Tohoku Mathematical Journal
2010
December
SECOND SERIES VOL. 62, NO. 4

Tohoku Math. J.
62 (2010), 527-558

Title ORBITS, RINGS OF INVARIANTS AND WEYL GROUPS FOR CLASSICAL $\Theta$-GROUPS

Author Takuya Ohta

(Received February 12, 2008, revised February 17, 2010)
Abstract. In this paper, we study the invariant theory of Viberg's $\Theta$-groups in classical cases. For a classical $\Theta$-group naturally contained in a general linear group, we show the restriction map, from the ring of invariants of the Lie algebra of the general linear group to that of the $\Theta$-representation defined by the $\Theta$-group, is surjective. As a consequence, we obtain explicitly algebraically independent generators of the ring of invariants of the $\Theta$-representation. We also give a description of the Weyl groups of the classical $\Theta$-groups.

2000 Mathematics Subject Classification. Primary 17B70; Secondary 13A50, 14R20.

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