Connected components of closed affine Deligne-Lusztig varieties (Q5919695)

Introduced by \textit{M. Rapoport} [Astérisque 298, 271--318 (2005; Zbl 1084.11029)], affine Deligne-Lusztig varieties play an important role in the study of certain properties of local Shimura varieties. The paper under review studies the connected components of closed affine Deligne-Lusztig varieties with parahoric level structure for split reductive algebraic groups. To state the main result, we first need some notation. Let \(G\) be an unramified reductive group over a non-archimedean local field \(F\). Denote by \(L\) the completion of a maximal unramified extension of \(F\). Let \(\sigma\) denote both the Frobenius automorphism of \(L/F\) and the induced automorphism on \(G(L)\). For a fixed maximal torus \(T\) and a Borel subgroup \(B \supseteq T\) over \(F\), consider the Iwahori-Weyl group \(\tilde{W}\). Let \(K\) be a \(\sigma\)-stable parahoric subgroup containing a fixed \(\sigma\)-invariant Iwahori subgroup in \(G(L)\). For \(b \in G(L)\) and \(x \in \tilde{W}\), denote by \[X_x(b)_K := \{ gK \in G(L): g^{-1} b \sigma(g) \in KxK \} /K\] the corresponding affine Deligne-Lusztig variety. Then, for a geometric cocharacter \(\lambda\) of \(T\), the closed affine Deligne-Lusztig variety is given by \[X(\lambda, b)_K := \bigcup_{x\in Adm(\lambda)}X_x(b)_K,\] where \(Adm(\lambda) \subset \tilde{W}\) is the so-called \(\lambda\)-admissible set, introduced by \textit{R. Kottwitz} and \textit{M. Rapoport} [Manuscr. Math. 102, No. 4, 403--428 (2000; Zbl 0981.17003)]. The main result of the paper under review is the following: when \(G\) splits over \(F\), then there exists a bijection \(\pi_0(X(\lambda, b)_K) \cong c_{\lambda,b} \pi_1(G)_{\sigma} ,\) induced by the natural projection \(\eta_G: G(L) \rightarrow \pi_1(G)\). Here \((\lambda, b)\) is Hodge-Newton irreducible, \(c_{\lambda,b}\) is any element of \(\pi_1(G)\) such that \((\sigma -1)(c_{\lambda,b})= \eta_G(t^{\lambda})-\eta_G(b)\) and \(\pi_1(G)_{\sigma}\) is the subset of \(\sigma\)-fixed elements of the algebraic fundamental group of \(G\). Viewing \(\pi_1(G)\) as the group of connected components of \(G(L)\), one therefore concludes that any two points of \(X(\lambda, b)_K\) are in the same connected component of \(X(\lambda, b)_K\) if and only if they are in the same connected component of \(G(L)/K\). The proof uses the notion of permissable roots and involves a case-by-case analysis for some properties of root systems.