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

He, Xuhua

Ann. Math. (2) 179, No. 1, 367-404 (2014).

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)].

Reviewer: Alessandro Ruzzi (Roma)

MSC:

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

Keywords:

\(\sigma\)-conjugacy classes; affine Deligne-Lusztig varieties

Citations:

Zbl 1252.20047; Zbl 1229.14036

References:

[1]	R. Bédard, ”On the Brauer liftings for modular representations,” J. Algebra, vol. 93, iss. 2, pp. 332-353, 1985. · Zbl 0607.20024
[2]	R. Bédard, ”The lowest two-sided cell for an affine Weyl group,” Comm. Algebra, vol. 16, iss. 6, pp. 1113-1132, 1988. · Zbl 1195.20048
[3]	N. Bourbaki, Lie Groups and Lie Algebras. Chapters 4-6, New York: Springer-Verlag, 2002. · Zbl 0983.17001
[4]	C. Bonnafé and R. Rouquier, ”Affineness of Deligne-Lusztig varieties for minimal length elements,” J. Algebra, vol. 320, iss. 3, pp. 1200-1206, 2008. · Zbl 1195.20048
[5]	F. Bruhat and J. Tits, ”Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée,” Inst. Hautes Études Sci. Publ. Math., vol. 60, pp. 197-376, 1984. · Zbl 0597.14041
[6]	P. Deligne and G. Lusztig, ”Representations of reductive groups over finite fields,” Ann. of Math., vol. 103, iss. 1, pp. 103-161, 1976. · Zbl 0336.20029
[7]	Q. R. Gashi, ”On a conjecture of Kottwitz and Rapoport,” Ann. Sci. Éc. Norm. Supér., vol. 43, iss. 6, pp. 1017-1038, 2010. · Zbl 1225.14037
[8]	U. Görtz and X. He, ”Dimensions of affine Deligne-Lusztig varieties in affine flag varieties,” Doc. Math., vol. 15, pp. 1009-1028, 2010. · Zbl 1248.20048
[9]	U. Görtz, T. J. Haines, R. E. Kottwitz, and D. C. Reuman, ”Dimensions of some affine Deligne-Lusztig varieties,” Ann. Sci. École Norm. Sup., vol. 39, iss. 3, pp. 467-511, 2006. · Zbl 1108.14035
[10]	U. Görtz, T. J. Haines, R. E. Kottwitz, and D. C. Reuman, ”Affine Deligne-Lusztig varieties in affine flag varieties,” Compos. Math., vol. 146, iss. 5, pp. 1339-1382, 2010. · Zbl 1229.14036
[11]	U. Görtz, X. He, and S. Nie, \(P\)-alcoves and nonemptiness of affine Deligne-Lusztig varieties. · Zbl 1326.14122
[12]	M. Geck and G. Pfeiffer, ”On the irreducible characters of Hecke algebras,” Adv. Math., vol. 102, iss. 1, pp. 79-94, 1993. · Zbl 0816.20034
[13]	T. J. Haines, ”Introduction to Shimura varieties with bad reduction of parahoric type,” in Harmonic Analysis, the Trace Formula, and Shimura Varieties, Providence, RI: Amer. Math. Soc., 2005, vol. 4, pp. 583-642. · Zbl 1148.11028
[14]	T. J. Haines and M. Rapoport, On parahoric subgroups.
[15]	X. He, ”Minimal length elements in some double cosets of Coxeter groups,” Adv. Math., vol. 215, iss. 2, pp. 469-503, 2007. · Zbl 1149.20035
[16]	X. He, ”Closure of Steinberg fibers and affine Deligne-Lusztig varieties,” Int. Math. Res. Not., vol. 2011, iss. 14, pp. 3237-3260, 2011. · Zbl 1264.20042
[17]	X. He, ”A subalgebra of 0-Hecke algebra,” J. Algebra, vol. 322, iss. 11, pp. 4030-4039, 2009. · Zbl 1186.20005
[18]	X. He, Minimal length elements of extended affine Weyl group, I. · Zbl 1335.20036
[19]	X. He and G. Lusztig, ”A generalization of Steinberg’s cross section,” J. Amer. Math. Soc., vol. 25, iss. 3, pp. 739-757, 2012. · Zbl 1252.20047
[20]	X. He and S. Nie, Minimal length elements of extended affine Weyl group, II. · Zbl 1335.20036
[21]	X. He and T. Wedhorn, On parahoric reductions of Shimura varieties of PEL type.
[22]	X. He and Z. Yang, ”Elements with finite Coxeter part in an affine Weyl group,” J. Algebra, vol. 372, pp. 204-210, 2012. · Zbl 1271.20052
[23]	U. Hartl and E. Viehmann, ”The Newton stratification on deformations of local \(G\)-shtukas,” J. Reine Angew. Math., vol. 656, pp. 87-129, 2011. · Zbl 1225.14036
[24]	N. Iwahori and H. Matsumoto, ”On some Bruhat decomposition and the structure of the Hecke rings of \({\mathfrak p}\)-adic Chevalley groups,” Inst. Hautes Études Sci. Publ. Math., iss. 25, pp. 5-48, 1965. · Zbl 0228.20015
[25]	R. E. Kottwitz, ”Isocrystals with additional structure,” Compositio Math., vol. 56, iss. 2, pp. 201-220, 1985. · Zbl 0597.20038
[26]	R. E. Kottwitz, ”Isocrystals with additional structure. II,” Compositio Math., vol. 109, iss. 3, pp. 255-339, 1997. · Zbl 0966.20022
[27]	R. E. Kottwitz, ”Dimensions of Newton strata in the adjoint quotient of reductive groups,” Pure Appl. Math. Q., vol. 2, iss. 3, Special Issue: In honor of Robert D. MacPherson. Part 1, pp. 817-836, 2006. · Zbl 1109.11033
[28]	R. E. Kottwitz and M. Rapoport, ”On the existence of \(F\)-crystals,” Comment. Math. Helv., vol. 78, iss. 1, pp. 153-184, 2003. · Zbl 1126.14023
[29]	D. Krammer, ”The conjugacy problem for Coxeter groups,” Groups Geom. Dyn., vol. 3, iss. 1, pp. 71-171, 2009. · Zbl 1176.20032
[30]	G. Lusztig, Characters of Reductive Groups over a Finite Field, Princeton, NJ: Princeton Univ. Press, 1984, vol. 107. · Zbl 0556.20033
[31]	C. Lucarelli, ”A converse to Mazur’s inequality for split classical groups,” J. Inst. Math. Jussieu, vol. 3, iss. 2, pp. 165-183, 2004. · Zbl 1054.14059
[32]	G. Pappas and M. Rapoport, ”Twisted loop groups and their affine flag varieties,” Adv. Math., vol. 219, iss. 1, pp. 118-198, 2008. · Zbl 1159.22010
[33]	M. Rapoport, ”A guide to the reduction modulo \(p\) of Shimura varieties,” in Automorphic Forms. I, , 2005, pp. 271-318. · Zbl 1084.11029
[34]	M. Rapoport and M. Richartz, ”On the classification and specialization of \(F\)-isocrystals with additional structure,” Compositio Math., vol. 103, iss. 2, pp. 153-181, 1996. · Zbl 0874.14008
[35]	T. Richarz, ”Schubert varieties in twisted affine flag varieties and local models,” J. Algebra, vol. 375, pp. 121-147, 2013. · Zbl 1315.14066
[36]	T. A. Springer, ”Regular elements of finite reflection groups,” Invent. Math., vol. 25, pp. 159-198, 1974. · Zbl 0287.20043
[37]	E. Viehmann, ”The dimension of some affine Deligne-Lusztig varieties,” Ann. Sci. École Norm. Sup., vol. 39, iss. 3, pp. 513-526, 2006. · Zbl 1108.14036
[38]	E. Viehmann and T. Wedhorn, ”Ekedahl-Oort and Newton strata for Shimura varieties of PEL type,” Math. Ann., vol. 356, iss. 4, pp. 1493-1550, 2013. · Zbl 1314.14047
[39]	B. Zbarsky, On Some Stratifications of Affine Deligne-Lusztig Varieties for \({ {SL}}(3)\), ProQuest LLC, Ann Arbor, MI, 2008.