Geometric and homological properties of affine Deligne-Lusztig varieties. (English) Zbl 1321.14039

The author studies affine Deligne-Lusztig varieties \(X_{\tilde{w}}(b)\) in the affine flag variety of a quasi-split tamely ramified group.
The term affine refers to the fact that the notion is defined in terms of affine root systems. Let \(G\) be a connected reductive group. For simplicity, suppose \(G\) split over \(\mathbb{F}_q\) and let \(L = k((\epsilon))\) be the field of the Laurent series. The Frobenius automorphism \(\sigma\) on \(G\) induces an automorphism \(\sigma\) on the loop group \(G(L)\). Let \(I\) be a \(\sigma\)-stable Iwahori subgroup of \(G(L)\). By definition, the affine Deligne-Lusztig variety associated with \(\tilde{w}\) in the extended affine Weyl group \(\tilde{W}\cong I \backslash G(L)/I\) and \(b \in G(L)\) is \[ X_{\tilde{w}}(b) = \{gI\in G(L)/I \;|\;g^{ -1} b\sigma(g)\in I\dot{\tilde{w}}I \}, \] where \(\dot{\tilde{w}}\in G(L)\) is a representative of \(\tilde{w}\in \tilde{W}\). Understanding the emptiness/nonemptiness pattern and dimension of affine Deligne-Lusztig varieties is fundamental to understand certain aspects of Shimura varieties with Iwahori level structures.
The affine Deligne-Lusztig variety \(X_{\tilde{w}}(b)\) for an arbitrary \(\tilde{w}\in\tilde{W}\) and \(b\in G(L)\) is very difficult to understand. One of the main goal of this paper is to develop a reduction method for studying the geometric and homological properties of \(X_{\tilde{w}}(b)\). In the finite case, Lang’s theorem implies that \(G\) is a single \(\sigma\)-conjugacy class. This is the reason why a (classical) Deligne-Lusztig variety depends only on the parameter \(w\in W\), with no need to choose an element \(b\in G\). However, in the affine setting, the analog of Lang’s theorem fails and \(X_{\tilde{w}}(b)\) depends on two parameters. Hence it is a challenging task even to describe when \(X_{\tilde{w}}(b)\) is nonempty . To overcome this difficulty, the author proves that Lang’s theorem holds “locally” for loop groups, using a reduction method.
Although the structure of arbitrary affine Deligne-Lusztig varieties is quite complicated, the varieties associated with minimal length elements \(\tilde{w}\in\tilde{W}\) have a very nice geometric structure. The author describes the geometric structure of \(X_{\tilde{w}}(b)\) for such a \(\tilde{w}\) generalizing one of the main results in [X. He and G. Lusztig, J. Am. Math. Soc. 25, No. 3, 739–757 (2012; Zbl 1252.20047)] to the affine case.
The emptiness/nonemptiness pattern and dimension formula for affine Deligne-Lusztig varieties can be obtained keeping track of the reduction step from an arbitrary element to a minimal length element. This is accomplished via the class polynomials of affine Hecke algebras. The affine Deligne-Lusztig variety \(X_{\tilde{w}}(b)\) is nonempty if and only if a certain class polynomial is nonzero. Moreover, the author establishes a connection between the dimension of \(X_{\tilde{w}}(b)\) and the degree of such class polynomial.
As a consequence, he proves a conjecture of U. Görtz et al. [Compos. Math. 146, No. 5, 1339–1382 (2010; Zbl 1229.14036)].


14M17 Homogeneous spaces and generalizations
14L30 Group actions on varieties or schemes (quotients)
20G99 Linear algebraic groups and related topics
Full Text: DOI arXiv


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