g∗ −→ g, λ 7→ Xλ characterized by λ(Y ) =< Xλ, Y >, Y ∈ g. Therefore the coadjoint G-orbits in g∗ may be identified with adjoint G-orbits in g. The philosophy of the orbit method says that we may attach the irreducible unitary representations of G to the coadjoint orbits in g∗. Historically the orbit method that was first initiated by A.A. Kirillov (cf. [K]) early in the 1960s in a real nilpote...

Let k be an algebraically closed field and let G be a reductive linear algebraic group over k. Let P be a parabolic subgroup of G, Pu its unipotent radical and pu the Lie algebra of Pu. A fundamental result of R. Richardson says that P acts on pu with a dense orbit (see [9]). The analogous result for the coadjoint action of P on pu is already known for char k = 0 (see [5]). In this note we prov...

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There is a natural action of SL(2, R) on the moduli space of translation surfaces, and this yields an action of the unipotent subgroup U = 1 * 0 1. We classify the U-invariant ergodic measures on certain special submanifolds of the moduli space. (Each submanifold is the SL(2, R)-orbit of the set of branched covers of a fixed Veech surface.) For the U-action on these submanifolds, this is an ana...

We give a new characterization of Lusztig’s canonical quotient, a finite group attached to each special nilpotent orbit of a complex semisimple Lie algebra. This group plays an important role in the classification of unipotent representations of finite groups of Lie type. We also define a duality map. To each pair of a nilpotent orbit and a conjugacy class in its fundamental group, the map assi...

There is a natural action of SL(2,R) on the moduli space of translation surfaces, and this yields an action of the unipotent subgroup U = {( 1 ∗ 0 1 )} . We classify the U -invariant ergodic measures on certain special submanifolds of the moduli space. (Each submanifold is the SL(2,R)-orbit of the set of branched covers of a fixed Veech surface.) For the U -action on these submanifolds, this is...

Let G be a simple algebraic group of classical type over an algebraically closed field k. Let P be a parabolic subgroup of G and let p = LieP be the Lie algebra of P with Levi decomposition p = l⊕ u, where u is the Lie algebra of the unipotent radical of P and l is a Levi complement. Thanks to a fundamental theorem of R. W. Richardson ([16]), P acts on u with an open dense orbit; this orbit is ...

Let G be a connected Lie group and let F be a lattice in G (not necessarily co-compact). We show that if («,) is a unipotent one-parameter subgroup of G then every ergodic invariant (locally finite) measure of the action of («,) on G/Y is finite. For 'arithmetic lattices' this was proved in [2]. The present generalization is obtained by studying the 'frequency of visiting compact subsets' for u...

1. General introduction, Birkhoff’s Ergodic Theorem vs. Ratner’s Theorems on unipotent flows; measure classification implies classification of orbit closures; uniform convergence and the theorem of Dani-Margulis; the statement of the Oppenheim Conjecture. 2. The case of SL(2, R) (the mixing argument). We will be loosely following Ratner’s paper [18]. 3. The classification of invariant measures ...

Let k be an algebraically closed field of characteristic exponent p ≥ 1. Let G be a connected reductive algebraic group over k and let g be the Lie algebra of G. Note that G acts on G and on g by the adjoint action. Let UG be the variety of unipotent elements of G. Let Ng be the variety of nilpotent elements of g. In [L2], we have proposed a definition of a partition of UG into smooth locally c...