A Splitting Result for Real Submanifolds of a Kähler Manifold

Let $$(Z,\omega )$$ ( Z , ω ) be a connected Kähler manifold with an holomorphic action of the complex reductive Lie group $$U^\mathbb {C}$$ U C , where U is a compact connected Lie group acting in a hamiltonian fashion. Let G be a closed compatible Lie group of $$U^\mathbb {C}$$ U C and let M be a G-invariant connected submanifold of Z. Let $$x\in M$$ x ∈ M . If G is a real form of $$U^\mathbb {C}$$ U C , we investigate conditions such that $$G\cdot x$$ G · x compact implies $$U^\mathbb {C}\cdot x$$ U C · x is compact as well. The vice-versa is also investigated. We also characterize G-invariant real submanifolds such that the norm-square of the gradient map is constant. As an application, we prove a splitting result for real connected submanifolds of $$(Z,\omega )$$ ( Z , ω ) generalizing a result proved in Gori and Podestà (Ann Global Anal Geom 26: 315–318, 2004), see also Bedulli and Gori (Results Math 47: 194–198, 2005), Biliotti (Bull Belg Math Soc Simon Stevin 16: 107–116 2009).

Parabolic Higgs Bundles for Real Reductive Lie Groups

This chapter explains the correspondence between local systems on a punctured Riemann surface with the structure group being a real reductive Lie group G, and parabolic G-Higgs bundles. The chapter describes the objects involved in this correspondence, taking some time to motivate them by recalling the definitions of G-Higgs bundles without parabolic structure and of parabolic vector bundles. Finally, it explains the relevant polystability condition and the correspondence between local systems and Higgs bundles.

Convexity theorems for the gradient map on probability measures

Given a Kähler manifold (Z, J, ω) and a compact real submanifold M ⊂ Z, we study the properties of the gradient map associated with the action of a noncompact real reductive Lie group G on the space of probability measures on M. In particular, we prove convexity results for such map when G is Abelian and we investigate how to extend them to the non-Abelian case.

A comparison of Paley–Wiener theorems for real reductive Lie groups

In this paper we make a detailed comparison between the Paley–Wiener theorems of J. Arthur and P. Delorme for a real reductive Lie group

THE CONTINUOUS SPECTRUM IN DISCRETE SERIES BRANCHING LAWS

If G is a reductive Lie group of Harish-Chandra class, H is a symmetric subgroup, and π is a discrete series representation of G, the authors give a condition on the pair (G, H) which guarantees that the direct integral decomposition of π|H contains each irreducible representation of H with finite multiplicity. In addition, if G is a reductive Lie group of Harish-Chandra class, and H ⊂ G is a closed, reductive subgroup of Harish-Chandra class, the authors show that the multiplicity function in the direct integral decomposition of π|H is constant along "continuous parameters". In obtaining these results, the authors develop a new technique for studying multiplicities in the restriction π|H via convolution with Harish-Chandra characters. This technique has the advantage of being useful for studying the continuous spectrum as well as the discrete spectrum.

Langlands parameters of derived functor modules and Vogan diagrams

Let $G$ be a linear reductive Lie group with finite center, let $K$ be a maximal compact subgroup, and assume that $\mathrm{rank } G = \mathrm {rank } K$. Let $\ g= l\oplus u$ be a $\theta$ stable parabolic subalgebra obtained by building $l$ from a subset of the compact simple roots and form $A_g(\lambda)$. Suppose $\Lambda=\lambda+2\delta( u\cap p)$ is $K$-dominant and the infinitesimal character, $\lambda+\delta$, of $A_{g}(\lambda)$ is nondominant due to a noncompact simple root. By interpreting these conditions on the level of Vogan diagrams, a conjecture by Knapp is (essentially) settled for the groups $G=SU(p,q),\, Sp(p,q)$, and $SO^*(2n)$, thereby determining the Langlands parameters of natural irreducible subquotient of $A_{ g}(\lambda)$. For the remaining classical groups, simple supplementary conditions are given under which the Langlands parameters may be determined.

Mellin Transforms of Whittaker Functions

In this note we show that for an arbitrary reductive Lie group and any admissible irreducible Banach representation the Mellin transforms of Whittaker functions extend to meromorphic functions. We locate the possible poles and show that they always lie along translates of walls of Weyl chambers.