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Hecke translates of orbit closures

The Proposition below bounds the support of \(V_\lambda*\mathrm{IC}_0\). Its key step (1): move an orbit closure in \(LX\) by all loops of size at most \(\lambda\); which orbit closure contains the result?

1 The Proposition

Statement explained on these pages

Proposition (support of convolution). Let \(X=G/G^\theta\) be a symmetric variety, \(\lambda\in\Lambda_T^+\) with \(-\theta\lambda\in\Lambda_T^+\), and \(\nu\in\Lambda_S^+\). Then:

\[ \begin{aligned} &(1)\ \ D_{\le\lambda}\cdot\overline{\mathcal O}_\nu\subset\overline{\mathcal O}_{\nu+r(\lambda)},\ \ \text{and } t^\lambda\cdot x_\nu\in\mathcal O_{\nu+r(\lambda)};\\ &(2)\ \ \operatorname{Supp}\mathrm{IC}_0=\mathcal O_0=L^+X;\\ &(3)\ \ \operatorname{Supp}(V_\lambda*\mathcal F)\subset D_{\le\lambda}\cdot\operatorname{Supp}\mathcal F\ \text{ for every }\mathcal F;\\ &\text{hence}\ \ \operatorname{Supp}(V_\lambda*\mathrm{IC}_0)\subset\overline{\mathcal O}_{r(\lambda)} . \end{aligned} \]

Step (1) is the orbit closure relation and the subject of these pages: proof on the theorem page, (3) in Part I, step 6. The draft's earlier wording of (1), in Sakellaridis–Venkatesh's conventions, is translated on the norm-balls page.

Spaces and groups

\(G\) connected reductive over \(\mathbb C\); \(F=\mathbb C(\!(t)\!)\), \(\mathfrak o=\mathbb C[\![t]\!]\); \(LG=G(F)\supset K=L^+G=G(\mathfrak o)\).
\(X=G/G^\theta\) with base point \(x_0=eG^\theta\); \(LX=X(F)\supset L^+X=X(\mathfrak o)\).
\(T\) a \(\theta\)-stable maximal torus containing a maximal \(\theta\)-split torus \(S\). \(\Lambda_T=\operatorname{Hom}(\mathbb G_m,T)\) and \(\Lambda_S=\operatorname{Hom}(\mathbb G_m,S)\) are the coweight lattices; \(\Lambda_T^+\) the dominant ones, \(\Lambda_S^+=\Lambda_S\cap\Lambda_T^+\); \(\le\) the dominance order.
\(D_{\le\lambda}\) \(=\bigsqcup_{\mu\le\lambda}Kt^\mu K\subset LG\): a set of loops, those of size \(\le\lambda\). It is where the Hecke operator can reach, not the operator itself.
\(\mathcal O_\nu=K\cdot x_\nu\subset LX\) and \(\overline{\mathcal O}_\nu=\bigsqcup_{\mu\le\nu}\mathcal O_\mu\); \(\mathrm{IC}_0\) is the IC sheaf of the closed orbit \(\mathcal O_0=L^+X\).

Maps, with domain and codomain

Table 1. map and from → to and rule
map	from → to	rule
\(\theta\)	\(G\to G\)	an involution; \(G^\theta\) its fixed points
action	\(LG\times LX\to LX\)	\((g,x)\mapsto g\cdot x\); so \(D_{\le\lambda}\cdot Y=\{g\cdot y: g\in D_{\le\lambda},\,y\in Y\}\)
\(t^{(\cdot)}\)	\(\Lambda_T\to T(F)\subset LG\)	\(\lambda\mapsto t^\lambda=\lambda(t)\), the coweight evaluated at \(t\in F^\times\)
\(\tau\)	\(X\to G\), and \(LX\to LG\)	\(gx_0\mapsto g\,\theta(g)^{-1}\); \(x_\nu\in LX\) is the point with \(\tau(x_\nu)=t^\nu\)
\(r\)	\(\Lambda_T\to\Lambda_S\)	\(\lambda\mapsto\lambda-\theta\lambda\); lands in \(\Lambda_S^+\) when \(-\theta\lambda\) is dominant
\(V_\lambda*-\)	\(\mathrm{Shv}(K\backslash LX)\to\mathrm{Shv}(K\backslash LX)\)	the Hecke operator: convolution with \(\mathrm{IC}(\overline{\mathrm{Gr}}_\lambda)\) (geometric Satake)
\(\operatorname{Supp}\)	\(\mathrm{Shv}(K\backslash LX)\to\{\text{closed } K\text{-stable subsets of } LX\}\)	where the sheaf is non-zero

In \(X=\mathrm{SL}_2/\mathrm{SO}_2\) with \(\lambda=(m,-m)\), \(\nu=(a,-a)\): \(r(\lambda)=(2m,-2m)\), and (1) reads \(D_{\le m}\cdot\overline{\mathcal O}_a\subset\overline{\mathcal O}_{a+2m}\). All symbols, with instances: notation page.

2 The easiest case at a glance

For X = SL2/SO2, points of LX are lines in the Bruhat–Tits tree; the orbits O0, O1, O2 are the lines at distance 0, 1, 2 from the base vertex v0 — **Figure 1.** For \(X=\mathrm{SL}_2/\mathrm{SO}_2\), points of \(LX\) are lines in the Bruhat–Tits tree (definition), and orbits of \(K\) are these lines sorted by their distance to the base vertex \(v_0\); a Hecke translate by \(D_{\le m}\) moves \(v_0\) by \(2m\), so \(\overline{\mathcal O}_a\mapsto\overline{\mathcal O}_{a+2m}\) (try it).

Why lines? The stabilizer of \(I\in LX\) is \(\mathrm{SO}_2(F)\), a split torus. A split torus fixes no vertex of the tree but preserves exactly one line, its apartment. So a point of \(LX\) is a line, just as a point of \(\mathrm{Gr}=G/K\) is a vertex because \(K\) fixes \(v_0\). Since \(K=L^+G\) fixes \(v_0\), a \(K\)-orbit of lines is determined by the distance to \(v_0\). Proof and literature: Example, Part I, step 4.

3 Map of the pages

Figure 2. The pages and how they depend on each other; numbers give a reading order. Short route (about 30 minutes): 2 → 3 → 4 → 5 → 6 → 9 → 10. Comparing with Sakellaridis–Venkatesh: 8 → 9.

4 After reading, you should be able to

Picture a point of \(LX\) for \(X=\mathrm{SL}_2/\mathrm{SO}_2\) as a line in the Bruhat–Tits tree, and read off its orbit as a distance.
State the closure relation for symmetric \(X\), prove it from three cited facts, and see it as a triangle inequality in the example.
Explain why odd orbits are never top orbits in the example, and why that matters for an induction over top orbits such as the draft's Sat = Full argument.

5 Status and labels

Status: (1) proved here for symmetric \(X\) from three cited facts; (2) holds since \(\mathcal O_0\) is closed; (3) proved in Part I. A TODO in the draft.
Prerequisites: Reductive groups, coweights, the dominance order. Sheaves appear only in Part I, steps 5–6.
Labels: DefinitionProved (cited)Proved hereHeuristicOpen / gap