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The Demazure-Tits subgroup of a simple Lie group. (English) Zbl 0661.22007

The Demazure-Tits subgroup, DT(G), of a simple Lie group, G, is a certain finite subgroup of G [J. Tits, J. Algebra 4, 96-116 (1966; Zbl 0145.247)] for which there exists a map onto the Weyl group, W(G), of G whose kernel is the group, \(Z_ 2^{\ell}\), of elements of order 2 in the maximal toroidal subgroup, \(U(1)^ l\), of G, where G has rank l. It is pointed out that the significance of DT(G) lies in the fact that it is the group of invariance of the table of Clebsch-Gordan coefficients of G relative to an appropriate basis. Its representations provide a generalization of charge conjugation operators [R. V. Moody and J. Patera, J. Math. Phys. 25, 2838-2847 (1984; Zbl 0562.22005)]. The group DT(G) is first defined by means of an explicit presentation involving the Cartan matrix of the Lie algebra, \({\mathfrak g}\), of G. Its structure is described in detail for \(A_{\ell}\), \(B_{\ell}\), \(C_{\ell}\), \(D_{\ell}\) and \(G_ 2\), and for these groups some low- dimensional representations of DT(G) are given. Each of the rank 2 Lie groups \(A_ 2\), \(B_ 2\) and \(G_ 2\) is then considered in turn: the character tables of both W(G) and DT(G) are given, and the orbits of W(G) on the weight lattice of G and those of DT(G) in any irreducible finite- dimensional representation space of G are decomposed into the sum of irreducible representations of W(G) and DT(G), respectively. In this analysis generating function methods are exploited to good effect and elements of finite order in G are used to describe G-conjugacy classes of the generators of DT(G).
Reviewer: R.C.King

MSC:

22E46 Semisimple Lie groups and their representations
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
22E70 Applications of Lie groups to the sciences; explicit representations
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