Let G be a connected Lie group, F be a lattice in G and U = {ut},~R be a unipotent one-parameter subgroup of G, viz. Adu is a unipotent linear transformation for all u ~ U. Consider the flow induced by the action of U (on the left) on G/F. Such a flow is referred as a unipotent flow on the homogeneous space G/F. The study of orbits of unipotent flows has been the subject of several papers. For ...

We prove a Lie-theoretic result for type A root systems. This allows us to determine the unipotent orbits attached to degenerate Eisenstein series on general linear groups, confirming a conjecture of David Ginzburg. In particular, this also shows that any unipotent orbit of general linear groups does occur as the unipotent orbit attached to a specific automorphic representation. The proof of th...

The theorems of M. Ratner, describing the finite ergodic invariant measures and the orbit closures for unipotent flows on homogeneous spaces of Lie groups, are extended for actions of subgroups generated by unipotent elements. More precisely: Let G be a Lie group (not necessarily connected) and Γ a closed subgroup of G. Let W be a subgroup of G such that AdG(W ) is contained in the Zariski clos...

Let G be a Lie group and Γ be a discrete subgroup. We show that if {μn} is a convergent sequence of probability measures on G/Γ which are invariant and ergodic under actions of unipotent one-parameter subgroups, then the limit μ of such a sequence is supported on a closed orbit of the subgroup preserving it, and is invariant and ergodic for the action of a unipotent one-parameter subgroup of G.

This article is based on lectures given by the authors in 2005 and 2006. Our first goal is to present an introduction to the orbit method with an emphasis on the character theory of finite nilpotent groups. The second goal (motivated by a recent work of G. Lusztig) is to explain several nontrivial aspects of character theory for finite groups of the form G(Fqn), where G is a unipotent algebraic...

We show that the theta representations on certain covers of general linear groups support certain types of unique functionals. The proof involves two types of Fourier coefficients. The first are semi-Whittaker coefficients, which generalize coefficients introduced by Bump and Ginzburg for the double cover. The covers for which these coefficients vanish identically (resp. do not vanish for some ...

We prove results about orbit closures and equidistribution for the SL(2,R) action on the moduli space of compact Riemann surfaces, which are analogous to the theory of unipotent flows. The proofs of the main theorems rely on the measure classification theorem of [EMi2] and a certain isolation property of closed SL(2,R) invariant manifolds developed in this paper.

Let G be a reductive linear algebraic group, P a parabolic subgroup of G and Pu its unipotent radical. We consider the adjoint action of P on the Lie algebra pu of Pu. Richardson’s dense orbit theorem says that there is a dense P -orbit in pu. We consider some instances when P acts with a dense orbit on terms p u of the descending central series of pu. In particular, we show (in good characteri...

Let G be the universal cover of the group of automorphisms of a symmetric tube domain and let P = L N be its Shilov boundary parabolic subgroup. This paper attaches an irreducible unitary representation of G to each of the (finitely many) L-orbits on n*. The Hilbert space of the representation consists of functions on the orbit which are square-integrable with respect to a certain L-equivariant...

By an additive action on algebraic variety X we mean a regular effective Gan×X→X with open orbit of the commutative unipotent group Gan. In this paper, give uniqueness criterion for complete toric variety.