Where the relation comes from
Three main events, marked with large dots, each with an obstacle, an advance, and what stayed open. Smaller dots give context. Each event links to the page where its mathematics is used.
The thread. The relation is a relative version of a fact about groups: Hecke operators move Schubert cells upward by at most \(\lambda\). The relative version needs (a) a replacement for Schubert cells in \(LX\), and (b) an inequality controlling how far a Hecke operator can move them. SV supplied (b) for \(p\)-adic points, and Chen–Yi supplied (a) for symmetric \(X\).
Satake computed the algebra of bi-\(K\)-invariant functions on \(G(F)\), \(F\) a \(p\)-adic field. The basis is the double cosets \(Kt^\lambda K\) of the Cartan decomposition \(G(F)=\bigsqcup_{\lambda\in\Lambda^+}Kt^\lambda K\). For split \(G\) the algebra is \(\cong\mathbb C[\Lambda]^W\). I. Satake, Publ. Math. IHÉS 18 (1963). General reductive \(G\): Bruhat–Tits, Publ. Math. IHÉS 41 (1972).
→ On these pages: Cartan decomposition · Hecke operators
For \(\mathrm{SL}_2\), the Bruhat–Tits building is a tree. Its vertices are lattices in \(F^2\) up to scaling. The cosets \(Kt^{(m,-m)}K\) become spheres of radius \(2m\) around a base vertex, and Hecke operators become averaging over spheres. This is the picture used on the example pages. J.-P. Serre, Arbres, amalgames, SL₂, Astérisque 46 (1977); English translation Trees (1980), Ch. II.1.
→ On these pages: the tree · Hecke widget
- Obstacle
- Satake's isomorphism is a computation, and it does not explain why the Langlands dual group appears.
- Advance
- Lusztig replaced the functions by IC sheaves on the affine Grassmannian \(\mathrm{Gr}=G(\!(t)\!)/G[\![t]\!]\). Ginzburg and Mirković–Vilonen made the sheaf category a tensor category equivalent to \(\mathrm{Rep}(G^\vee)\): the geometric Satake equivalence. Two geometric facts carry the combinatorics, and step (1) of the Proposition imitates them: \[ \overline{\mathrm{Gr}}_\lambda=\bigsqcup_{\mu\le\lambda}\mathrm{Gr}_\mu,\qquad D_{\le\lambda_1}\cdot D_{\le\lambda_2}\subset D_{\le\lambda_1+\lambda_2}. \] Key example: \(G=\mathrm{SL}_2\) and the tree. This is the contrast example.
- Unresolved
- What replaces Schubert cells and their closure order when the group is replaced by a \(G\)-variety \(X\)?
G. Lusztig, Astérisque 101–102 (1983). V. Ginzburg, arXiv:alg-geom/9511007. I. Mirković, K. Vilonen, Ann. of Math. 166 (2007). X. Zhu, arXiv:1603.05593, Prop. 2.1.5 and (2.1.17).
→ On these pages: \(D_{\le\lambda}\) and “sizes add” · \(V_\lambda*-\) · group case
They studied perverse sheaves on a finite-dimensional model of the loop space of an affine spherical \(X\) (meromorphic quasimaps). By the Tannakian formalism they obtained a subgroup \(G_X^\vee\subset G^\vee\). This was the first geometric-Satake statement for a \(G\)-variety \(X\) rather than a group. D. Gaitsgory, D. Nadler, Spherical varieties and Langlands duality, Moscow Math. J. 10 (2010), arXiv:math/0611323.
→ On these pages: the loop space LX
- Obstacle
- Harmonic analysis on a spherical variety \(X(F)\), \(F\) a \(p\)-adic field, requires controlling how far a Hecke operator can push the support of a function.
- Advance
- They introduced norm balls \(X_{\succeq\mu}\), cut out by the sizes of \(B\)-eigenfunctions, and proved \(X_{\succeq\mu}\cdot Ka_\lambda K\subset X_{\succeq\mu+\overline\lambda}\) (Lemma 5.5.2). The proof is a one-line submultiplicativity. Step (1) of the Proposition is modelled on this lemma. Key example: \(\mathrm{SL}_2/\mathrm{SO}_2\), where the norm is the largest entry of a symmetric matrix (norm balls).
- Unresolved
- Norm balls are sets of \(F\)-points cut out by inequalities, with no sheaves involved. Whether they are orbit closures in the ind-scheme \(LX\) is not addressed.
Y. Sakellaridis, A. Venkatesh, Periods and harmonic analysis on spherical varieties, Astérisque 396 (2017), §5.5, pp. 85–86; arXiv:1203.0039. Extracted pages: sources/SV17-section-5.5-pp85-86.pdf.
→ On these pages: the norm · norm balls · Lemma 5.5.2
- Obstacle
- \(LX\) is an infinite-dimensional ind-scheme. Its \(L^+G\)-orbits are not double cosets of a group, so the Cartan decomposition gives nothing directly.
- Advance
- For symmetric \(X=G/G^\theta\), the Cartan embedding \(\tau(g)=g\theta(g)^{-1}\) places \(X\) inside \(G\). The orbits are indexed by \(\Lambda_S^+\) (Prop. 17), and the closure order is the dominance order (Thm. 18). They also prove placidity and study the singularities. Key examples: \(\mathrm{SL}_n/\mathrm{SO}_n\) and the group case.
- Unresolved
- General spherical \(X\): a footnote of the draft asks “is this true in general?”. §§4–10 of Chen–Yi assume \(G^\theta\) connected. The Hecke-translate bound of step (1) still has to be assembled from these facts; see the theorem page.
T.-H. Chen, L. Yi, Singularities of orbit closures in loop spaces of symmetric varieties, arXiv:2310.20006 (v4), §2.1.
→ On these pages: orbits · closures · cited facts F1–F2
Ben-Zvi, Sakellaridis and Venkatesh's relative Langlands duality predicts a spectral description of sheaves on \(L^+G\backslash LX\). The draft proves a tamely ramified version, assuming the spherical category is generated by \(V_\lambda * \mathrm{IC}_0\) (“Sat = Full”). That is why it needs the support of \(V_\lambda*\mathrm{IC}_0\) (the Proposition), and therefore its step (1). arXiv:2409.04677.
→ On these pages: the statement