Kostant's Convexity Research Articles

Admissible restrictions of irreducible representations of reductive Lie groups: symplectic geometry and discrete decomposability

Let $G$ be a real reductive Lie group, $L$ a compact subgroup, and $\pi$ an irreducible admissible representation of $G$. In this article we prove a necessary and sufficient condition for the finiteness of the multiplicities of $L$-types occurring in $\pi$ based on symplectic techniques. This leads us to a simple proof of the criterion for discrete decomposability of the restriction of unitary representations with respect to noncompact subgroups (the author, Ann. Math. 1998), and also provides a proof of a reverse statement which was announced in [Proc.ICM 2002, Thm.D]. A number of examples are presented in connection with Kostant's convexity theorem and also with non-Riemannian locally symmetric spaces.

Weight distribution of Iwasawa projection

In the setting of a connected noncompact semisimple Lie group with a finite center, we prove a property of the Iwasawa decomposition, and use it to show that, in Kostant's convexity result [7], the Iwasawa projection puts almost all its weight near a single vertex of the convex hull. As another application, we will revisit the results of Huang–Tam [10,5] on the iteration of group elements under the Cartan and the Iwasawa decompositions.

A convexity theorem for noncommutative gradient flows

In this survey we shall prove a convexity theorem for gradient actions of reductive Lie groups on Riemannian symmetric spaces. After studying general properties of gradient maps, this proof is established by (1) an explicit calculation on the hyperbolic plane followed by a transfer of the results to general reductive Lie groups, (2) a reduction to a problem on abelian spaces using Kostant's Convexity Theorem, (3) an application of Fenchel's Convexity Theorem. In the final section the theorem is applied to gradient actions on other homogeneous spaces and we show, that Hilgert's Convexity Theorem for moment maps can be derived from the results.

Kostant's convexity theorem and the compact classical groups

The relationship between the classical Schur-Horn's theorem on the diagonal elements of a Hermitian matrix with prescribed eigenvalues and Kostant's convexity theorem in the context of Lie groups. By using Kostant's convexity theorem, we work out the statements on the special orthogonal group and the symplectic group explicitly. Schur-Horn's result can be stated in terms of a set of inequalities. The counterpart in the Lie-theoretic context is related to a partial ordering, introduced by Atiyah and Bott, defined on the closed fundamental Weyl chamber. Some results of Thompson on the diagonal elements of a matrix with prescribed singular values are recovered. Thompson-Poon's theorem on the convex hull of Hermitian matrices with prescribed eigenvalues is also generalized. Then a result of Atiyah-Bott is recovered.

A convexity theorem for semisimple symmetric spaces

In this paper we prove a convexity theorem for semisimple symmetric spaces which generalizes Kostant's convexity theorem for Riemannian symmetric spaces. Let τ be an involution on the semisimple connected Lie group G and H = GQ the 1-component of the group of fixed points. We choose a Cartan involution θ of G which commutes with τ and write K = Gθ for the group of fixed points. Then there exists an abelian subgroup A of G, a subgroup M of K commuting with A, and a nilpotent subgroup N such that HMAN is an open subset of G and there exists an analytic mapping L: HMAN -* α = h(A) with L(hman) = log a. The set of all elements in A for which aH C HMAN is a closed convex cone. Our main result is the description of the projections L(aH) C α for these elements as the sum of the convex hull of the Weyl group orbit of log a and a certain convex cone in α. 0. Introduction. If G is a connected semisimple Lie group and G — KAN an Iwasawa decomposition, then the convexity theorem of Kostant describes the image of the sets aK under the projection G = KA'N -> α' = L(A') ,kexpXn*-+X as the convex hull of the Weyl group orbit through log a. Recently van den Ban proved a generalization of this theorem to the following situation. Let τ be an involution on the semisimple Lie group G with finite center, G = KA'N a compatible Iwasawa decomposition, i.e., K is τ-invariant, and o! = α^ + αq the corresponding decomposition of a' = L(^4/) into 1 and — 1 eigenspaces for τ. Suppose that H c Gτ is an essentially connected subgroup (see §1 for the definition). Then he describes the image of the sets aH, a £ exp αq under the projection F: G —• α q defined by g e K exp(a^) exp F(g)N. This set is the sum of the convex hull of the orbit of log a under a certain Weyl group and a convex cone in α q. We generalize Kostant's theorem into another direction. We consider the projection L: HMAN -> α defined by g e HMexpL(g)N, where H c Gτ is essentially connected and M, A, and N are defined in §1. This makes sense because the ^4-component in a product hman is unique and HMAN is open in G. So the main new difficulties are the non-compactness of H and the fact that the projection L