ABSTRACT
More generally, let w be the straight extension of some element w0 in the Weyl group of a semisimple Lie algebra CS0 . Construct as above the w-twisted loop algebra CS associated to CS0 • Then CS is isomorphic to the untwisted loop algebra CSL0 , and is naturally provided with a Z-gradation induced by w, a priori different from the homogeneous gradation. Similarly in the construction of Section 5. 2. 2, replacing the diagram automorphism, Po by the product of Po by some inner automorphism w0 of the root system of CS0 leads to a twisted loop algebra CS isomorphic to cs~~>, with a natural gradation a priori different from the homogeneous
5.2.3 Unitary Highest Weight Representations Preliminaries Let C§ be a complex Lie algebra, and assume that there is a consistent conjugation w on C§ and a Cfi-subalgebra lJ (parabolic subalgebra) such that
1T[UU(C§)] . v A = 'V' 1r(b) · vA = A(b)vA, V bE lJ.
