Ratner's Work on Unipotent Flows and Impact

Ratner’s Work on Unipotent Flows and Its Impact Elon Lindenstrauss, Peter Sarnak, and Amie Wilkinson Dani above. As the name suggests, these theorems assert that the closures, as well as related features, of the orbits of such flows are very restricted (rigid). As such they provide a fundamental and powerful tool for problems connected with these flows. The brilliant techniques that Ratner in- troduced and developed in establishing this rigidity have been the blueprint for similar rigidity theorems that have been proved more recently in other contexts. We begin by describing the setup for the group of 푑×푑 matrices with real entries and determinant equal to 1 — that is, SL(푑, ℝ). An element 푔 ∈ SL(푑, ℝ) is unipotent if 푔−1 is a nilpotent matrix (we use 1 to denote the identity element in 퐺), and we will say a group 푈 < 퐺 is unipotent if every element of 푈 is unipotent. Connected unipotent subgroups of SL(푑, ℝ), in particular one-parameter unipo- Ratner presenting her rigidity theorems in a plenary tent subgroups, are basic objects in Ratner’s work. A unipo- address to the 1994 ICM, Zurich. tent group is said to be a one-parameter unipotent group if there is a surjective homomorphism defined by polyno- In this note we delve a bit more into Ratner’s rigidity theo- mials from the additive group of real numbers onto the rems for unipotent flows and highlight some of their strik- group; for instance ing applications, expanding on the outline presented by Elon Lindenstrauss is Alice Kusiel and Kurt Vorreuter professor of mathemat- 1 푡 푡2/2 1 푡 ics at The Hebrew University of Jerusalem. His email address is elon@math 푢(푡) = ( ) and 푢(푡) = ⎜⎛ 1 푡 ⎟⎞ . .huji.ac.il 1 . ⎝ 1 ⎠ Peter Sarnak is Eugene Higgins professor of mathematics at Princeton Univer- sity and also a professor at the Institute for Advanced Study. His email address is [email protected]. In both cases it is easy to verify directly that these poly- Amie Wilkinson is a professor of mathematics at the University of Chicago. Her nomials do indeed define a homomorphism: i.e., for any email address is [email protected]. 푠, 푡 ∈ ℝ it holds that 푢(푡 + 푠) = 푢(푡) ⋅ 푢(푠). While For permission to reprint this article, please contact: there is essentially no loss of generality in discussing only [email protected]. the case of SL(푑, ℝ), a more natural context is that of lin- DOI: https://doi.org/10.1090/noti/1829 ear algebraic groups — subvarieties of SL(푑, ℝ) defined MARCH 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 373 by polynomial equations that are closed under multiplica- of Numbers is how I have called this book, since I ar- tions and taking inverses (this notion actually makes sense rived at the methods, which deliver in it proofs of arith- for more general fields than the real numbers; if we wantto metic theorems, through spatial considerations. emphasize that we are working with the field of real num- Ratner’s work is a remarkable contribution in the general bers we will call such groups linear algebraic groups over theme of applying “analysis of the infinite” and “spatial ℝ). Connected unipotent subgroups of SL(푑, ℝ) are al- considerations” to number theory. ways linear algebraic groups. Another nice class of exam- So what did Ratner prove in these remarkable papers? ples are the orthogonal groups. Given a quadratic form Perhaps the easiest to explain is her Orbit Closure Classi- 푄(퐱) over ℝ (positive definite or not) in 푑 variables, one fication Theorem, confirming an important conjecture of can consider the group SO(푄) of all matrices in SL(푑, ℝ) M. S. Raghunathan: that preserve this form, i.e. 푑×푑-matrices 푀 so that 푄(푀퐱) = 푄(퐱) for all 푥 ∈ ℝ푑. This group will be compact if and Theorem 1 (Ratner’s Orbit Closure Theorem [M3]). Let only if 푄 is a positive definite or a negative definite form. 퐺 be a real linear algebraic group as above, Γ a lattice in 퐺 Ratner’s theorems on rigidity of unipotent group actions and 푈 < 퐺 a connected unipotent group. Then for any point deal with the action of a unipotent group 푈 on a quotient 푥 ∈ 퐺/Γ the closure of its 푈-orbit is a very nice object: a space of 퐺 by a discrete subgroup. An important exam- single orbit of some closed connected group 퐿 that is sandwiched ple of such a quotient space is when 퐺 = SL(푑, ℝ) and between 푈 and 퐺 (and may coincide with either). Moreover, Γ = SL(푑, ℤ), in which case 퐺/Γ can be identified with this single orbit of 퐿 has finite volume. 푑 the space lattices in ℝ that have unit covolume. A lattice Recall that the 푈-orbit of a point 푥 is simply the set 푑 in ℝ can be specified by giving 푑 linearly independent vec- {푢.푥 ∶ 푢 ∈ 푈}. Note that in particular this shows that any 푑 tors that generate it — i.e. vectors 푣1,..., 푣푑 ∈ ℝ (that 푈-orbit closure has a natural 푈-invariant probability mea- we prefer to think of as column vectors) so that Λ = ℤ푣1 + sure attached to it. We also remark that one can loosen the ⋯ + ℤ푣푑, and the condition that the lattice has unit covol- requirement that 푈 be unipotent to 푈 being generated by ume amounts to requiring that det(푣1, … , 푣푑) = 1, or one-parameter unipotent groups — the passage from The- in other words that the matrix 푔 = (푣1, … , 푣푑) obtained orem 1 to this more general statement is not very difficult. by joining together these 푑 vectors be in SL(푑, ℝ). The Unlike previous work towards Raghunathan’s Conjecture, generators of the lattice Λ are not uniquely determined: in particular Margulis’ proof in the mid 1980s of the (then) ′ ′ 푣1,..., 푣푑 generate the same lattice as 푣1,..., 푣푑 if and fifty year old Oppenheim Conjecture using a special case ′ ′ only if (푣1, … , 푣푑) = (푣1, … , 푣푑)훾 for 훾 ∈ SL(푑, ℤ), in of Raghunathan’s Conjecture, Ratner’s route to classifying other words, lattices of unit covolume in ℝ푑 are in one-to- orbit closures was not direct but by via a measure classifi- one correspondence with elements of SL(푑, ℝ)/ SL(푑, ℤ). cation result: Any matrix ℎ ∈ SL(푑, ℝ) acts on this space by left mul- Theorem 2 (Ratner’s Measure Classification Theorem tiplication; in terms of lattices this amounts to the map [M2, M1]). Let 퐺, Γ and 푈 be as in Theorem 1. Then the from the space of unit covolume lattices to itself taking a only (Borel) probability measures on 퐺/Γ that are invariant lattice Λ < ℝ푑 to the lattice {ℎ.푣 ∶ 푣 ∈ Λ}. and ergodic under 푈 are the natural measures on the orbit clo- This quotient space has the important property of hav- sures described in Theorem 1. ing finite volume, or more precisely an SL(푑, ℝ)-invariant probability measure. A subgroup Γ of a topological group This requires a bit of explanation: We equip 푋 = 퐺/Γ 퐺 which is discrete and such that 퐺/Γ has finite volume with the Borel 휎-algebra ℬ, and consider probability mea- is called a lattice (admittedly, this can be a bit confusing sures on the measurable space (푋, ℬ). Such a measure 휇 at first since our basic example of such 퐺/Γ is the space is 푈-invariant if the push forward of it under left multipli- of lattices in ℝ푑.. ., though this terminology is consistent). cation by every 푢 ∈ 푈 remains the same; 휇 is 푈-ergodic if Hermann Minkowski seems to have been the first to real- every 푈-invariant Borel subset of 푋 is either null or conull. ize the importance of such quotients, and in particular the Every 푈-invariant probability measure can be presented as space of lattices in ℝ푑, to number theory at the turn of the an average of ergodic ones, hence classifying the 푈-ergodic 19th century. In the introduction to his book Geometrie measures gives a description of all 푈-invariant probability der Zahlen, Minkowski writes1 measures on 푋. Dani conjectured this measure classifica- This book contains a new kind of applications of analy- tion result in the same paper where Raghunathan’s Conjec- sis of the infinite to the theory of numbers or, better, cre- ture first appeared. ates a new bond between these two areas. Geometry It is possible to reduce both Theorem 1 and Theorem 2 to the case where 푈 is a one-parameter unipotent group. 1Translated from the original German to English. The following theorem implies both of the theorems quoted 374 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 3 above in the one-parameter case, but is used by Ratner as a digression. It would have been interesting to have such bridge allowing her to pass from the measure classification rigidity results also for local fields of positive characteristic theorem (which, as we said in the outset, is the heart of her such as 픽푞((푡)) — the field of formal Laurent series with work on unipotent flows) to the obit closure theorem: coefficients in the finite field 픽푞 with 푞 elements — but there seem to be serious technical obstacles to doing so Theorem 3 (Ratner’s Distribution Rigidity Theorem and only partial results in this direction are known. [M3]). Let 퐺, Γ, 푈 be as above, and let 푥 ∈ 퐺/Γ.

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