Iwahori level · 2

The criterion and the obstruction

Replace \(K\) by the Iwahori subgroup \(I\). The basic function is still \(\one_{X(\cO)}\), but now it must generate the larger space \(\CI\) over the larger algebra \(\HI\). Sakellaridis's Theorem decides the spherical half of this; the other half is a single obstruction module, detected by representations such as the Steinberg representation.

Sections · arrows: “is used by”

SettingWhat changes when \(K\) shrinks to \(I\)?

[Iwahori–Matsumoto, Prop. 2.4, Thm. 2.5; HKP, §1.2–1.3] Definition

\[K=G(\cO)\xrightarrow{\ \mathrm{ev}\ }G(\Fq),\qquad I:=\mathrm{ev}^{-1}\big(B(\Fq)\big),\qquad\CK:=\Sw(X)^K\ \subseteq\ \CI:=\Sw(X)^I,\] \[\HI:=C_c(I\backslash G(F)/I),\qquad\MIK\ \text{(the Iwahori–spherical bimodule)},\qquad T_w:=\one_{IwI}\ \ (w\in\Wa).\]

Here \(\Wa:=\LT\rtimes W\) is the extended affine Weyl group, \(G(F)=\bigsqcup_{w\in\Wa}IwI\), and \(\ell:\Wa\to\Z_{\ge0}\) its length, with \(|IwI/I|=q^{\ell(w)}\). Inside \(K\),

\[K=\bigsqcup_{w\in W}IwI,\qquad\one_K=\sum_{w\in W}T_w,\qquad[K:I]=|G(\Fq)/B(\Fq)|=\sum_{w\in W}q^{\ell(w)} .\]

Conventions: \(X=G/H\) with left actions; left \(\HI\)-modules.

Convolution. Haar measure on \(G(F)\) with \(\vol I=1\). For \(f\in\HI\) and \(\varphi\in\CI\),

\[(f\ast\varphi)(x):=\int_{G(F)}f(g)\,\varphi(g^{-1}x)\,dg=\sum_{g\in G(F)/I}f(g)\,\varphi(g^{-1}x).\]

The same formula with \(\vol K=1\), summed over \(G(F)/K\), is written \(\ast_K\); this is the convolution of the spherical level. The two differ by \([K:I]\): \(f\ast g=[K:I]\,f\ast_Kg\) whenever \(f\) is right \(K\)-invariant and \(g\) left \(K\)-invariant, since the \([K:I]\) cosets \(hI\) inside one \(hK\) contribute equal terms.

Well defined; \(\CI\) is a unital \(\HI\)-module
  • The summand depends only on \(gI\), and finitely many cosets contribute. \((f\ast\varphi)(ix)=(f\ast\varphi)(x)\) for \(i\in I\), by the substitution \(g\mapsto i^{-1}g\). \(\supp(f\ast\varphi)\subseteq\supp f\cdot\supp\varphi\).
  • The same formula makes \(\HI\) an associative algebra with unit \(T_e=\one_I\), and \((f_1\ast f_2)\ast\varphi=f_1\ast(f_2\ast\varphi)\). With \(\vol K=1\) instead, \(\CK\) is a unital \(\HK\)-module with unit \(\one_K\) [Cartier, §1.3–1.4; HKP, §1.2–1.3].
  • For \(f\in\HI\), \(\varphi\mapsto f\ast\varphi\) is \(\pi(f)=\int f(g)\pi(g)\,dg\) for the representation \((\pi(g)\varphi)(x)=\varphi(g^{-1}x)\) of \(G(F)\) on \(\Sw(X)\). So every \(G(F)\)-map \(\Sw(X)\to V\) restricts to an \(\HI\)-map \(\CI\to V^I\).

The three questions about the same \(\phiz\):

\[\text{(SG)}\ \ \HK\ast_K\phiz=\CK;\qquad\HI\ast\phiz\subseteq\CI;\qquad\text{(IG)}\ \ \HI\ast\phiz=\CI .\]

The spherical idempotentHow does \(\HK\) sit inside \(\HI\)?

(the spherical idempotent) [HKP, §4.1; Iwahori–Matsumoto, Prop. 2.4, Thm. 2.5, Prop. 3.2] Proved (cited)

Put \(\eK:=[K:I]^{-1}\one_K=[K:I]^{-1}\sum_{w\in W}T_w\in\HI\).

  1. \(\eK\ast\eK=\eK\). For \(f\in\HI\) and \(\varphi\in\CI\), right convolution averages \(f\) over \(xK\), left convolution averages \(\varphi\) over \(Kx\): \[(f\ast\eK)(x)=\frac1{[K:I]}\sum_{k\in K/I}f(xk),\qquad(\eK\ast\varphi)(x)=\frac1{[K:I]}\sum_{k\in K/I}\varphi(k^{-1}x).\]
  2. \(\HI\ast\eK=\MIK\) and \(\eK\ast\CI=\CK\). Both maps are projections: \(f\ast\eK=f\) for right \(K\)-invariant \(f\), and \(\eK\ast\varphi=\varphi\) for \(K\)-invariant \(\varphi\).
  3. \(\eK\HI\eK:=\{\eK\ast f\ast\eK:\ f\in\HI\}\) is the space of bi-\(K\)-invariant functions in \(\HI\); it is closed under \(\ast\), with unit \(\eK\), and \[\Psi:(\HK,\ast_K)\xrightarrow{\ \sim\ }(\eK\HI\eK,\ast),\qquad\Psi(f)=[K:I]^{-1}f,\qquad\Psi^{-1}(h)=[K:I]\,h,\qquad\Psi(\one_K)=\eK,\] is an isomorphism of algebras. For \(f\in\HK\) and \(\varphi\in\CK\), \(\Psi(f)\ast\varphi=f\ast_K\varphi\); hence \(\eK\HI\eK\ast\varphi=\HK\ast_K\varphi\).
Proof
  1. Averaging. In \((f\ast\eK)(x)=\sum_{g\in G(F)/I}f(g)\,\eK(g^{-1}x)\) only the cosets \(gI\subseteq xK\) contribute, each with weight \([K:I]^{-1}\), and they are the \(xkI\), \(k\in K/I\). Likewise \((\eK\ast\varphi)(x)=[K:I]^{-1}\sum_{k\in K/I}\varphi(k^{-1}x)\). The average over \(xK\) is right \(K\)-invariant in \(x\) and equals \(f(x)\) when \(f\) already is; the same on the left for \(\varphi\). For \(f=\eK\), which is right \(K\)-invariant, this reads \(\eK\ast\eK=\eK\).
  2. Images. By 1, \(f\ast\eK\) is right \(K\)-invariant, and a right \(K\)-invariant \(f\) equals \(f\ast\eK\); so \(\HI\ast\eK\) is the right \(K\)-invariant part of \(\HI\), which is \(\MIK\). In the same way \(\eK\ast\CI\) is the \(K\)-invariant part of \(\CI\), which is \(\CK\).
  3. The corner. \(\eK\ast f\ast\eK\) is bi-\(K\)-invariant by 1, and a bi-\(K\)-invariant \(f\) equals \(\eK\ast f\ast\eK\); so \(\eK\HI\eK\) is the space of bi-\(K\)-invariant functions, which as a vector space is \(C_c(K\backslash G(F)/K)=\HK\). It is closed under \(\ast\) because \(\eK\) is idempotent, \((\eK\ast f\ast\eK)\ast(\eK\ast g\ast\eK)=\eK\ast(f\ast\eK\ast g)\ast\eK\), and \(\eK\) is a two-sided unit for it. For bi-\(K\)-invariant \(f,g\), \(f\ast g=[K:I]\,f\ast_Kg\) (convolution). Hence \(\Psi(f)\ast\Psi(g)=[K:I]^{-2}f\ast g=[K:I]^{-1}f\ast_Kg=\Psi(f\ast_Kg)\), and \(\Psi(\one_K)=\eK\). The same count gives \(f\ast\varphi=[K:I]\,f\ast_K\varphi\) for \(f\in\HK\), \(\varphi\in\CK\), that is \(\Psi(f)\ast\varphi=f\ast_K\varphi\).
  4. Where these facts are in the literature. \(K=\bigsqcup_WIwI\) with \(|IwI/I|=q^{\ell(w)}\), hence \(\one_K=\sum_WT_w\) and \([K:I]=\sum_Wq^{\ell(w)}\): Iwahori–Matsumoto, Prop. 2.4, Thm. 2.5, Prop. 3.2; Cartier, formula (22). The idempotent and the corner \(\HK=\eK\HI\eK\): [HKP, §4.1]. The same facts located line by line: the note's functions page, steps 4–5 and 7.

So \(\HK\) is the corner \(\eK\HI\eK\), with unit \(\eK\ne T_e\). By Bernstein, the centre of \(\HI\) is \(\C[\As]^W\) again, compatibly with Satake: \(\HK\simeq\C[\As]^W\simeq Z(\HI)\) [HKP, §2.3, §4.6]. So the centre acts on \(\CI\), and on the corner \(\eK\ast\CI=\CK\) through the action the Theorem describes.

The criterionHow do (IG) and (SG) relate?

(what \(\phiz\) generates; workspace “Lemma A”) Proved here

\[\HI\ast\phiz=(\HI\ast\eK)\ast\phiz=\MIK\ast\phiz :\]

only the Iwahori–spherical translates of \(\phiz\) are needed.

Proof

\(\phiz\) is \(K\)-invariant, so \(\phiz=\eK\ast\phiz\) and \(f\ast\phiz=(f\ast\eK)\ast\phiz\) (, part 2); then \(\HI\ast\eK=\MIK\), by the same part.

(criterion; workspace “Proposition B”) Proved here

\[\HI\ast\phiz=\CI\iff\Big(\HK\ast_K\phiz=\CK\ \text{ and }\ \HI\ast\CK=\CI\Big).\]

Moreover \(\HI\ast\phiz=\HI\ast\CK\) iff (SG) holds, and then both equal \(\MIK\ast\CK\), the image of \(\MIK\otimes_{\HK}\CK\to\CI\). In particular (IG) implies (SG).

Proof
  1. (⇒) Apply \(\eK\), using \(\phiz=\eK\ast\phiz\) and : \(\CK=\eK\ast\CI=\eK\ast\HI\ast\phiz=(\eK\HI\eK)\ast\phiz=\HK\ast_K\phiz\). And \(\phiz\in\CK\) gives \(\CI=\HI\ast\phiz\subseteq\HI\ast\CK\).
  2. (⇐) \(\HI\ast\CK=\HI\ast(\HK\ast_K\phiz)=\HI\ast(\eK\HI\eK)\ast\phiz\subseteq\HI\ast\phiz\), so \(\CI=\HI\ast\CK\subseteq\HI\ast\phiz\).
  3. The “moreover”, (⇐): under (SG), step 2 gives \(\HI\ast\CK\subseteq\HI\ast\phiz\), and \(\phiz\in\CK\) gives the other inclusion. For \(\varphi\in\CK\), \(f\ast\varphi=(f\ast\eK)\ast\varphi\), which gives the form \(\MIK\ast\CK\).
  4. The “moreover”, (⇒): if \(\HI\ast\phiz=\HI\ast\CK\), apply \(\eK\): \(\HK\ast_K\phiz=(\eK\HI\eK)\ast\phiz=\eK\ast\HI\ast\phiz=\eK\ast\HI\ast\CK\supseteq\eK\ast\CK=\CK\).

The obstruction moduleWhat is left over once (SG) holds?

Definition

\[\QX:=\CI\big/\HI\ast\CK,\qquad Q_{X,\phiz}:=\CI\big/\HI\ast\phiz,\qquad0\to\frac{\HI\ast\CK}{\HI\ast\phiz}\to Q_{X,\phiz}\to\QX\to0 .\]

\(\eK\QX=0\), because \(\eK\ast\CI=\CK\subseteq\HI\ast\CK\).

\(Q_{X,\phiz}=0\) is (IG). \(\QX\ne0\) says more: no \(K\)-invariant function at all generates \(\CI\). Under (SG) the two agree (the criterion's “moreover”). For split symmetric \(X\), \(\CI\) is finitely generated over \(\HI\) [Shtotland 2024, Thm. 5.9], so \(\QX\ne0\) is then equivalent to a witness representation as in , with \(\Sw(X)\) replaced by the subrepresentation generated by \(\CI\) (the last step of its proof).

(obstruction principle; workspace “Lemma C”) Proved here

Let \(f:\CI\to Q\) be a nonzero \(\HI\)-linear map.

  1. If \(f(\phiz)=0\), then \(\HI\ast\phiz\subseteq\ker f\subsetneq\CI\).
  2. If \(\eK Q=0\), then \(f\) kills \(\HI\ast\CK\); so \(f(\phiz)=0\), part 1 applies, and \(f\) factors through \(\QX\).
  3. Conversely, if (SG) holds and \(\HI\ast\phiz\ne\CI\), then the projection \(\CI\to\QX\) is such a map.
Proof
  1. \(f(a\ast\phiz)=a\ast f(\phiz)=0\) for \(a\in\HI\).
  2. \(f(\CK)=f(\eK\ast\CI)=\eK\ast f(\CI)\subseteq\eK Q=0\); then \(\HI\)-linearity.
  3. Under (SG), \(\HI\ast\phiz=\HI\ast\CK\) (), so \(\QX=Q_{X,\phiz}\ne0\), and \(\eK\QX=0\).

(idempotent ideals) Proved here

For a unital algebra \(A\) and an idempotent \(e\in A\): \(AeA=A\) iff \(eL\ne0\) for every nonzero \(A\)-module \(L\).

Proof
  1. (⇒) If \(eL=0\), then \(AeA\cdot L=Ae\cdot(AL)\subseteq A\cdot eL=0\); with \(AeA=A\) this gives \(L=A\cdot L=0\).
  2. (⇐) If \(AeA\ne A\), then \(L:=A/AeA\ne0\), and \(e\cdot(a+AeA)=ea+AeA=0\), since \(ea=1\cdot e\cdot a\in AeA\).

With \(A=\HI\) and \(e=\eK\): the modules killed by \(\eK\) are exactly what separates \(\HI\eK\HI\) from \(\HI\), and \(\QX\) is the universal one reached from \(\CI\). At the spherical level \(e\) is the unit, and nothing is killed.

(representation witnesses; workspace “Corollary D”) [Borel 1976, Thm. 4.10, Cor. 4.11] Proved here, from cited results

Let \(\pi\) be an irreducible smooth representation of \(G(F)\) with

\[\pi^I\ne0,\qquad\pi^K=0,\qquad\Hom_{G(F)}\big(\Sw(X),\pi\big)\ne0 .\]

Then \(\QX\ne0\) and \(\HI\ast\phiz\subsetneq\CI\): (IG) fails for every \(K\)-invariant basic function. For \(X=G/H\) with \(H\) reductive, \(\Hom_{G(F)}(\Sw(X),\pi)\ne0\) iff the contragredient \(\tilde\pi\) carries a nonzero \(H(F)\)-invariant functional (\(\tilde\pi\) is \(H\)-distinguished).

Proof
  1. The map. A nonzero \(G(F)\)-map \(f:\Sw(X)\to\pi\) is onto, since \(\pi\) is irreducible. Invariants under the compact open \(I\) are exact (\(\pi(T_e)\) projects onto them), so \(f(\CI)=\pi^I\ne0\), and \(f|_{\CI}\) is \(\HI\)-linear (setting, last bullet). \(\eK\pi^I=\pi^K=0\), so , part 2: \(f\) factors through \(\QX\ne0\) and kills \(\phiz\).
  2. Distinction. The smooth dual of \(\Sw(G/H)\) is \(C^\infty(G/H)\): a distribution invariant under a compact open \(J\) is a function on the discrete set \(J\backslash G/H\). So \(\Hom_G(\Sw(X),\pi)=\Hom_G(\tilde\pi,C^\infty(G/H))=\Hom_H(\tilde\pi,\C)\) for admissible \(\pi\) and unimodular \(H\) (Frobenius reciprocity).
  3. Every obstruction has this shape. Borel's dictionary [Borel 1976, Thm. 4.10, Cor. 4.11]: \(\pi\mapsto\pi^I\) is an equivalence between representations generated by their \(I\)-fixed vectors and \(\HI\)-modules, and \(\pi^K=\eK\pi^I\). So the irreducible \(\HI\)-modules killed by \(\eK\) are exactly the \(\pi^I\) above. If \(\QX\) has an irreducible quotient (for instance when \(\CI\) is finitely generated over the Noetherian algebra \(\HI\)), the obstruction is of this shape, with \(\Sw(X)\) replaced by the subrepresentation it generates from \(\CI\).

The Steinberg representation of \(\GL_2\)What does a module killed by \(\eK\) look like?

(\(\St\) of \(\GL_2\))

\[\St:=C^\infty(\mathbb P^1(F))\big/\C\one,\qquad\St^K=0,\qquad\St^I=\C\,[\one_{U_1}],\qquad T_s=-1\ \text{on}\ \St^I .\]
Proof
  1. \(K\) is transitive on \(\mathbb P^1(F)\), so \(C^\infty(\mathbb P^1)^K=\C\one\) and \(\St^K=0\).
  2. \(I\) has two orbits \(U_1,U_2\) on \(\mathbb P^1(F)\) (orbits), so \(C^\infty(\mathbb P^1)^I=\C\one_{U_1}\oplus\C\one_{U_2}\) and \(\St^I\) is the line of \([\one_{U_1}]=-[\one_{U_2}]\).
  3. \(\one_K=T_e+T_s\), \((T_s+1)(T_s-q)=0\); \(T_s=q\) on the \(K\)-fixed line, \(-1\) on a complement, and \(\eK=0\) on \(\St^I\). So \(T_s=-1\) there.

\(K\)-averaging forgets the difference between the two \(I\)-orbits of \(\mathbb P^1(F)\); \(\St\) is where that difference survives once the constants are removed. Every failure of (IG) on the group and \(\GL_2\) page and the symmetric page is witnessed by \(\St\), a twist, or a relative of it.