. \(\operatorname{vol}I=1\). Coefficients are a commutative ring \(R\); the main case is \(R=\mathbb C\).

• Function spaces and the basic function

  \(X\) is an affine \(G\)-variety over \(\mathcal O\) (the paper's \(X\), spread out). \(X(F)\) is totally disconnected and locally compact, and \(X(\mathcal O)=X(F)\cap\mathcal O^n\) is compact, open and \(K\)-stable. For \(J\in\{K,I\}\) and a \(G(F)\)-set \(Y\) with open stabilizers:

  $$C_c(J\backslash Y):=\{\psi:Y\to R\ \text{locally constant, compactly supported, }\psi(jy)=\psi(y)\},$$ $$\Phi_K:=C_c(K\backslash X(F)),\quad\Phi_I:=C_c(I\backslash X(F)),\quad\mathcal H_K:=C_c(K\backslash G(F)/K),\quad\mathcal H_I:=C_c(I\backslash G(F)/I),\quad\mathcal M:=C_c(I\backslash G(F)/K).$$ $$\varphi_0:=\mathbf 1_{X(\mathcal O)}\in\Phi_K\ \ (\text{the basic function, shadow of }\mathrm{IC}_0),\qquad T_w:=\mathbf 1_{IwI}\in\mathcal H_I\ (w\in\widetilde W),\qquad\mathbf 1_K=\sum_{w\in W}T_w .$$

  \(\mathcal H_K\) and \(\mathcal H_I\) are the spherical and the Iwahori Hecke algebras; \(\mathcal M\) is the shadow of \(\mathcal D_c(I\backslash\mathrm{Gr})\), \(\Phi_K\) of \(\mathcal D_c(K\backslash LX)\) and \(\Phi_I\) of \(\mathcal D_c(I\backslash LX)\).

• Convolution

  For \(f\in C_c(I\backslash G(F)/J)\) and \(\psi\in C_c(J\backslash Y)\), a finite sum over the fibre of the action map:

  \(I\backslash G(F)/J\times J\backslash Y\) \(I\backslash G(F)\overset{J}{\times}Y\) \(I\backslash Y\) \(\tilde p_J\) \(m_J\)

  \(\tilde p_J[g,y]=(gJ,\,Jy)\) and \(m_J[g,y]=gy\); the fibre of \(m_J\) over \(y\) is \(\{[g,g^{-1}y]:g\in G(F)/J\}\).

  $$(f\ast_J\psi)(y):=\sum_{g\in G(F)/J}f(g)\,\psi(g^{-1}y)=\sum_{[g,y']\mapsto y}f(g)\psi(y')\in C_c(I\backslash Y).$$

  The summand depends only on \(gJ\); the sum is finite because \(f\) has finite support in \(G(F)/J\); the result is \(I\)-invariant because \(f\) is. Under the functions–sheaves dictionary this is \(m_{J*}\) of the twisted product. Convolution is associative,

  $$(f\ast_Jf')\ast_{J'}\psi=f\ast_J(f'\ast_{J'}\psi)\qquad(f\in C_c(I\backslash G/J),\ f'\in C_c(J\backslash G/J'),\ \psi\in C_c(J'\backslash Y)),$$

  both sides being \(\sum_{g\in G/J}\sum_{g'\in G/J'}f(g)f'(g^{-1}g')\psi(g'^{-1}y)\). Instances: \(\ast_K:\mathcal M\times\Phi_K\to\Phi_I\); \(\ast_I:\mathcal H_I\times\Phi_I\to\Phi_I\); \(\ast_I:\mathcal H_I\times\mathcal M\to\mathcal M\); \(\ast_K:\mathcal M\times\mathcal H_K\to\mathcal M\); and the products of \(\mathcal H_K\) and \(\mathcal H_I\). A function on \(I\backslash G(F)/K\) cannot be convolved on the right with one on \(I\backslash G(F)/I\): the middle group must agree.

• Inflation, integration over \(K/I\), averaging

  $$q^*:\mathcal M\to\mathcal H_I,\ \ f\mapsto f;\qquad q_*:\mathcal H_I\to\mathcal M,\ \ (q_*h)(g):=\sum_{k\in K/I}h(gk);\qquad p^*:\Phi_K\hookrightarrow\Phi_I .$$ $$q_*q^*f=[K:I]\,f,\qquad q_*h=h\ast_I\mathbf 1_K,\qquad q^*(\mathcal M)=\{h\in\mathcal H_I:\ h\ast_I\mathbf 1_K=[K:I]\,h\}.$$

  When \([K:I]\in R^\times\), the spherical idempotent and the averaging projector are

  $$e_K:=\tfrac{1}{[K:I]}\mathbf 1_K=\tfrac{1}{[K:I]}\sum_{w\in W}T_w\in\mathcal H_I,\qquad e_K\ast_Ie_K=e_K,\qquad\operatorname{Av}:=e_K\ast_I(-):\Phi_I\to\Phi_K,\quad\operatorname{Av}\circ p^*=\mathrm{id}.$$

  \(\operatorname{Av}\psi(x)=\tfrac{1}{[K:I]}\sum_{k\in K/I}\psi(k^{-1}x)\). The same averaging on \(\mathcal M\) lands in \(\mathcal H_K\): \(e_K\ast_If\in\mathcal H_K\) for \(f\in\mathcal M\).

• The three spans

  $$\mathcal S_T^{\mathrm{fun}}:=\mathcal M\ast_K\varphi_0,\qquad\mathcal S_H^{\mathrm{fun}}:=\mathcal H_I\ast_I\varphi_0,\qquad\mathcal S_E^{\mathrm{fun}}:=\operatorname{Im}\big(\mathcal M\otimes_{\mathcal H_K}\Phi_K\to\Phi_I,\ f\otimes\varphi\mapsto f\ast_K\varphi\big)=\mathcal M\ast_K\Phi_K,$$

  all \(R\)-submodules of \(\Phi_I\) (\(f\mapsto f\ast_K\varphi_0\) is linear, so \(\mathcal M\ast_K\varphi_0=\{f\ast_K\varphi_0\}\)); \(\mathcal M\ast_K\Phi_K\) means the span of all \(f\ast_K\varphi\). The thick span of the sheaf statement becomes the linear span: functions have no cones and no retracts. \(\mathcal S_H^{\mathrm{fun}}\) is the cyclic \(\mathcal H_I\)-submodule generated by \(\varphi_0\); \(\mathcal S_T^{\mathrm{fun}}\) is an \(\mathcal H_I\)-submodule by associativity.

Hypotheses

• F1 · Spherical generation for functions.

  The basic function generates the unramified functions on \(X(F)\) under the spherical Hecke algebra: $$\mathcal H_K\ast_K\varphi_0=\Phi_K=C_c(K\backslash X(F)).$$

  Discussion

  The function shadow of H1. It holds in the group case (\(\varphi_0\) is the unit of \(\mathcal H_K\)). For \(X=H\backslash G\) affine homogeneous spherical over a characteristic-zero local field, under the hypotheses of §8 of Sakellaridis, Spherical functions on spherical varieties, Theorem 8.0.2 gives \(C_c^\infty(X)^K\simeq\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X}\) with \(\mathbf 1_{X(\mathfrak o)}\mapsto1\), the spherical Hecke algebra acting through its image under restriction; so F1 holds exactly when the restriction map \(\mathbb C[A^*]^W\to\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X}\) is surjective (Corollary 8.0.4), and Example 8.0.1 there is a case where it fails. The sheaf version is the paper's Proposition 4.30 under its orbit hypotheses, recorded in the Spherical generation workspace.

• F2 · Invertible index.

  The index of the Iwahori subgroup is invertible in the coefficients: $$[K:I]=\sum_{w\in W}q^{\ell(w)}\in R^\times .$$

  Discussion

  Automatic for \(R=\mathbb C\). It is the shadow of S1, \(|W|\in k^\times\): the number \([K:I]\) is the Poincaré polynomial of \(G/B\) at \(q\), and \(|W|\) is its value at \(q=1\), the Euler characteristic.

Proof

Step 1 sets up the dictionary. Steps 2–4 prove (i)–(ii) and identify \([K:I]\). Step 5 proves (iii), step 6 proves (iv) by averaging, steps 7–8 compute the group case, and step 9 records what fails when \([K:I]\) is not invertible.

1. Purpose The statement for functions

   Replace each category by its space of functions and each thick span by a linear span:

   $$\mathcal D_c(I\backslash\mathrm{Gr})\rightsquigarrow\mathcal M,\qquad\mathcal H_{c,I}\rightsquigarrow\mathcal H_I,\qquad\mathcal D_c(K\backslash LX)\rightsquigarrow\Phi_K,\qquad\mathcal D_c(I\backslash LX)\rightsquigarrow\Phi_I,\qquad\mathrm{IC}_0\rightsquigarrow\varphi_0 .$$
   Proof

   • The dictionary sends an \(\ell\)-adic complex to its trace-of-Frobenius function; pushforward along the action map becomes the sum over its fibre and the twisted product becomes the product of values (Zhu, Lemma 5.6.1). The \(\mathcal D\)-module setting has no Frobenius; only the shape of the statement is transported, and nothing below is a proof about sheaves.
   • Retracts of sheaves become divisions by scalars: \(q_*q^*f=[K:I]\,f\) is the shadow of \(q_*q^*\mathcal F\simeq\mathcal F\otimes^!\gamma^!\mathcal A\) with \(\chi(\mathcal A)=|W|\).
2. Result Convolution is a finite sum, and associative

   \(f\ast_J\psi\in C_c(I\backslash Y)\) for \(f\in C_c(I\backslash G(F)/J)\), \(\psi\in C_c(J\backslash Y)\), and \((f\ast_Jf')\ast_{J'}\psi=f\ast_J(f'\ast_{J'}\psi)\).

   Proof

   • \(f(gj)\psi(j^{-1}g^{-1}y)=f(g)\psi(g^{-1}y)\), so the summand is a function of \(gJ\). \(f\) is compactly supported and right \(J\)-invariant with \(J\) open, so its support meets finitely many cosets. \((f\ast_J\psi)(iy)=\sum_gf(g)\psi(g^{-1}iy)=\sum_{g'}f(ig')\psi(g'^{-1}y)=(f\ast_J\psi)(y)\) with \(g=ig'\). A finite sum of translates of \(\psi\) is locally constant with compact support.
   • Associativity: \((f\ast_Jf')(g')=\sum_{g\in G/J}f(g)f'(g^{-1}g')\), hence \(((f\ast_Jf')\ast_{J'}\psi)(y)=\sum_{g'}\sum_gf(g)f'(g^{-1}g')\psi(g'^{-1}y)\); on the other side substitute \(g'=gg''\) in \(\sum_gf(g)\sum_{g''}f'(g'')\psi(g''^{-1}g^{-1}y)\).
3. Method (i) · Fubini along \(G(F)/I\to G(F)/K\)

   For \(h\in\mathcal H_I\) and \(\varphi\in\Phi_K\),

   $$h\ast_Ip^*\varphi=(q_*h)\ast_K\varphi,\qquad\text{so}\qquad\mathcal S_H^{\mathrm{fun}}\subseteq\mathcal S_T^{\mathrm{fun}}\ \text{ over any }R.$$
   Proof

   • The fibres of \(G(F)/I\to G(F)/K\) are the cosets \(gK/I\cong K/I\). Group the sum: \(\sum_{g\in G/I}h(g)\varphi(g^{-1}x)=\sum_{g'\in G/K}\sum_{k\in K/I}h(g'k)\varphi(k^{-1}g'^{-1}x)\), and \(\varphi(k^{-1}y)=\varphi(y)\) because \(\varphi\) is \(K\)-invariant. The inner sum is \((q_*h)(g')\).
   • This is the shadow of \(\mathcal G\ast_Ip^!\mathcal M\simeq(q_*\mathcal G)\ast_K\mathcal M\) (Proposition, step 5): base change along the comparison square is the regrouping of the sum.
4. Method (ii) · Inflate and integrate: the factor \([K:I]\)

   For \(f\in\mathcal M\) and \(\varphi\in\Phi_K\),

   $$(q^*f)\ast_Ip^*\varphi=[K:I]\cdot f\ast_K\varphi,\qquad[K:I]=\sum_{w\in W}q^{\ell(w)}.$$
   Proof

   • Apply (i) to \(h=q^*f\): \((q_*q^*f)(g)=\sum_{k\in K/I}f(gk)=[K:I]\,f(g)\) since \(f\) is right \(K\)-invariant.
   • \(K/I\to G(\mathbb F_q)/B(\mathbb F_q)\) is a bijection (reduction modulo \(\varpi\) is onto \(G(\mathbb F_q)\) by smoothness and Hensel's lemma, and \(I\) is the preimage of \(B(\mathbb F_q)\)). The Bruhat decomposition of \(G/B\) into cells \(C_w\cong\mathbb A^{\ell(w)}\) (Brion, Proposition 1.2.1) is defined over \(\mathbb F_q\), so \(|G(\mathbb F_q)/B(\mathbb F_q)|=\sum_wq^{\ell(w)}\); equivalently \(K=\bigsqcup_{w\in W}IwI\) (Iwahori–Matsumoto, Proposition 2.4 and Theorem 2.5) with \(|IwI/I|=q^{\ell(w)}\) (their Proposition 3.2); in Cartier's survey this is \((K:B)=\sum_{w\in W}q(w)\), formula (22).
   • This is the shadow of the key lemma: \(f=\tfrac{1}{[K:I]}q_*q^*f\) is the retraction, with \([K:I]\) in place of \(\chi(G/B)=|W|\).
5. Result (iii) · The two spans are \(\mathcal H_I\cdot\varphi_0\)

   Assume F2. Then

   $$\mathcal M\ast_K\varphi_0=\mathcal H_I\ast_I\varphi_0=\mathcal H_I\cdot\varphi_0 .$$
   Proof

   • \(\supseteq\) is (i): \(h\ast_I\varphi_0=(q_*h)\ast_K\varphi_0\) with \(q_*h\in\mathcal M\).
   • \(\subseteq\) is (ii): \(f\ast_K\varphi_0=\tfrac{1}{[K:I]}(q^*f)\ast_I\varphi_0\) with \(q^*f\in\mathcal H_I\).
   • Both sides are \(\mathcal H_I\)-submodules of \(\Phi_I\): \(h\ast_I(f\ast_K\varphi_0)=(h\ast_If)\ast_K\varphi_0\) by associativity. The sheaf statement \(\mathcal S_T=\mathcal S_H\) has the same two halves, with the division replaced by the retraction.
6. Result (iv) · Averaging: the image equals the span exactly when F1 holds

   Assume F2. Then \(\mathcal M\ast_K\varphi_0\subseteq\mathcal M\ast_K\Phi_K\), the \(K\)-invariant part of the span is the spherical span,

   $$\operatorname{Av}\big(\mathcal M\ast_K\varphi_0\big)=\mathcal H_K\ast_K\varphi_0,\qquad p^*\Phi_K\subseteq\mathcal M\ast_K\Phi_K,$$

   and \(\mathcal M\ast_K\Phi_K=\mathcal M\ast_K\varphi_0\) if and only if \(\mathcal H_K\ast_K\varphi_0=\Phi_K\).

   Proof

   • \(\Leftarrow\): \(\mathcal M\ast_K\Phi_K=\mathcal M\ast_K(\mathcal H_K\ast_K\varphi_0)=(\mathcal M\ast_K\mathcal H_K)\ast_K\varphi_0=\mathcal M\ast_K\varphi_0\), because \(f\ast_K\mathbf 1_K=f\) for \(f\in\mathcal M\).
   • \(p^*\varphi=\mathbf 1_K\ast_K\varphi\) with \(\mathbf 1_K\in\mathcal M\) (the shadow of the unit-like object \(\delta\) on \(\mathbb BI\)), so \(p^*\Phi_K\subseteq\mathcal M\ast_K\Phi_K\).
   • Averaging: for \(f\in\mathcal M\), \(\operatorname{Av}(f\ast_K\varphi_0)=e_K\ast_I(f\ast_K\varphi_0)=(e_K\ast_If)\ast_K\varphi_0\), and \(e_K\ast_If=\tfrac{1}{[K:I]}\sum_{k\in K/I}f(k^{-1}\,\cdot)\) is left \(K\)-invariant, hence in \(\mathcal H_K\). Conversely \(\mathcal H_K\ast_K\varphi_0\subseteq\mathcal M\ast_K\varphi_0\) is \(K\)-invariant, so fixed by \(\operatorname{Av}\).
   • \(\Rightarrow\): if \(\mathcal M\ast_K\Phi_K=\mathcal M\ast_K\varphi_0\), then for \(\varphi\in\Phi_K\), \(p^*\varphi=f\ast_K\varphi_0\) for some \(f\), and \(\varphi=\operatorname{Av}(p^*\varphi)=(e_K\ast_If)\ast_K\varphi_0\in\mathcal H_K\ast_K\varphi_0\). This is the shadow of step 4 of the Proposition: \(\operatorname{Av}\) is \(p_*\) up to the factor \([K:I]\).
7. Result (v) · The group case: the two-sided ideal \(\mathcal H_Ie_K\mathcal H_I\)

   For \(X=G\) under \(G\times G\): \(\Phi_K=\mathcal H_K\), \(\varphi_0=\mathbf 1_K\) its unit, \(\Phi_I=\mathcal H_I\), and

   $$\mathcal S_T^{\mathrm{fun}}=\mathcal S_H^{\mathrm{fun}}=\mathcal S_E^{\mathrm{fun}}=\mathcal H_Ie_K\mathcal H_I .$$
   Proof

   • The groups are \(K\times K\) and \(I\times I\); \(X(\mathcal O)=K\), \(X(F)=G(F)\). \(\Phi_K=C_c(K\backslash G(F)/K)=\mathcal H_K\) with \(\varphi_0=\mathbf 1_K\), and \(\Phi_I=\mathcal H_I\). F1 holds: \(\mathcal H_K\ast\mathbf 1_K=\mathcal H_K\).
   • \(\mathcal M_{G\times G}=C_c(I\backslash G/K)\otimes C_c(I\backslash G/K)\) acts by \((f_1\otimes f_2)\ast\varphi=f_1\ast_K\varphi\ast_Kf_2^\vee\) with \(f_2^\vee(g)=f_2(g^{-1})\in C_c(K\backslash G/I)\): the sum over \((g_1,g_2)\in(G\times G)/(K\times K)\) of \(f_1(g_1)f_2(g_2)\varphi(g_1^{-1}xg_2)\).
   • Inside \(\mathcal H_I\): \(C_c(I\backslash G/K)=\mathcal H_Ie_K\) and \(C_c(K\backslash G/I)=e_K\mathcal H_I\) (right, resp. left \(K\)-invariance is \(h\ast e_K=h\), resp. \(e_K\ast h=h\)), and \(f_1\ast_Kf_2'=\tfrac{1}{[K:I]}f_1\ast_If_2'\) for such functions. Hence \(\mathcal S_T^{\mathrm{fun}}=\mathcal H_Ie_K\cdot e_K\mathcal H_I=\mathcal H_Ie_K\mathcal H_I\), \(\mathcal S_H^{\mathrm{fun}}=\mathcal H_I\ast_I\mathbf 1_K\ast_I\mathcal H_I=\mathcal H_Ie_K\mathcal H_I\), and \(\mathcal S_E^{\mathrm{fun}}=\mathcal H_Ie_K\mathcal H_Ke_K\mathcal H_I=\mathcal H_Ie_K\mathcal H_I\) since \(e_K\mathcal H_Ke_K=\mathcal H_K\ni\mathbf 1_K\).
   • \(G=\mathrm{SL}_2\): \(W=\{1,s\}\), \(K/I=\mathbb P^1(\mathbb F_q)\), \([K:I]=q+1\), \(\mathbf 1_K=T_1+T_s\), \(e_K=\tfrac{1}{q+1}(T_1+T_s)\); \(q_*(T_1)=\mathbf 1_K\) and \(q_*(T_s)=q\,\mathbf 1_K\), since for \(x\in K\) exactly one point of \(xK/I\) lies in \(I\) and the other \(q\) lie in \(IsI\).
8. Conclusion The ideal is proper: the sign character kills \(e_K\)

   The sign character of the Iwahori–Hecke algebra vanishes on \(\mathbf 1_K\), so

   $$\varepsilon:\mathcal H_I\to R,\ \ T_w\mapsto(-1)^{\ell(w)},\qquad\varepsilon(\mathbf 1_K)=\sum_{w\in W}(-1)^{\ell(w)}=0,\qquad\mathcal H_Ie_K\mathcal H_I\subseteq\ker\varepsilon\subsetneq\mathcal H_I .$$
   Proof

   • In the Iwahori–Matsumoto presentation \(T_s^2=(q-1)T_s+q\) and the braid relations are homogeneous in length, so \(T_s\mapsto-1\) for every simple affine reflection extends to a character \(\varepsilon\) (\((-1)^2=(q-1)(-1)+q\)). It is Borel's special character \(\sigma_0\), equal to \(-1\) on every \(e_s\) (Borel, 3.5), the character of the Iwahori-fixed line of the special representation of Matsumoto and Shalika (Steinberg); the presentation is Iwahori–Matsumoto, Theorem 3.5.
   • \(w\mapsto ws\) is a bijection of \(W\) changing the parity of \(\ell\), so \(\sum_{w\in W}(-1)^{\ell(w)}=0\) when \(W\ne1\); hence \(\varepsilon(e_K)=0\) and \(\varepsilon\) kills the two-sided ideal, while \(\varepsilon(T_1)=1\).
   • So the function-level Iwahori Satake module of the group is a proper ideal of \(\mathcal H_I\): it is the part of \(\mathcal H_I\) seen by representations with a \(K\)-fixed vector, and the sign character (Steinberg) is not in it. This matches the paper's remark that the conjecture is formulated for the Satake subcategory and not for all of \(\mathcal D_c(I\backslash LX)\).
9. Assumption What fails without F2

   The only division in the argument is by \([K:I]\). If \([K:I]=0\) in \(R\), then (ii) reads \((q^*f)\ast_I\varphi=0\), \(e_K\) and \(\operatorname{Av}\) are undefined, and the proofs of \(\mathcal S_T^{\mathrm{fun}}\subseteq\mathcal S_H^{\mathrm{fun}}\) and of (iv) \(\Rightarrow\) break; (i) and (iv) \(\Leftarrow\) survive.

   Proof

   • Example: \(G=\mathrm{SL}_2\), \(R=\mathbb F_\ell\) with \(\ell\mid q+1\). Then \(q^*(f)\ast_I\varphi_0=0\) for every \(f\); the inclusion \(\mathcal S_T^{\mathrm{fun}}\subseteq\mathcal S_H^{\mathrm{fun}}\) still holds for \(f=\mathbf 1_K\) because \(q_*(T_1)=\mathbf 1_K\) gives \(\mathbf 1_K\ast_K\varphi_0=T_1\ast_I\varphi_0\), but no longer by inflation.
   • The sheaf counterpart is S1: the key lemma divides by \(\chi(G/B)=|W|\), the value of \([K:I]=\sum_wq^{\ell(w)}\) at \(q=1\). Whether the inclusions themselves fail without F2 or S1 is not examined here.