Proposition 4.6 · Proposition: all structure constants and Satake coefficients agree over close fields

Proof.

1. Orbit test. Under (H-Cartan), \(y\in Kx_\nu\iff\tau(y)\in Kt^\nu K\): if \(y\in Kx_{\nu'}\) then \(\tau(y)\in Kt^{\nu'}K\) by (E4) of Proposition 4.5, and the double cosets of distinct dominant \(\nu'\) are disjoint.
2. Cosets. \(Kt^\lambda K/K\cong K/K_\lambda\), \(K_\lambda=K\cap t^\lambda Kt^{-\lambda}\supseteq K(c(\lambda))\), \(c(\lambda)=\max_\alpha|\langle\alpha,\lambda\rangle|\) (Iwahori factorization of \(K(c)\) into \(T(1+\varpi^c\mathcal O)\) and root groups \(U_\alpha(\varpi^c\mathcal O)\), as in Lemma 1.7). \(k\in K_\lambda\iff\operatorname{val}\rho_i(t^{-\lambda}kt^\lambda)=0\) for all \(i\) (Lemma 4.2(c)), a condition of type Lemma 4.2(e); so \(K_\lambda/K(D)\subseteq G(\mathcal O/\varpi^D)\) and its order are determined by \((\mathcal O/\varpi^D,\bar\varpi)\) for \(D\ge c(\lambda)\) large.
3. Incidence. For \(g=kt^\lambda\): \(\tau(g^{-1}x_\nu)=t^{-\lambda}k^{-1}\tau(x_\nu)\theta(k)\theta(t^\lambda)=t^{-\lambda}k^{-1}t^\nu\theta(k)t^{\theta\lambda}\). By step 1 and Lemma 4.2(c),

The matrix coefficients are \(\varpi^{-\langle\lambda,\chi\rangle+\langle\theta\lambda,\chi'\rangle}\sum_{\chi''}\rho_i(k^{-1})_{\chi\chi''}\varpi^{\langle\nu,\chi''\rangle}\rho_i(\theta(k))_{\chi''\chi'}\), polynomials over \(\Lambda\) in the coordinates of \(k\) and \(\varpi^{\pm1}\) (\(\theta\) is defined over \(\Lambda\)); the condition is right \(K_\lambda\)-invariant and of type Lemma 4.2(e).

4. Counting. \(c^\nu_{\lambda\mu}(E)=\#\{\bar k\in G(\mathcal O/\varpi^D):\text{condition}\}\big/\#(K_\lambda/K(D))\) with \(D=D(\lambda,\mu,\nu)\); numerator and denominator are determined by \((\mathcal O/\varpi^D,\bar\varpi)\). Taking \(D(\lambda,\mu)=\max_{\nu\in N(\lambda,\mu)}D(\lambda,\mu,\nu)\) gives (b), once (a) shows that only \(\nu\in N(\lambda,\mu)\) occur.
5. Support, (a). \(T_\lambda*\mathbf 1_{Kx_\mu}\) is \(K\)-invariant, supported in \(Kt^\lambda Kx_\mu\), and \(\tau(Kt^\lambda Kx_\mu)\subseteq Kt^\lambda Kt^\mu Kt^{-\theta\lambda}K\); applying Lemma 4.2(c) twice gives \(\nu\preceq\lambda+\mu+(-\theta\lambda)^+\) for every \(Kx_\nu\) in the support. Under (H-Cartan) the \(\mathbf 1_{Kx_\nu}\) form a basis of \(C_K(E)\).
6. Satake, (c). With \((\mathcal Sf)(t^\nu)=\delta_B(t^\nu)^{1/2}\int_{U(E)}f(t^\nu u)\,du\), \(\operatorname{vol}U(\mathcal O)=1\), \(\delta_B(t^\nu)^{1/2}=q^{-\langle\rho,\nu\rangle}\): \(\int_UT_\lambda(t^\nu u)\,du\) counts the cosets \(t^\nu uK\subseteq Kt^\lambda K\), i.e. \(n_{\lambda\nu}\) (two \(u\) give the same coset iff they differ by \(U(E)\cap K=U(\mathcal O)\)). Parametrizing \(Kt^\lambda K/K\) by \(K/K_\lambda\) as in step 2, \(kt^\lambda K\subseteq U(E)t^\nu K\iff\operatorname{val}(\ell_i\rho_i(kt^\lambda))=\langle\nu,w_0\omega_i\rangle\ \forall i\) (Lemma 4.2(d)), a condition of type Lemma 4.2(e); only \(\nu\) with \(\nu^+\preceq\lambda\) occur. So \(n_{\lambda\nu}\) agrees over \(D'(\lambda)\)-close fields. The normalization of \(\mathcal S\) is the same formula for both fields, so \(\mathcal S(T_\lambda)\) is the same element of \(\mathbb C[\Lambda_T]^W\). (For \(\mathbb F_q(\!(t)\!)\) this is consistent with Zhu, Lemma 5.6.3, which expresses \(\mathcal S^{-1}(\chi_\mu)\) through Kazhdan–Lusztig polynomials evaluated at \(q\).)
7. (d) is the case \(\mu=0\), since \(\mathbf 1_{Kx_0}=\varphi_0\) by (E2). Lemma 1.7 used a closed embedding \(X\subseteq V\) to test \(g^{-1}x_\mu\in X(\mathcal O)\); here the test is \(\tau(g^{-1}x_\mu)\in K\), valid under (H-Cartan) (or by Audit 4.3 item 1 in general). \(\square\)