2 · Background · Part I of II
Before the theorem I: the group side and Hecke operators
Steps 1–4 involve only \(G\). Steps 5–6 let \(LG\) act on \(LX=X(F)\) for a \(G\)-variety \(X\) (step 2); the symmetric \(X\) and its orbits are Part II. Each step gives a definition and its name in the literature, then its instance in \(G=\mathrm{SL}_2\) in a blue box. Click a step title to open or close it.
1 The formal disc
Definition
\(\operatorname{Spec}\mathfrak o\) is a small disc and \(\operatorname{Spec}F\) the punctured disc. A negative valuation is a pole at the puncture.
| \(F=\mathbb C(\!(t)\!)\) | formal Laurent series \(\sum_{n\ge n_0}c_nt^n\), \(n_0\in\mathbb Z\) |
| \(\mathfrak o=\mathbb C[\![t]\!]\) | formal power series \(\sum_{n\ge0}c_nt^n\) |
| \(\operatorname{val}:F^\times\to\mathbb Z\) | valuation: \(\operatorname{val}\big(\sum_{n\ge n_0}c_nt^n\big)=n_0\) when \(c_{n_0}\neq0\); \(\operatorname{val}0=\infty\) |
Instance
\(\operatorname{val}(t^{-2}+3)=-2\): a pole of order 2. \(\operatorname{val}(1+t)=0\): a unit of \(\mathfrak o\).2 Loop group and arc group
Definition
For a variety \(Y\): the loop space \(LY\) has points \(Y(F)\), and the arc space \(L^+Y\) has points \(Y(\mathfrak o)\). For a group, \(LG=G(F)\) is the loop group and \(K:=L^+G=G(\mathfrak o)\) is the arc group (the draft writes \(L^+G\)).
Instance
\(LG=\mathrm{SL}_2(F)\): matrices of Laurent series with determinant 1. \(K=\mathrm{SL}_2(\mathfrak o)\): those without poles.3 Cartan decomposition and the affine Grassmannian
Proved (cited)
\[
LG=\bigsqcup_{\lambda\in\Lambda_T^+}Kt^\lambda K,\qquad \mathrm{Gr}:=LG/K,\qquad \mathrm{Gr}_\lambda:=Kt^\lambda K/K,\qquad \overline{\mathrm{Gr}}_\lambda=\bigsqcup_{\mu\le\lambda}\mathrm{Gr}_\mu .
\]
\(\Lambda_T^+\) is the set of dominant coweights, and \(\mu\le\lambda\) means \(\lambda-\mu\) is a non-negative integral sum of simple coroots (the dominance order). \(\mathrm{Gr}\) is the affine Grassmannian, \(\mathrm{Gr}_\lambda\) are its Schubert cells, and \(\overline{\mathrm{Gr}}_\lambda\) are the Schubert varieties [Zhu, Prop. 2.1.5].
Definition, used in the instances below
The Bruhat–Tits tree of \(\mathrm{SL}_2\) [Serre, Trees, II.1]:
- vertices \([M]\): lattices \(M=g\,\mathfrak o^2\subset F^2\) (\(g\in\mathrm{GL}_2(F)\)) up to scaling \(M\sim t^jM\); the base vertex is \(v_0=[\mathfrak o^2]\);
- edges: \([M]\) — \([M']\) when \(tM\subsetneq M'\subsetneq M\) for suitable representatives; the graph is a tree;
- \(\operatorname{dist}(v,w)\) = number of edges on the unique path from \(v\) to \(w\);
- \(\mathrm{SL}_2(F)\) acts by isometries, \(g[M]=[gM]\), and \(K\) is the stabilizer of \(v_0\).
Built and drawn in the example: vertices, edges, distance, spheres, the action.
Instance
\(\Lambda_T^+=\{(m,-m):m\ge0\}\), and \(g\in Kt^{(m,-m)}K\) exactly when the worst pole among the entries of \(g\) has order \(m\). Reason: multiplying by \(K\) on either side cannot make the worst pole worse, nor (by symmetry) better; then evaluate on \(t^{(m,-m)}=\operatorname{diag}(t^m,t^{-m})\). In the tree, \(\mathrm{Gr}_{(m,-m)}\) is the sphere of radius \(2m\) about \(v_0\): \(g\in Kt^{(m,-m)}K\iff\operatorname{dist}(v_0,gv_0)=2m\) [Serre, II.1; example, Part II, step 6].
4 \(D_{\le\lambda}\): the loops of size at most \(\lambda\)
Definition
\[
D_{\le\lambda}:=\bigsqcup_{\mu\le\lambda}Kt^\mu K\ \subset\ LG .
\]
It is the preimage of the Schubert variety \(\overline{\mathrm{Gr}}_\lambda\) under \(LG\to\mathrm{Gr}\). Equivalently it is the closure of one double coset, which the draft writes \(\overline{L^+Gt^\lambda L^+G}\). It is a set of group elements, closed and stable under \(K\) on both sides.
Proved (cited)
Sizes add. \(D_{\le\lambda}\cdot D_{\le\mu}\subset D_{\le\lambda+\mu}\) for \(\lambda,\mu\in\Lambda_T^+\) [Zhu, (2.1.17)].
Instance
Write \(D_{\le m}:=D_{\le(m,-m)}\). Then
\[
D_{\le m}=\{g\in\mathrm{SL}_2(F):\ \min_{i,j}\operatorname{val}(g_{ij})\ge -m\}=\{g:\ \operatorname{dist}(v_0,gv_0)\le 2m\},
\]
the second equality by step 3 (\(\operatorname{dist}\) as in the tree). “Sizes add” is the triangle inequality \(\operatorname{dist}(v_0,ghv_0)\le\operatorname{dist}(v_0,gv_0)+\operatorname{dist}(v_0,hv_0)\).
5 Hecke operators
Definition
Functions (over \(\mathbb F_q(\!(t)\!)\), Satake 1963): the Hecke operator \(T_\lambda\) averages a \(K\)-invariant function over the translates by \(Kt^\lambda K\),
\[
(T_\lambda f)(x)=\sum_{gK\,\subset\, Kt^\lambda K} f(g^{-1}x).
\]
Sheaves (geometric Satake): \(V_\lambda\in\mathrm{Rep}(G^\vee)\) corresponds to the IC sheaf \(\mathrm{IC}_\lambda\) of \(\overline{\mathrm{Gr}}_\lambda\); the draft writes \(V_\lambda\) for both. The Hecke operator is convolution
\[
\mathcal F\ \longmapsto\ V_\lambda*\mathcal F:=a_!\big(\mathrm{IC}_\lambda\,\widetilde\boxtimes\,\mathcal F\big),\qquad a:LG\times^{K}LX\to LX,\ \ [g,x]\mapsto g\cdot x
\]
[Chen–Yi, (21) and §10.2]; the twisted product \(\widetilde\boxtimes\) is recalled in step 6.
Instance
For a \(K\)-invariant function \(f\) on the even vertices of the tree (\(G=\mathrm{SL}_2\) over \(\mathbb F_q(\!(t)\!)\), so that the sum is finite), \((T_{(1,-1)}f)(w)\) is the sum of \(f\) over the \(q(q+1)\) vertices at distance 2 from \(w\).Do not confuse
| \(D_{\le\lambda}\) | Hecke operator \(V_\lambda*-\) | |
|---|---|---|
| what it is | a closed subset of \(LG\) | a functor on sheaves (an operator on functions) |
| remembers | only where the kernel lives | the kernel \(\mathrm{IC}_\lambda\): stalks, shifts, multiplicities |
6 Where a Hecke operator can reach
Proved here from cited definitions
Proposition (support of a Hecke translate). For \(\lambda\in\Lambda_T^+\) and \(\mathcal F\in D(K\backslash LX)\),
\[
\operatorname{Supp}(V_\lambda*\mathcal F)\ \subset\ D_{\le\lambda}\cdot\operatorname{Supp}\mathcal F:=\{g\cdot x:\ g\in D_{\le\lambda},\ x\in\operatorname{Supp}\mathcal F\}.
\]
If \(\operatorname{Supp}\mathcal F\) is closed, so is the right-hand side. This is step (3) of the Proposition.
Literature.
- The Hecke action on \(LX\) is defined in Chen–Yi, arXiv:2310.20006, §8.1.1, eq. (21) (for \(I_0\)), and §10.2 (the same with \(L^+G\)): \(\mathcal M\star\mathcal F:=a_!(\mathcal M\,\widetilde\boxtimes\,\mathcal F)\). Their Appendix A sets up sheaves on placid ind-schemes such as \(LX\).
- Group case (\(X=G\), i.e. sheaves on \(\mathrm{Gr}\)): Zhu, arXiv:1603.05593, (5.1.3) defines \(\mathcal A_1\star\mathcal A_2=m_!(\mathcal A_1\widetilde\boxtimes\mathcal A_2)\). §A.1.4 shows the twisted product of sheaves supported on \(X_i\) and \(Y_j\) lives on \(X_i\widetilde\times Y_j\), and (2.1.17) gives \(m(\mathrm{Gr}_{\le\mu_1}\widetilde\times\mathrm{Gr}_{\le\mu_2})\subset\mathrm{Gr}_{\le\mu_1+\mu_2}\). Together these say \(\operatorname{Supp}(\mathrm{IC}_{\mu_1}\star\mathrm{IC}_{\mu_2})\subset\overline{\mathrm{Gr}}_{\mu_1+\mu_2}\).
- I did not find the relative statement above as a numbered lemma in these sources. The proof below is the group-case argument with \(LX\) in place of \(\mathrm{Gr}\), and uses only the definitions above plus base change for \(a_!\).
Proof
Setup (Chen–Yi (21) with \(L^+G\)). Consider
\[
LG/K\times LX\ \xleftarrow{\ p\times\mathrm{id}\ }\ LG\times LX\ \xrightarrow{\ q\ }\ LG\times^{K}LX\ \xrightarrow{\ a\ }\ LX,\qquad a([g,y])=g\cdot y .
\]
For \(\mathcal M\) on \(\mathrm{Gr}=LG/K\), the twisted product \(\mathcal M\,\widetilde\boxtimes\,\mathcal F\) is the unique complex with \(q^*(\mathcal M\,\widetilde\boxtimes\,\mathcal F)\cong(p\times\mathrm{id})^*(\mathcal M\boxtimes\mathcal F)\), and \(V_\lambda*\mathcal F:=a_!(\mathrm{IC}_\lambda\,\widetilde\boxtimes\,\mathcal F)\).
- External product. Stalks multiply: \((\mathcal M\boxtimes\mathcal F)_{(y,x)}=\mathcal M_y\otimes\mathcal F_x\), which vanishes exactly when one factor does (coefficients in a field). So \(\operatorname{Supp}(\mathcal M\boxtimes\mathcal F)=\operatorname{Supp}\mathcal M\times\operatorname{Supp}\mathcal F\).
- Pullback. A pullback has the same stalk at a point as the original at its image, so \(\operatorname{Supp}(p\times\mathrm{id})^*(\mathcal M\boxtimes\mathcal F)=p^{-1}(\operatorname{Supp}\mathcal M)\times\operatorname{Supp}\mathcal F\).
- Descent along \(q\). \(q\) is a surjective \(K\)-torsor, and \(q^*\) preserves stalks, so \(\operatorname{Supp}(\mathcal M\,\widetilde\boxtimes\,\mathcal F)=q\big(p^{-1}(\operatorname{Supp}\mathcal M)\times\operatorname{Supp}\mathcal F\big)=p^{-1}(\operatorname{Supp}\mathcal M)\times^K\operatorname{Supp}\mathcal F .\) For \(\mathcal M=\mathrm{IC}_\lambda\): \(\operatorname{Supp}\mathrm{IC}_\lambda=\overline{\mathrm{Gr}}_\lambda\) and \(p^{-1}(\overline{\mathrm{Gr}}_\lambda)=D_{\le\lambda}\). So \(\mathcal E:=\mathrm{IC}_\lambda\,\widetilde\boxtimes\,\mathcal F\) is supported on \(D_{\le\lambda}\times^K\operatorname{Supp}\mathcal F\).
- Stalks of \(a_!\). By base change for \(!\)-pushforward (on finite-type pieces of the placid presentation), \((a_!\mathcal E)_x\cong R\Gamma_c\big(a^{-1}(x),\ \mathcal E|_{a^{-1}(x)}\big)\), with fibre \(a^{-1}(x)=\{[g,\,g^{-1}x]:\ gK\in\mathrm{Gr}\}\).
- Conclude. If \((a_!\mathcal E)_x\neq0\), some point \([g,g^{-1}x]\) of the fibre lies in \(\operatorname{Supp}\mathcal E\). Then \(g\in D_{\le\lambda}\) and \(g^{-1}x\in\operatorname{Supp}\mathcal F\), so \(x=g\cdot(g^{-1}x)\in D_{\le\lambda}\cdot\operatorname{Supp}\mathcal F\).
- Closedness. \([g,y]\mapsto(gK,\,g\cdot y)\) identifies \(D_{\le\lambda}\times^K Y\) with a closed subset of \(\overline{\mathrm{Gr}}_\lambda\times LX\) when \(Y\) is closed. Then \(a\) becomes the projection to \(LX\), which is proper because \(\overline{\mathrm{Gr}}_\lambda\) is projective. So the image \(D_{\le\lambda}\cdot Y\) is closed. \(\square\)
Instance
Step 4 is the sheaf version of \((T_\lambda f)(x)=\sum_{gK\subset Kt^\lambda K}f(g^{-1}x)\) from step 5: the stalk at \(x\) only sees the points \(g^{-1}x\). Compare Zhu, Lemma 5.6.1, whose proof uses the same fibre \(m^{-1}(x)=\{(y,y^{-1}x)\}\).So a statement about the sets \(D_{\le\lambda}\cdot\overline{\mathcal O}_\nu\) bounds where Hecke operators can put sheaves. Computing those sets needs the orbits of \(K\) on \(LX\): Part II.