Lemma 6.3 · $\mathrm{IC}(Y_n)=\nu_{n*}k$ with stalk rank $n+1-d(v_0,\gamma)$

$\mathrm{IC}(Y_n)=\nu_{n*}k$ with stalk rank $n+1-d(v_0,\gamma)$

1. Fibres. \(\nu_n[g,x]=\gamma\) iff \(g^{-1}\gamma\in L^+X\) iff \(\gamma\ni g v_0=:u\) (Setup 6.1(a)), with \(u\in\mathrm{Gr}_{\le n}\). If \(d(v_0,\gamma)=m\) with projection point \(p\in\gamma\), the vertices of \(\gamma\) at distance \(m+j\) from \(v_0\) are the \(1\) (if \(j=0\)) or \(2\) (if \(j>0\)) points at distance \(j\) from \(p\); those with \(m+j\le n\), \(m+j\equiv n\) number \(n-m+1\) (if \(m\le n\)), and \(0\) otherwise. So the image is \(Y_n\) and fibres are finite; over \(\{d(v_0,\gamma)=n\}\) the fibre is the single point \(p\). Checked by brute force on tree balls (tree_check.py).
2. Finite. \(\mathrm{Gr}_{\le n}\) is proper, so \(\nu_n\) is proper (as for the convolution map; at a finite level of Setup 6.1 it is a proper morphism of finite type schemes); proper and quasi-finite is finite. Over the \(K\)-orbit \(O_n\) the map \(\nu_n^{-1}(O_n)\to O_n\) is a \(K\)-equivariant bijection of homogeneous spaces, hence a universal homeomorphism, so \(\nu_{n*}k|_{O_n}=k\) (étale sheaves do not see universal homeomorphisms).
3. Source rationally smooth. \(\widetilde Y_n\to\mathrm{Gr}_{\le n}\) is locally trivial with pro-smooth fibre \(L^+X\). \(\mathrm{Gr}_{\le n}\) for \(\mathrm{PGL}_2\) is rationally smooth: its Iwahori preimage in \(\mathrm{Fl}\) is a Schubert variety of the infinite dihedral group \(W_{\mathrm{aff}}\), all of whose Kazhdan–Lusztig polynomials equal \(1\) (Kazhdan–Lusztig 1979–80; equivalently Lusztig's \(q\)-analogue \(m^{n}_{n-2j}(q)=q^{j}\) for \(\mathrm{SL}_2\), Lusztig 1983, Theorem 6.1). Hence \(\mathrm{IC}_{\widetilde Y_n}=k[\dim]\) and \(\mathrm{IC}^{\mathrm{Gr}}_{(n,0)}=k[n]\).
4. Small map. For finite \(\nu\), \(\mathcal H^i(\nu_*\mathcal F)_y=\bigoplus_{x\in\nu^{-1}(y)}\mathcal H^i(\mathcal F)_x\) and \(\nu_*=\nu_!\) commutes with \(\mathbb D\); so \(\nu_*\) carries the strict support and cosupport conditions of \(\mathrm{IC}_{\widetilde Y_n}\) to those on \(Y_n\), and \(\nu_{n*}k[n+1]=j_{!*}(\nu_{n*}k|_{O_n})[n+1]=\mathrm{IC}(Y_n)\) (BBD 2.1.11 characterization; de Cataldo–Migliorini, small maps). Stalks: step 1. Pointwise pure: stalks are \(k^{N}\) placed in one degree with trivial Frobenius after the twist. And \(\nu_{n*}(k_{\mathrm{Gr}_{\le n}}[n]\tilde\boxtimes k_{L^+X}[1])\) is by definition \(\mathrm{IC}^{\mathrm{Gr}}_{(n,0)}\ast_K\mathrm{IC}_0\).
5. Class. Trace of \(\nu_{n*}k[n+1]((n+1)/2)\) at \(\gamma\) is \((-q^{-1/2})^{n+1}f_n(\gamma)=v^{n+1}f_n(\gamma)\). In the basis \(M_O=v^{\delta(O)}\mathbf 1_O\): coefficient of \(M_O\) is \(f_n(O)\,v^{\,n+1-\delta(O)}\), exponent \(\ge1\) for \(O\ne O_{0,n}\) in \(Y_n\) (all have \(\delta\le n\)). Bar-invariance follows from Verdier self-duality of \(\mathrm{IC}\); independently it follows from Lemma 6.4 by induction (\(C_{1,n+1}=\underline H_{s_0}C_{0,n}-C_{1,n-1}\), \(C_{0,0}=M_{O_{0,0}}+v(M_{A_+}+M_{A_-})\) checked directly). The bar involution used is Chen–Yi's \(D_\delta\): it satisfies their Theorem 2(i)–(iii) (checked to depth \(8\) in lv_checks.py) and is determined by (i), (ii) and \(D_\delta(M_{A_\pm})=M_{A_\pm}\), since \(\mathcal M\) is generated over the affine Hecke algebra by \(M_{A_\pm}\) (\(\mathbf 1_{O_{0,0}}=T_{s_1}\mathbf 1_{A_+}-\mathbf 1_{A_-}\), \(\mathbf 1_{O_{1,0}}=T_{s_0}\mathbf 1_{A_+}-\mathbf 1_{A_-}\), \(\mathbf 1_{O_{1,m}}=T_{s_0}\mathbf 1_{O_{0,m-1}}\), \(\mathbf 1_{O_{0,m}}=T_{s_1}\mathbf 1_{O_{1,m-1}}\)). Uniqueness in Chen–Yi, Theorem 3, and Theorem 4(iii) (from Theorems 81, 82) give the same \([\mathrm{IC}]\): a second proof of (c) not using step 3.
6. (d). \(K\)-orbits on \(LX\) are the \(O_n\), \(\overline{O_n}=Y_n\), and \(\mathrm{IC}(Y_n)\) is the Satake action of \(\mathrm{IC}^{\mathrm{Gr}}_{(n,0)}\) on \(\mathrm{IC}_0\) by (b), with stabilizers connected; the satake-two-levels summand criterion (Proposition I) then gives \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\).