On measure rigidity of unipotent subgroups of semisimple groups.

*(English)*Zbl 0745.28010[Joint review with the above cited paper.]

These two important papers are the first two parts of a three part series concerned with measure rigidity properties of unipotent subgroups of real Lie groups.

Various notions of rigidity have been investigated in the theory of real Lie groups and their discrete subgroups. In particular, a subgroup \(U\) of a real Lie group \(G\) is called topologically rigid if for all lattices \(\Gamma\subseteq G\) and \(x\to\Gamma\setminus G\) the closure of the \(U\)- orbit \(xU\) in \(\Gamma\setminus G\) is a homogeneous set, i.e., of the shape \(\Gamma yH\) for some \(y\in G\) and some closed subgroup \(H\subseteq G\) such that \(yHy^{-1}\cap\Gamma\) is again a lattice in \(yHy^{-1}\). One of the outstanding open problems in this field is the question of the validity of Raghunathan’s conjecture, which asserts that every unipotent subgroup of a connected real Lie group \(G\) is topologically rigid. A modification of this conjecture, called Raghunathan’s measure conjecture, replaces topological rigidity by measure rigidity (where \(U\subseteq G\) is called measure rigid if every ergodic \(U\)-invariant Borel probability measure on \(\Gamma\setminus G\) is algebraic).

The proof of this conjecture is accomplished in the series of papers under review. Part I (“Strict measure rigidity \(\ldots\)”) proves strict measure rigidity (assuming \(\Gamma\) only to be discrete) for unipotent subgroups of connected solvable real Lie groups. In fact, a more general theorem (Theorem 1) on algebraicity of measures invariant under the action of unipotent elements is proved which implies (among other results) strict measure rigidity for the connected solvable case. The key steps towards the proof consist in the development of an ergodic theory for measure-preserving actions of nilpotent Lie groups and in establishing the so-called \(R\)-property, a dynamical property of a unipotent subgroup \(N\) (with Lie algebra \({\mathfrak N}\)) of a Lie group \(G\) (with Lie algebra \({\mathfrak G}\)). It allows (roughly) to control the projection onto a complement of \({\mathfrak N}\) in \({\mathfrak G}\) of the adjoint action on \({\mathfrak N}\) of elements of \({\mathfrak G}\) obtained as \(\exp(t_ 1b_ 1)\ldots\exp(t_ rb_ r)\) with suitably restricted \(t_ i\) and a “triangular” basis \(\{b_ 1,\ldots,b_ r\}\) of \({\mathfrak N}\).

The main theorem of Part II (“On measure rigidity \(\ldots\)”) proves the algebraicity of a measure \(\mu\) on an arbitrary Lie group \(G\) which admits an ergodic action of a nilpotent horocyclic element of \({\mathfrak G}\) with certain additional properties. As a corollary of the main theorem the authoress obtains measure rigidity of actions of unipotent elements of a connected semisimple \(G\) on \(G/\Gamma\) for a compatible lattice \(\Gamma\) of \(G\). Other important consequences of the main theorem concern ergodic joinings of unipotent elements and give a classification of such joinings generalizing the results of the authoress and Witte for \(SL_ 2(\mathbb{R})\) [the authoress, Ann. Math., II. Ser. 118, 277-313 (1983; Zbl 0556.28020); D. Witte, Am. J. Math. 109, 927-961 (1987; Zbl 0653.22005)].

These two important papers are the first two parts of a three part series concerned with measure rigidity properties of unipotent subgroups of real Lie groups.

Various notions of rigidity have been investigated in the theory of real Lie groups and their discrete subgroups. In particular, a subgroup \(U\) of a real Lie group \(G\) is called topologically rigid if for all lattices \(\Gamma\subseteq G\) and \(x\to\Gamma\setminus G\) the closure of the \(U\)- orbit \(xU\) in \(\Gamma\setminus G\) is a homogeneous set, i.e., of the shape \(\Gamma yH\) for some \(y\in G\) and some closed subgroup \(H\subseteq G\) such that \(yHy^{-1}\cap\Gamma\) is again a lattice in \(yHy^{-1}\). One of the outstanding open problems in this field is the question of the validity of Raghunathan’s conjecture, which asserts that every unipotent subgroup of a connected real Lie group \(G\) is topologically rigid. A modification of this conjecture, called Raghunathan’s measure conjecture, replaces topological rigidity by measure rigidity (where \(U\subseteq G\) is called measure rigid if every ergodic \(U\)-invariant Borel probability measure on \(\Gamma\setminus G\) is algebraic).

The proof of this conjecture is accomplished in the series of papers under review. Part I (“Strict measure rigidity \(\ldots\)”) proves strict measure rigidity (assuming \(\Gamma\) only to be discrete) for unipotent subgroups of connected solvable real Lie groups. In fact, a more general theorem (Theorem 1) on algebraicity of measures invariant under the action of unipotent elements is proved which implies (among other results) strict measure rigidity for the connected solvable case. The key steps towards the proof consist in the development of an ergodic theory for measure-preserving actions of nilpotent Lie groups and in establishing the so-called \(R\)-property, a dynamical property of a unipotent subgroup \(N\) (with Lie algebra \({\mathfrak N}\)) of a Lie group \(G\) (with Lie algebra \({\mathfrak G}\)). It allows (roughly) to control the projection onto a complement of \({\mathfrak N}\) in \({\mathfrak G}\) of the adjoint action on \({\mathfrak N}\) of elements of \({\mathfrak G}\) obtained as \(\exp(t_ 1b_ 1)\ldots\exp(t_ rb_ r)\) with suitably restricted \(t_ i\) and a “triangular” basis \(\{b_ 1,\ldots,b_ r\}\) of \({\mathfrak N}\).

The main theorem of Part II (“On measure rigidity \(\ldots\)”) proves the algebraicity of a measure \(\mu\) on an arbitrary Lie group \(G\) which admits an ergodic action of a nilpotent horocyclic element of \({\mathfrak G}\) with certain additional properties. As a corollary of the main theorem the authoress obtains measure rigidity of actions of unipotent elements of a connected semisimple \(G\) on \(G/\Gamma\) for a compatible lattice \(\Gamma\) of \(G\). Other important consequences of the main theorem concern ergodic joinings of unipotent elements and give a classification of such joinings generalizing the results of the authoress and Witte for \(SL_ 2(\mathbb{R})\) [the authoress, Ann. Math., II. Ser. 118, 277-313 (1983; Zbl 0556.28020); D. Witte, Am. J. Math. 109, 927-961 (1987; Zbl 0653.22005)].

Reviewer: R.Schulze-Pillot (Bonn)

##### MSC:

28D05 | Measure-preserving transformations |

22E40 | Discrete subgroups of Lie groups |

37A99 | Ergodic theory |

##### Keywords:

unipotent transformations; measure rigidity; unipotent subgroups of real Lie groups; Raghunathan’s conjecture; topological rigidity; ergodic theory; measure-preserving actions; \(R\)-property; ergodic action; ergodic joinings
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##### References:

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