# zbMATH — the first resource for mathematics

A generalization of Springer theory using nearby cycles. (English) Zbl 0938.22011
Let $$f: {\mathbb C}^d \rightarrow {\mathbb C}$$ be a non-constant polynomial. Then the complex cohomology groups of the Milnor fibre at each point fit together to give a constructible bounded complex, $$P$$, of sheaves, which is called the sheaf of nearby cycles on $$f$$. It is a perverse sheaf in the sense of intersection homology [M. Goresky and R. MacPherson, Astérisque 101-102, 135-192 (1983; Zbl 0524.57022)]. The definition of the sheaf of nearby cycles, $$P$$, generalises to the case of a dominant map, $$f: {\mathbb C}^d \rightarrow {\mathbb C}^r$$ and the author studies the structure of $$P$$ when the components of $$f$$ are given by homogeneous polynomials. The motivation comes from Springer theory, which studies the singularities of the adjoint quotient map, $${\mathcal G} \rightarrow G \backslash\backslash {\mathcal G}$$ of a complex semisimple Lie algebra, $${\mathcal G}$$. For the class of polar representations, $$V$$, of $$G$$ (satisfying an extra orbit finiteness condition called visibility) the author’s main result states that the nearby cycles complex for the quotient map, $$V \rightarrow G \backslash\backslash V$$, satisfies ${\mathcal F}P \cong {\mathcal IC}((V^{*})^{rs} , {\mathcal L}) .$ Here $${\mathcal F}$$ is the Fourier transform functor of R. Hotta and M. Kashiwara [Invent. Math. 75, 327-358 (1984; Zbl 0538.22013)]. Using this description the author establishes a number of other interesting results, including a description of the holonomy of the local system, $${\mathcal L}$$, and the monodromy action on $$P$$.

##### MSC:
 22E46 Semisimple Lie groups and their representations 14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
Full Text:
##### References:
 [1] A. Beĭlinson and J. Bernstein, A proof of Jantzen conjectures, I. M. Gel$$^{\prime}$$fand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993, pp. 1 – 50. · Zbl 0790.22007 [2] A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5 – 171 (French). · Zbl 0536.14011 [3] T. Braden and M. Grinberg, Perverse sheaves on rank stratifications, to appear in the Duke J. Math. · Zbl 0958.14009 [4] Walter Borho and Robert MacPherson, Représentations des groupes de Weyl et homologie d’intersection pour les variétés nilpotentes, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 15, 707 – 710 (French, with English summary). · Zbl 0467.20036 [5] Walter Borho and Robert MacPherson, Partial resolutions of nilpotent varieties, Analysis and topology on singular spaces, II, III (Luminy, 1981) Astérisque, vol. 101, Soc. Math. France, Paris, 1983, pp. 23 – 74. · Zbl 0576.14046 [6] Jean-Luc Brylinski, Transformations canoniques, dualité projective, théorie de Lefschetz, transformations de Fourier et sommes trigonométriques, Astérisque 140-141 (1986), 3 – 134, 251 (French, with English summary). Géométrie et analyse microlocales. · Zbl 0624.32009 [7] Jiri Dadok and Victor Kac, Polar representations, J. Algebra 92 (1985), no. 2, 504 – 524. · Zbl 0611.22009 [8] V. Ginzburg, Intégrales sur les orbites nilpotentes et représentations des groupes de Weyl, C.R. Acad. Sci. Paris 296 (1983), Sèrie I, pp. 249-253. · Zbl 0544.22009 [9] M. Grinberg, On the Specialization to the Asymptotic Cone, preprint, math.AG/9805031. · Zbl 1078.14516 [10] M. Grinberg, Morse groups in symmetric spaces corresponding to the symmetric group, preprint, math.AG/9802091. [11] Mark Goresky and Robert MacPherson, Intersection homology. II, Invent. Math. 72 (1983), no. 1, 77 – 129. · Zbl 0529.55007 [12] Mark Goresky and Robert MacPherson, Morse theory and intersection homology theory, Analysis and topology on singular spaces, II, III (Luminy, 1981) Astérisque, vol. 101, Soc. Math. France, Paris, 1983, pp. 135 – 192. · Zbl 0524.57022 [13] Mark Goresky and Robert MacPherson, Stratified Morse theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 14, Springer-Verlag, Berlin, 1988. · Zbl 0639.14012 [14] Heisuke Hironaka, Stratification and flatness, Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976) Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, pp. 199 – 265. · Zbl 0424.32004 [15] R. Hotta and M. Kashiwara, The invariant holonomic system on a semisimple Lie algebra, Invent. Math. 75 (1984), no. 2, 327 – 358. · Zbl 0538.22013 [16] Ryoshi Hotta, On Springer’s representations, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 863 – 876 (1982). · Zbl 0584.20033 [17] V. G. Kac, Some remarks on nilpotent orbits, J. Algebra 64 (1980), no. 1, 190 – 213. · Zbl 0431.17007 [18] David Kazhdan and George Lusztig, A topological approach to Springer’s representations, Adv. in Math. 38 (1980), no. 2, 222 – 228. · Zbl 0458.20035 [19] George Kempf and Linda Ness, The length of vectors in representation spaces, Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), Lecture Notes in Math., vol. 732, Springer, Berlin, 1979, pp. 233 – 243. · Zbl 0407.22012 [20] B. Kostant and S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971), 753 – 809. · Zbl 0224.22013 [21] Masaki Kashiwara and Pierre Schapira, Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 292, Springer-Verlag, Berlin, 1994. With a chapter in French by Christian Houzel; Corrected reprint of the 1990 original. Masaki Kashiwara and Pierre Schapira, Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 292, Springer-Verlag, Berlin, 1990. With a chapter in French by Christian Houzel. [22] Lê Dũng Tráng, Some remarks on relative monodromy, Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976) Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, pp. 397 – 403. · Zbl 0428.32008 [23] Lê Dũng Tráng, The geometry of the monodromy theorem, C. P. Ramanujam — a tribute, Tata Inst. Fund. Res. Studies in Math., vol. 8, Springer, Berlin-New York, 1978, pp. 157 – 173. · Zbl 0434.32010 [24] G. Lusztig, Green polynomials and singularities of unipotent classes, Adv. in Math. 42 (1981), no. 2, 169 – 178. · Zbl 0473.20029 [25] G. Lusztig, Left cells in Weyl groups, Lie group representations, I (College Park, Md., 1982/1983) Lecture Notes in Math., vol. 1024, Springer, Berlin, 1983, pp. 99 – 111. [26] Robert MacPherson, Global questions in the topology of singular spaces, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) PWN, Warsaw, 1984, pp. 213 – 235. · Zbl 0612.57012 [27] John Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. · Zbl 0184.48405 [28] È. B. Vinberg , Lie groups and Lie algebras, III, Encyclopaedia of Mathematical Sciences, vol. 41, Springer-Verlag, Berlin, 1994. Structure of Lie groups and Lie algebras; A translation of Current problems in mathematics. Fundamental directions. Vol. 41 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1990 [ MR1056485 (91b:22001)]; Translation by V. Minachin [V. V. Minakhin]; Translation edited by A. L. Onishchik and È. B. Vinberg. [29] T. A. Springer, Trigonometric sums, Green functions of finite groups and representations of Weyl groups, Invent. Math. 36 (1976), 173 – 207. · Zbl 0374.20054 [30] T. A. Springer, A construction of representations of Weyl groups, Invent. Math. 44 (1978), no. 3, 279 – 293. · Zbl 0376.17002 [31] Jean-Luc Brylinski, (Co)-homologie d’intersection et faisceaux pervers, Bourbaki Seminar, Vol. 1981/1982, Astérisque, vol. 92, Soc. Math. France, Paris, 1982, pp. 129 – 157 (French). T. A. Springer, Quelques applications de la cohomologie d’intersection, Bourbaki Seminar, Vol. 1981/1982, Astérisque, vol. 92, Soc. Math. France, Paris, 1982, pp. 249 – 273 (French). [32] Peter Slodowy, Simple singularities and simple algebraic groups, Lecture Notes in Mathematics, vol. 815, Springer, Berlin, 1980. · Zbl 0441.14002 [33] G. Shephard and J. Todd, Finite unitary reflection groups, Canadian J. Math. 6 (1954), pp. 410-431. · Zbl 0055.14305
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.