Generalized Verma modules, the Cartan-Helgason theorem, and the Harish-Chandra homomorphism

Verma modules , Harish - Chandra ’ s homomorphism

The Harish-Chandra homomorphism is due to [Harish-Chandra 1951]. Attention to universal modules with highest weights is in ]Harish-Chandra 1951], [Cartier 1955], as well as [Verma 1968], [BernsteinGelfand-Gelfand 1971a], [Bernstein-Gelfand-Gelfand 1971b], [Bernstein-Gelfand-Gelfand 1975]. See also [Jantzen 1979]. [1℄ We treat sl(2) in as simple a style as possible, to highlight ideas. Then sl(3...

Harish-Chandra’s homomorphism, Verma modules

The Harish-Chandra homomorphism is due to [Harish-Chandra 1951]. Attention to universal modules with highest weights is in ]Harish-Chandra 1951], [Cartier 1955], as well as [Verma 1968], [BernsteinGelfand-Gelfand 1971a], [Bernstein-Gelfand-Gelfand 1971b], [Bernstein-Gelfand-Gelfand 1975]. See also [Jantzen 1979]. [1] We treat sl(2) in as simple a style as possible, to highlight ideas. Then sl(3...

Generalized Harish-Chandra Modules

Cramped Subgroups and Generalized Harish-chandra Modules

Let G be a reductive complex Lie group with Lie algebra g. We call a subgroup H ⊂ G cramped if there is an integer b(G,H) such that each finite-dimensional representation of G has a non-trivial invariant subspace of dimension less than b(G,H). We show that a subgroup is cramped if and only if the moment map T ∗(K/L) → k∗ is surjective, where K and L are compact forms of G and H. We will use thi...

A Capelli Harish-chandra Homomorphism

For a real reductive dual pair the Capelli identities define a homomorphism C from the center of the universal enveloping algebra of the larger group to the center of the universal enveloping algebra of the smaller group. In terms of the Harish-Chandra isomorphism, this map involves a ρ-shift. We view a dual pair as a Lie supergroup and offer a construction of the homomorphism C based solely on...