Intertwining Functors and Irreducibility of Standard Harish—Chandra Sheaves

Let g be a complex semisimple Lie algebra and σ an involution of g. Denote by t the fixed point set of this involution. Let K be a connected algebraic group and ϕ a morphism of K into the group G = Int(g) of inner automorphisms of g such that its differential is injective and identifies the Lie algebra of K with t. Let X be the flag variety of g, i.e. the variety of all Borel subalgebras in g. Then K acts algebraically on X, and it has finitely many orbits which are locally closed smooth sub varieties. The typical situation is the following: g is the complexification of the Lie algebra of a connected real semisimple Lie group G₀ with finite center, K is the complexification of a maximal compact subgroup of G₀, and σ the corresponding Cartan involution.