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On the Hilbert polynomials and Hilbert series of homogeneous projective varieties. (English) Zbl 1310.14044

Cogdell, James (ed.) et al., Arithmetic geometry and automorphic forms. Festschrift dedicated to Stephen Kudla on the occasion of his 60th birthday. Somerville, MA: International Press; Beijing: Higher Education Press (ISBN 978-1-57146-229-9/pbk). Advanced Lectures in Mathematics (ALM) 19, 253-263 (2011).
Let \(G\) be a connected semisimple complex algebraic group, let \(V\) be a simple \(G\)-module, and let \({\mathbb P}(V)\) be the projective space of all hyperplanes in \(V\) endowed with the natural action of \(G\). The space \({\mathbb P}(V)\) contains the unique closed \(G\)-orbit \(X\). Using Weyl’s dimension formula, the authors compute the Hilbert polynomial, the Hilbert series, the dimension, and the degree of \(X\), and consider several examples.
Reviewer’s remark. These results are the special cases of the known results about Hilbert polynomials, degrees, and dimensions of arbitrary normal spherical varieties; see [M. Brion, Duke Math. J. 58, No. 2, 397–424 (1989; Zbl 0701.14052); Lect. Notes Math. 1296, 177–192 (1987; Zbl 0667.58012); A. Yu. Okounkov, Funct. Anal. Appl. 31, No. 2, 138–140 (1997); translation from Funkts. Anal. Prilozh. 31, No. 2, 82–85 (1997; Zbl 0928.14032); and also D. Panyushev, Transform. Groups 2, No. 1, 91–115 (1997; Zbl 0891.22013)].
For the entire collection see [Zbl 1245.00028].

MSC:

14M17 Homogeneous spaces and generalizations
14L30 Group actions on varieties or schemes (quotients)
20G05 Representation theory for linear algebraic groups
22E46 Semisimple Lie groups and their representations
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
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