Shoji, T.; Sorlin, K. Exotic symmetric space over a finite field. III. (English) Zbl 1311.14053 Transform. Groups 19, No. 4, 1149-1198 (2014). This is a joint review of the second [T. Shoji and K. Sorlin, Transform. Groups 19, No. 3, 887–926 (2014; Zbl 1319.14056)] and the third 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. Reviewer: Balasubramanian Sury (Bangalore) Cited in 1 ReviewCited in 3 Documents MSC: 14M27 Compactifications; symmetric and spherical varieties 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14G17 Positive characteristic ground fields in algebraic geometry Keywords:character sheaves; exotic symmetric space; intersection cohomology; double partitions Citations:Zbl 1308.14050; Zbl 1319.14056; Zbl 1273.14093 PDFBibTeX XMLCite \textit{T. Shoji} and \textit{K. Sorlin}, Transform. Groups 19, No. 4, 1149--1198 (2014; Zbl 1311.14053) Full Text: DOI arXiv References: [1] P. Achar, Green functions via hyperbolic localization, preprint, arXiv:1004.4412. · Zbl 1252.20044 [2] P. Achar, A. Henderson, Orbit closures in the enhanced nilpotent cone, Adv. in Math. 219 (2008), 27-62, Corrigendum 228 (2011), 2984-2988. · Zbl 1205.14061 [3] A. A. Beilinson, J. Bernstein, P. Deligne, Faisceaux Pervers, Astérisque 100 (1982). [4] 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 [5] T. Braden, Hyperbolic localization of intersection cohomology, Transform. Groups 8 (2003), 209-216. · Zbl 1026.14005 [6] J. Bernstein, P. Lunts, Equivariant Sheaves and Functors, Lecture Note in Mathematics, Vol. 1578, Springer-Verlag, 1994. · Zbl 0808.14038 [7] M. Finkelberg, V. Ginzburg, Cherednik algebras for algebraic curves, in: Representation Theory of Algebraic Groups and Quantum Groups, Progress in Mathematics, Vol. 284, Birkhäuser/Springer, New York, 2010, pp. 121-153. · Zbl 1242.14005 [8] M. Finkelberg, V. Ginzburg, R. Travkin, Mirabolic affine Grassmannian and character sheaves, Selecta Math. (N.S.) 14 (2009), 607-628. · Zbl 1215.20041 [9] V. Ginzburg, Admissible modules on a symmetric space, Astérisque 173-174 (1989), 199-255. [10] I. Grojnowski, Character Sheaves on Symmetric Spaces, PhD thesis, MIT, 1992, available at www.dpmms.cam.ac.uk/ groj/thesis.ps. · Zbl 0815.20029 [11] A. Henderson, Fourier transform, parabolic induction, and nilpotent orbits, Transform. Groups 6 (2001), 353-370. · Zbl 1035.22003 [12] A. Henderson, P. E. Trapa, The exotic Robinson-Schensted correspondence, J. Algebra 370 (2012), 32-45. · Zbl 1273.14093 [13] S. Kato, An exotic Deligne-Langlands correspondence for symplectic groups, Duke Math. J. 148 (2009), 306-371. · Zbl 1183.20002 [14] S. Kato, Deformations of nilpotent cones and Springer correspondence, Amer. J. of Math. 133 (2011), 519-553. · Zbl 1242.20056 [15] G. Lusztig, Character sheaves, I, Adv. in Math. 56 (1985), 193-227. · Zbl 0586.20018 [16] T. A. Springer, A purity result or fixed point varieties in flag manifolds, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31 (1984) 271-282. · Zbl 0581.20048 [17] T. Shoji, K. Sorlin, Exotic symmetric space over a finite field, I, Transform. Groups 18 (2013), 877-929. · Zbl 1308.14050 [18] T. Shoji, K. Sorlin, Exotic symmetric space over a finite field, II, Transform. Groups 19 (2014), 887-926. · Zbl 1319.14056 [19] R. Travkin, Mirabolic Robinson-Schensted-Knuth correspondence, Selecta Mathematica (New series) 14 (2009), 727-758. · Zbl 1230.20047 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.