Holomorphic representations of nilpotent Lie groups

Let G be a connected, simply-connected complex nilpotent Lie group, and G_r ?( G a real form of G. Motivated by the problem of analytic continuation of Banach-space representations of G_R to holomorphic representations of G, we construct translation-invariant locally-convex algebras of entire functions on G (generalizing the classical spaces of entire functions of finite exponential order). The dual spaces of these algebras are naturally identified with algebras of left-invariant differential operators of infinite order on G. In connection with analytic continuation of unitary representations of G_R, we study the convex cone of entire functions on G whose restrictions to G_R are positive-definite, and determine the minimal order of growth at infinity of such functions.     

Modular Lie representations of groups of prime order

Let K be a field of prime characteristic p and let G be a group of order p. For any finite-dimensional KG-module V and any positive integer n let L ⁿ (V) denote the nth homogeneous component of the free Lie K-algebra generated by (a basis of) V. Then L ⁿ (V) can be considered as a KG-module, called the nth Lie power of V. The main result of the paper is a formula which describes the module structure of L ⁿ (V) up to isomorphism. Mathematics Subject Classification (2000): 17B01, 20C20.     

On extensions of representations for compact Lie groups

Let H be a closed normal subgroup of a compact Lie group G such that G/H is connected. This paper provides a necessary and sufficient condition for every complex representation of H to be extendible to G, and also for every complex G-vector bundle over the homogeneous space G/H to be trivial. In particular, we show that the condition holds when the fundamental group of G/H is torsion free.     

Convexity and unitary representations of nilpotent Lie groups

Summary The moment map of symplectic geometry is extended to associate to any unitary representation of a nilpotent Lie group aG-invariant subset of the dual of the Lie algebra. We prove that this subset is the closed conex hull of the Kirillov orbit of the representation.Supported by NSERC research grant no. A7918     

Stepwise square integrable representations of nilpotent Lie groups

We study the conditions for a nilpotent Lie group to be foliated into subgroups that have square integrable (relative discrete series) unitary representations, that fit together to form a filtration by normal subgroups. Then we use that filtration to construct a class of “stepwise square integrable” representations on which Plancherel measure is concentrated. Further, we work out the character formulae for those stepwise square integrable representations, and we give an explicit Plancherel formula. Next, we use some structure theory to check that all these constructions and results apply to nilradicals of minimal parabolic subgroups of real reductive Lie groups. Finally, we develop multiplicity formulae for compact quotients $N/\varGamma $ where $\varGamma $ respects the filtration.     

Operators arising from representations of nilpotent Lie groups

The following two results are obtained for an irreducible multiplier representation T of a connected nilpotent Lie group. First, T_f is a Hilbert-Schmidt operator if f is square-integrable with compact support. Second, T_f is of trace class if f has derivatives with sufficiently many moments. An application is made of the latter result to show that T_f can be of trace class even when f is not continuous.     

Branching rules for representations of classical Lie groups

Leningrad State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 84, No. 1, pp. 3–12, July, 1990.     

Separate and joint analyticity in Lie groups representations

We prove that for any Lie group there exists a basis of its Lie Algebra in which for any representation of the Lie group in a Hilbert space, a vector which is analytic for every operator representing that basis is an analytic vector for the representation.     

Partially harmonic spinors and representations of reductive Lie groups

Let G be a reductive Lie group subject to some minor technical restrictions. The Plancherel Theorem for G uses several series of unitary representation classes, one series for each conjugacy class of Cartan subgroups of G. Given a Cartan subgroup H ? G, we construct a G-homogeneous family X → Y of oriented riemannian symmetric spaces, some G-homogeneous bundles , and some Hilbert spaces of partially harmonic spinors with values in . Then G acts on by a unitary representation π_μ,σ^±. We then show that these π_μ,σ^± realize the series of representation classes of G associated to the conjugacy class of H.