N ยท Reference
Notation
One symbol, one meaning, on every page. The running instance is \(X=\mathrm{SL}_2/\mathrm{SO}_2\) with \(\lambda=(m,-m)\) and \(\nu=(a,-a)\).
1 Disc and loops
| symbol | meaning | instance |
|---|---|---|
| \(k,\ q\) | the coefficient field; \(k=\mathbb C\) for the Proposition, \(k=\mathbb F_q\) (\(q\) elements) in the pictures of the tree (example, Part II) | \(\mathbb F_3\), \(q=3\) |
| \(F,\ \mathfrak o,\ \operatorname{val}\) | \(k(\!(t)\!)\), \(k[\![t]\!]\), order of vanishing (I.1); \(k=\mathbb C\) unless a page says otherwise | \(\operatorname{val}(t^{-2}+3)=-2\) |
| \(\kappa\) | an element of \(K\) (not to be confused with the field \(k\)) | \(\begin{pmatrix}0&1\\-1&0\end{pmatrix}\) |
| \(LY,\ L^+Y\) | \(Y(F)\), \(Y(\mathfrak o)\) (I.2) | |
| \(K\) | \(L^+G=G(\mathfrak o)\); the draft writes \(L^+G\) | \(\mathrm{SL}_2(\mathfrak o)\) |
2 Group side
| symbol | meaning | instance |
|---|---|---|
| \(\Lambda_T^+,\ \le\) | dominant coweights, dominance order (I.3) | \((m,-m)\), \(m\ge0\) |
| \(\mathrm{Gr},\ \mathrm{Gr}_\lambda,\ \overline{\mathrm{Gr}}_\lambda\) | affine Grassmannian, Schubert cell, Schubert variety | even vertices; sphere of radius \(2m\) |
| \(D_{\le\lambda}\) | \(\bigsqcup_{\mu\le\lambda}Kt^\mu K\subset LG\), a set; the draft's \(\overline{L^+Gt^\lambda L^+G}\) (I.4) | \(D_{\le m}\): poles of order \(\le m\) |
| \(V_\lambda*-\) | Hecke operator: convolution with \(\mathrm{IC}_\lambda\), the sheaf geometric Satake attaches to \(V_\lambda\) (I.5) |
3 Symmetric side
| symbol | meaning | instance |
|---|---|---|
| \(\theta,\ X,\ x_0\) | involution, \(G/G^\theta\), base point (II.1) | \((g^t)^{-1}\); \(\{A=A^t,\det A=1\}\); \(I\) |
| \(\tau\) | Cartan embedding \(gx_0\mapsto g\theta(g)^{-1}\) | \(\tau(A)=A\) |
| \(S,\ \Lambda_S^+\) | maximal \(\theta\)-split torus, \(\Lambda_S\cap\Lambda_T^+\) | \(S=T\); \((a,-a)\) |
| \(\mathcal O_\nu,\ x_\nu\) | \(K\)-orbit \(K\cdot x_\nu\), \(\tau(x_\nu)=t^\nu\) (II.3) | \(\mathcal O_a\ni\operatorname{diag}(t^a,t^{-a})\) |
| \(\overline{\mathcal O}_\nu\) | orbit closure \(=\bigsqcup_{\mu\le\nu}\mathcal O_\mu\) (II.5) | pole order \(\le a\) |
| \(r(\lambda),\ \lambda^\theta\) | \(\lambda-\theta\lambda\); \((-\theta\lambda)^+\) (theorem) | \(2m\); \(m\) |
| \(\mathrm{IC}_0\) | IC sheaf of the closed orbit \(\mathcal O_0=L^+X\) |
4 Tree (instance only)
| symbol | meaning |
|---|---|
| \(v_0\) | base vertex, the lattice class \([\mathfrak o^2]\) (example, Part II) |
| \(\ell_A\) | line joining the isotropic points of \(q_A(v)=v^tA^{-1}v\) (example, Part III) |
| \(\operatorname{dist}\) | number of edges on the path (tree, Part I); \(\operatorname{dist}(v_0,\ell_A)=a\) for \(A\in\mathcal O_a\) |
5 Norm balls and the draft's earlier wording
| symbol | meaning | instance |
|---|---|---|
| \(\mathcal X(X)^+,\ f_\chi\) | weights of \(B\)-eigenfunctions in \(k[X]\); a chosen eigenfunction (norm balls 3) | \(\mathbb N\chi\), \(\chi=2\varpi\); \(f_{n\chi}=A_{22}^n\) |
| \(X^*(T),\ \Lambda_T,\ \langle\lambda,\chi\rangle\) | characters, cocharacters; \(\chi(\lambda(s))=s^{\langle\lambda,\chi\rangle}\) (norm balls 3a) | \(\langle(m,-m),k\varpi\rangle=mk\) |
| \(\mathcal X(X)\) | weights of \(B\)-eigenfunctions in \(k(X)\); \(\subset X^*(T)\) (norm balls 3c) | \(2\mathbb Z\varpi\) |
| \(\mathfrak a_X,\ \overline\lambda\) | \(\operatorname{Hom}_{\mathrm{Grp}}(\mathcal X(X),\mathbb Q)\); image of \(\lambda\), \(\overline\lambda(\chi)=\langle\lambda,\chi\rangle\) | \(\mathbb Q\); \(\overline{(m,-m)}=2m\) |
| \(N_\chi\) | norm \(-\min_{{\kappa}\in K}\operatorname{val}f_\chi({\kappa}^{-1}x)\) (norm balls 4); SV's \(\log_q\|x\|_\chi\) | pole order |
| \(X_{\succeq\mu}\) | norm ball (norm balls 6); the draft's \(X_{\ge\mu}\) | \(\overline{\mathcal O}_a\) for \(\mu=-a\) |
| \(\lambda\in\Lambda_G^-\) | the draft's anti-dominant convention (SV's \(a_\lambda\)) | \((-m,m)\) |