Wilfried Schmid

Variation of hodge structure: The singularities of the period mapping

Table of

Locally homogeneous complex manifolds

In this paper we discuss some geometric and analytic properties of a class of locally homogeneous complex manifolds. Our original motivation came from algebraic geometry where certain non-compact, homogeneous complex manifolds arose natural ly from the period matrices of general algebraic varieties in a similar fashion to the appearance of the Siegel upper-half-space from the periods of algebraic curves. However, these manifolds arc generally not Hermit ian symmetric domains and, because of this, several interesting new phenomena turn up. The following is a description of the manifolds we have in mind. Let Gc be a connected, complex semi-simple Lie group and B c Gc a parabolic subgroup. Then, as is well known, the coset space X = Gc/B is a compact, homogeneous algebraic manifold. I f G ~ Gc is a connected real form of Gc such tha t G N B = V is compact, then the G-orbit of the origin in X is a connected open domain D ~ X, and D = G/V is therefore a homogeneous complex mani. /o/d. Let F c G be a discrete subgroup such tha t the normalizcr N(F) intersects V only in the identity. Since F acts properly discontinuously without fixed points on D, the quotient space Y = F \ D inherits the structure of a complex manifold. We shall refer to a manifold of this type as a locally homogeneous complex mani]old. One case is when G=M is a maximal compact subgroup of Gc. Then necessarily F ={e), and D = X is the whole compact algebraic manifold. These varieties have been the subject of considerable study, and their basic properties are well known. The opposite extreme occurs when G has no compact factors. These non-compact homogeneous domains D then include the Hermit ian symmetric spaces, about which quite a bit is known, and also include important and interesting non-classical domains which have been discussed relatively little. I t is these manifolds which are our main interest; however, since the

A Geometric Construction of the Discrete Series for Semisimple Lie Groups

In the representation theory of a compact group K, a major role is played by the Peter-Weyl theorem, which asserts that the regular representation L 2(K) decomposes as a countable direct sum of irreducibles with finite multiplicity. For compact connected Lie groups this becomes much more concrete: the irreducibles are explicitly known, their characters are given by the famous Hermann Weyl formula, and there is a uniform geometrical construction for them due to Borel and Weil.

Singular unitary representations and indefinite harmonic theory

Abstract Square-integrable harmonic spaces are defined and studied in a homogeneous indefinite metric setting. In the process, Dolbeault cohomologies are unitarized, and singlar unitary representations are obtained and studied.

Quasi-Equivariant D-Modules, Equivariant Derived Category, and Representations of Reductive Lie Groups

In this note, we describe proofs of certain conjectures on functorial, geometric constructions of representations of a reductive Lie group G R . Our methods have applications beyond the conjectures themselves: unified proofs of the basic properties of the maximal and minimal globalizations of Harish-Chandra modules, and a criterion which insures that the solutions of a G R -invariant system of linear differential equations constitute a representation of finite length.

The Highly Oscillatory Behavior of Automorphic Distributions for SL(2)

Automorphic distributions for SL(2) arise as boundary values of modular forms and, in a more subtle manner, from Maass forms. In the case of modular forms of weight one or of Maass forms, the automorphic distributions have continuous first antiderivatives. We recall earlier results of one of us on the Holder continuity of these continuous functions and relate them to results of other authors; this involves a generalization of classical theorems on Fourier series by S. Bernstein and Hardy and Littlewood. We then show that the antiderivatives are non-differentiable at all irrational points, as well as all, or in certain cases, some rational points. We include graphs of several of these functions, which clearly display a high degree of oscillation. Our investigations are motivated in part by properties of ‘Riemann’s nondifferentiable function’, also known as ‘Weierstrass’ function’.

Two geometric character formulas for reductive Lie groups

In this paper we prove two formulas for the characters of representations of reductive groups. Both express the character of a representation π in terms of the same geometric data attached to π. When specialized to the case of a compact Lie group, one of them reduces to Kirillov’s character formula in the compact case, and the other, to an application of the Atiyah-Bott fixed point formula to the Borel-Weil realization of the representation π. To set the stage, let us first recall the Borel-Weil theorem for a connected, compact Lie group GR. For simplicity, we assume that GR is simply connected. We let gR denote its Lie algebra and g = C ⊗R gR the complexified Lie algebra. Via the adjoint action, GR operates on igR∗, the space of all R-linear functions λ : gR → iR. Every GR-orbit Ω in igR∗ – “coadjoint orbit” for short – carries a canonical GRinvariant symplectic structure. The orbit Ω is said to be integral if some, or equivalently any, λ ∈ Ω exponentiates to a character e of the isotropy subgroup (GR)λ. In that case, the one-dimensional representation spaces of the characters e, λ ∈ Ω, fit together into a GR-equivariant, real algebraic, Hermitian line bundle LΩ → Ω. The pair (Ω,LΩ → Ω) carries a unique “positive polarization”: a GR-invariant complex structure on the manifold Ω and the structure of a GR-equivariant holomorphic line bundle on LΩ, positive in the sense of algebraic geometry. The hypothesis of simple connectivity ensures that the square root √ KΩ of the canonical bundle KΩ exists as a GR-equivariant holomorphic line bundle. According to the Borel-Weil theorem, the natural action of GR on the space of holomorphic LΩ-valued half forms H0(Ω,O(K Ω ⊗ LΩ)) is irreducible; moreover, the association Ω H0(Ω,O(K Ω ⊗ LΩ)) (1.1)

A VANISHING THEOREM FOR OPEN ORBITS ON COMPLEX FLAG MANIFOLDS

A real reductive Lie group G acts on complex flag manifolds Gc/(parabolic subgroup). The open orbits D = G(x) are precisely the ho- mogeneous complex manifolds G/H, where H is the centralizer of a torus. We prove that D is (s + Incomplete in the sense of Andreotti and Grauert, with s = complex dimension of a maximal compact subvariety of D. Thus Hq{D,J) = 0 for q > s and any coherent sheaf 7 —♦ D. This vanishing theorem is needed for the realization of certain unitary representations on Dolbeault cohomology groups of homogeneous vector bundles.

Summation formulas, from Poisson and Voronoi to the present

Summation formulas have played a very important role in analysis and number theory, dating back to the Poisson summation formula. The modern formulation of Poisson summation asserts the equality