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Exotic symmetric space over a finite field. II. (English) Zbl 1319.14056

This is a joint review of the second and the third [T. Shoji and K. Sorlin, Transform. Groups 19, No. 4, 1149–1198 (2014; Zbl 1311.14053)] part.
Let \(k\) be an algebraic closure of a finite field of odd characteristic. Let \(V\) be a \(k\)-vector space of dimension \(2n\). Let \(H\) be the symplectic group \(G^{\theta} \cong \mathrm{Sp}_{2n}\) where \(\theta\) is an involutive automorphism on \(G = \mathrm{GL}(V)\). The symmetric space \(G/H\) can be identified with \(G_{-}^{\theta} := \{g \in G : \theta(g)=g^{-1} \}\). Further, \(H\) acts diagonally on the variety \(\mathbb{X} := G_{-}^{\theta} \times V\).
In the first two papers in this series [T. Shoji and K. Sorlin, Transform. Groups 18, No. 3, 877–929 (2013; Zbl 1308.14050); Transform. Groups 19, No. 3, 887–926 (2014; Zbl 1319.14056)], the intersection cohomology complexes associated to \(H\)-orbits on \(\mathbb{X}\) are studied. In the first one, the set of character sheaves on \(\mathbb{X}\) is defined as a certain set of \(H\)-equivariant simple perverse sheaves on \(\mathbb{X}\). The authors consider the \(\mathbb{F}_q\)-structure on \(\mathbb{X}\) with Frobenius map \(F\). If \(\hat{\mathbb{X}}^F\) denotes the set of character sheaves \(A\) such that \(F^{\ast}A \cong A\), then in the second paper, the authors prove that if \(q\) is large enough, the set of characteristic functions of character sheaves in it forms a basis of the space of \(H^F\)-invariant functions on \(\mathbb{X}^F\). A more general definition of character sheaves modelled on a method of Ginzburg had been suggested by A. Henderson and P. E. Trapa [J. Algebra 370, 32–45 (2012; Zbl 1273.14093)]. In the third paper, it is shown that the two definitions coincide; this was conjectured in the second paper. At the end of the third paper, the authors point out the corrections needed to be made in the first two papers.

MSC:

14M27 Compactifications; symmetric and spherical varieties
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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[1] P. Achar, A. Henderson, Orbit closures in the enhanced nilpotent cone, Adv. in Math. 219 (2008), 27-62, Corrigendum, ibid. 228 (2011), 2984-2988. · Zbl 1205.14061
[2] E. Bannai, N. Kawanaka, S.-Y. Song, The character table of the Hecke algebra ℋ(GL2n(Fq); Sp2n(Fq)), J. Algebra 129 (1990), 320-366. · Zbl 0761.20013
[3] C. De Concini, C. Procesi, Symmetric functions, conjugacy classes, and the flag variety, Invent. Math. 64 (1981), 203-230. · Zbl 0475.14041
[4] I. Grojnowski, Character Sheaves on Symmetric Spaces, Ph.D. thesis, MIT 1992. · Zbl 1074.20031
[5] A. Henderson, Fourier transform, parabolic induction, and nilpotent orbits, Transform. groups 6, (2001), 353-370. · Zbl 1035.22003
[6] A. Henderson, P. E. Trapa, The exotic Robinson-Schensted correspondence, J. Algebra 370 (2012), 32-45. · Zbl 1273.14093
[7] S. Kato, An exotic Deligne-Langlands correspondence, Duke Math. J. 148 (2009), 306-371. · Zbl 1183.20002
[8] S. Kato, An algebraic study of extension algebras, preprint, arXiv:1207.4640. · Zbl 1406.14013
[9] E. Letellier, Fourier Transforms of Invariant Functions of Finite Reductive Lie Algebras, Lecture Note in Mathematics, Vol. 1859, Springer-Verlag, Berlin, 2005. · Zbl 1076.43001
[10] G. Lusztig, Green polynomials and singularities of unipotent classes, Adv. in Math. 42 (1981), 169-178. · Zbl 0473.20029
[11] G. Lusztig, Character sheaves, II, Adv. in Math. 57 (1985), 226-265, V, ibid. 61 (1986), 103-155. · Zbl 0586.20019
[12] G. Lusztig, Character sheaves on disconnected groups, III, Represent. Theory 8 (2004), 125-144. · Zbl 1074.20031
[13] T. Shoji, Green functions attached to limit symbols, in: Representation Theory of Algebraic Groups and Quantum Groups, Adv. Stud. Pure Math., Vol. 40, Math. Soc. Japan, Tokyo 2004, pp. 443-467. · Zbl 1065.05096
[14] T. Shoji, Green functions associated to complex reflection groups, J. Algebra 245 (2001), 650-694. · Zbl 0997.20044
[15] T. Shoji, K. Sorlin, Exotic symmetric space over a finite field, I, Transform. Groups 18 (2013), 877-929. · Zbl 1308.14050
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