Little Weyl groups and variety of degenerate horospheres

V. S. Zhgoon

Čebyševskij sbornik, Tome 16 (2015) no. 4, pp. 164-187.

Résumé

Let $G$ be a connected reductive group acting on an irreducible normal algebraic variety $X$. We study equivariant geometry of the cotangent vector bundle $X$, and we apply these results to study of a little Weyl group. The aim of this paper is to extend various results of E. B. Vinberg, who constructed a rational Galois cover of $T^*_X$ of quasiaffine $X$ by means of cotangent bundle to the so-called variety of generic horosheres. It is well-known that the example of a flag variety shows that these results could not be generalized directly. We develop the results of D. A. Timashev [18], who obtained the generalizations of the results of Vinberg to the class of varieties wider than quasiaffine but smaller than quasiprojective. We construct a family of horospheres of a smaller dimension in $X$ which are called degenerate, and a variety $\mathcal{H}or$ parameterizing this family, which has the same dimension as the variety parametrizing generic horosheres. Moreover in the quasiaffine case our construction shows that the familly of degenerate horosheres coincides with the familly of generic ones. We show that for constructed family of horosheres there exists a rational $G$-equivariant symplectic Galois covering of cotangent vector bundles $T^*_{\mathcal{H}or} \dashrightarrow T^*_X$. It is proved that the extension of the fields of rational functions corresponding to this cover is a finite Galois extension with the Galois group isomorphic to the little Weyl group. As an application we get the description of the image of the moment map of $T^*_X$ and the image of the normalized moment map by means of purely geometric methods. The first description of the image of the normalized moment map was obtained by F. Knop, nevertheless his proof is non-elementary since it involves the methods of differential operators. Bibliography: 21 titles.

Détail

Keywords: cotangent bundle, moment map, horosphere, local structure theorem, little weyl group.

Bibliographie

[1] Berstein I. N., Gelfand I. M., Gelfand S. I., “Schubert cells and cohomology of the spaces $G/P$”, Russian Math. Surveys, 171:37 (3) (1973), 3–26 | MR

[2] Brion M., “The cone of effective one-cycles of certain G-varieties”, A Tribute to C. S. Seshadri, Hindustan Book Agency, 2003, 180–198 | MR

[3] Brion M., Luna D., Vust Th., “Espaces homogènes sphériques”, Invent. Math., 84 (1986), 617–632 | DOI | MR | Zbl

[4] McGovern W. M., “The adjoint representation and adjoint action”, Algebraic Quotients. Torus Actions and Cohomology. The Adjoint Representation and the Adjoint Action, Encyclopaedia of Mathematical Sciences, 131, Springer-Verlag, 2002 | MR | Zbl

[5] Grosshans F. D., “Constructing invariant polynomials via Tchirnhaus transformations”, Invariant theory, Lecture notes in Math., 1278, Springer, Berlin, 1987, 95–102 | DOI | MR

[6] Hartshorne R., Algebraic geometry, Springer-Verlag, New York–Heidelberg–Berlin, 1977 | MR | Zbl

[7] Humphreys J. E., Linear algebraic groups, Springer-Verlag, New York–Heidelberg–Berlin, 1975 | MR | Zbl

[8] Knop F., Kraft H., Vust T., “The Picard group of a G-variety”, Algebraische Transformationsgruppen und Invariantentheorie, DMV-Seminar, 13, Birkhauser Verlag, Basel–Boston, 1989, 77–88 | MR

[9] Knop F., “Weylgruppe und Momentabbildung”, Invent. Math., 99 (1990), 1–23 | DOI | MR | Zbl

[10] Knop F., “Uber Bewertungen, welche unter einer reduktiven Gruppe invariant sind”, Math. Ann., 295 (1993), 333–363 | DOI | MR | Zbl

[11] Knop F., “The asymptotic behavior of invariant collective motion”, Invent. math., 116 (1994), 309–328 | DOI | MR | Zbl

[12] Knop F., “A Harish-Chandra Homomorphism for Reductive Group Actions”, Annals of Mathematics, Series II, 140 (1994), 253–288 | DOI | MR | Zbl

[13] Knop F., “On the Set of Orbits for a Borel Subgroup”, Commentarii Mathematici Helvetici, 70 (1995), 285–309 | DOI | MR | Zbl

[14] Kraft H., Geometrishe Methoden in der Invariantentheorie, Aspects of Mathematics, Friedr. Vieweg Sohn, Braunschweig, 1984 | MR

[15] Luna D., “Grosses cellules pour les varietes spheriques”, Algebraic Groups and Lie Groups, Austral. Math. Soc. Lect. Ser., 9, Cambridge Univ. Press, Cambridge, 1997, 267–280 | MR | Zbl

[16] Popov V. L., “Contractions of the actions of reductive algebraic groups”, Mat. Sb., 130:3 (1986), 310–334 | MR | Zbl

[17] Springer T. A., Linear Algebraic Groups, Progress in Mathematics, 2nd ed., Springer, 1998 | MR

[18] Timashev D. A., “Equivariant symplectic geometry of cotangent bundles, II”, Moscow Math. J., 6:2 (2006), 389–404 | MR | Zbl

[19] Vinberg E. B., “Equivariant symplectic geometry of cotangent bundles”, Moscow Math. J., 1:2 (2001), 287–299 | MR | Zbl

[20] Vinberg E. B., Popov V. L., Invariant theory. Algebraic geometry IV, Encyclopaedia of Mathematical Sciences, 55, Springer-Verlag, Berlin, 1994 | MR

[21] Zhgoon V. S., “On the Local Structure Theorem and Equivariant Geometry of Cotangent Bundles”, Journal of Lie Theory, 23:3 (2013), 607–638 | MR | Zbl