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\(b\)-functions of regular 2-simple prehomogeneous vector spaces associated to the symplectic group and the orthogonal group. (English) Zbl 1081.11076

The \(b\)-function is an interesting invariant of a prehomogeneous vector space. Its calculation has been done according to the classification of prehomogeneous vector spaces, and several methods have been developed.
In the present paper, the author gives explicit expressions of a certain class of regular 2-simple prehomogeneous vector spaces, in a similar method to K. Ukai [J. Algebra 237, No. 1, 358–381 (2001; Zbl 1040.11092); Compos. Math. 135, No. 1, 49–102 (2003; Zbl 1053.20040)], which is based on functional equations of \(b\)-functions and reduction to simpler functions.

MSC:

11S90 Prehomogeneous vector spaces
20G05 Representation theory for linear algebraic groups
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