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Universal formula for the Hilbert series of minimal nilpotent orbits

Authors: A. Matsuo and A. P. Veselov
Journal: Proc. Amer. Math. Soc. 145 (2017), 5123-5130
MSC (2010): Primary 17B20; Secondary 17B08
Published electronically: August 1, 2017
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Abstract: We show that the Hilbert series of the projective variety $ X=P(O_{min}),$ corresponding to the minimal nilpotent orbit $ \mathcal O_{min},$ is universal in the sense of Vogel: it is written uniformly for all simple Lie algebras in terms of Vogel's parameters $ \alpha ,\beta ,\gamma $ and represents a special case of the generalized hypergeometric function $ {}_{4}F_{3}.$ A universal formula for the degree of $ X$ is then deduced.

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A. Matsuo
Affiliation: Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Tokyo, 153-8914, Japan

A. P. Veselov
Affiliation: Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, United Kingdom — and — Moscow State University, Moscow 119899, Russia

Received by editor(s): January 9, 2017
Published electronically: August 1, 2017
Additional Notes: The work of the first author was partially supported by JSPS KAKENHI Grant Number JP26610004
Communicated by: Jerzy Weyman
Article copyright: © Copyright 2017 American Mathematical Society

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