Computing Discrete Series Multiplicities in the Connected Case Atlas of Lie Groups AIM Workshop III
نویسنده
چکیده
A. (Forgive my choice of) Notation. Let G be a complex, connected, semisimple (or reductive) Lie group, T a maximal torus in G, and K the identity component of the fixed points of an involution in G. Equivalently, K is the complexification of a maximal compact subgroup of the identity component of a real form of G. Of course we assume T ⊂ K. We let ΛG denote the lattice of T -characters (the weight lattice), ΦG ⊂ ΛG the root system, Φ∨G the co-roots, Λ ∨ G the co-weights, and W (G) the Weyl group. Fix a choice of positive roots, say Φ+G, and let ρG denote half the sum of Φ + G. The root system of K and its Weyl group W (K) are determined by a Z/2Z-grading of ΦG; i.e., a map ε : ΦG → Z/2Z such that ε(α + β) = ε(α) + ε(β) whenever α, β, α + β are G-roots. In these terms,
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