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Tube domain and an orbit of a complex triangular group

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概要 Let $ w $ be a complex symmetric matrix of order $ r $, and $ Delta_1(w), cdots , Delta_r(w) $ the principal minors of $ w $. If $ w $ belongs to the Siegel right half space, then it is known that $ R...e(Delta_k(w)/Delta_{k-1}(w)) > 0 $ for $ k = 1,cdots,r $. In this paper we study this property in three directions. First we show that this holds for general symmetric right half spaces. Second we present a series of non-symmetric right half spaces with this property. We note that case-by-case verifications up to dimension 10 tell us that there is only one such irreducible non-symmetric tube domain. The proof of the property reduces to two lemmas. One is entirely generalized to non-symmetric cases as we prove in this paper. This is the third direction. As a byproduct of our study, we show that the basic relative invariants associated to a homogeneous regular open convex cone $ Omega $ studied earlier by the first author are characterized as the irreducible factors of the determinant of right multiplication operators in the complexification of the clan associated to $ Omega $.続きを見る

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登録日 2009.04.22
更新日 2018.02.21

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