Recall that the main goal of this series of posts is the proof of the following result:

Theorem 1Let be a semisimple algebraic Lie group of real rank . Denote by a horospherical subgroup of . If is a discrete Zariski-dense and irreducible subgroup such that is cocompact, then is commensurable to an arithmetic lattice .

Last time, we discussed the first half of the proof of this theorem in the particular case of , , and . Actually, we saw that this specific form of is not very important: all results from the previous post hold whenever

- is
*reflexive*: in the context of the example above, this is the fact that is conjugate to the opposite horospherical subgroup ; - is
*commutative*.

Indeed, we observed that reflexive allows to also assume that is cocompact in . Then, this property and the commutativity of were exploited to establish the closedness of the -orbit of in , where , , and

is the common Levi subgroup of the parabolic subgroups and normalizing and .

Today, we will discuss the second half of the proof of Theorem 1 in the particular case of , , and : in other terms, our goal below is to obtain the arithmeticity of from the closedness of in the homogenous space . This step is due to Hee Oh (see Proposition 3.4.4 of her paper).

**1. From closedness to infinite stabilizer**

Let and , so that . In particular, the closedness of implies that

is closed in .

The next proposition asserts that the stabilizer of this orbit is large whenever is not compact:

This proposition is a direct consequence of the closedness of the in and the following general fact:

Proposition 3Let be a Lie group, a lattice in , and . Suppose that is a semisimple subgroup with finite center such that is closed, then is a lattice in .

*Proof:* The first ingredient of the argument is *Howe–Moore’s mixing theorem*: it asserts that if is a semisimple group with finite center and is an unitary representation of with , then

for all . (Here, means that the projection of to any simple factor of diverges.)

The second ingredient of the argument is *Dani–Margulis recurrence theorem*: it says that if is a lattice in a Lie group and is a one-parameter unipotent subgroup of , then, given and , there exists a compact subset such that

for all .

The basic idea to obtain the desired proposition is to apply these ingredients to , where is a -invariant measure on , and is a one-parameter unipotent subgroup in the product of non-compact simple factors of . Here, we observe that is a *bona fide* Radon measure because we are assuming that is closed, and, if we take not contained in proper normal subgroups of , then as thanks to the absence of compact factors in . In this setting, our task is reduced to prove that is a *finite* measure.

In this direction, we apply Dani–Margulis recurrent theorem to get with and a compact subset such that

for all and . In this way, we obtain that the characteristic functions and of and are two elements of with , and, hence, by Fubini’s theorem,

for all . It follows from Howe–Moore’s mixing theorem that

that is, there exists a non-zero function which is -invariant. By averaging over the product of the compact factors of if necessary, we obtain a non-zero function which is -invariant. By ergodicity of , we have that is constant and, *a fortiori*, is a finite measure.

**2. From infinite stabilizer to infinite**

Our previous discussions (about -actions) paved the way to understand . Intuitively, it is important to get some information about in our way towards showing the arithmeticity of because we already know that and are lattices (i.e., projects to lattices in “other directions”).

The intuition in the previous paragraph is confirmed by *Margulis construction of -forms*:

Theorem 4 (Margulis)If is infinite, then is contained in some -form of .

We will discuss the proof of this result in the next section. For now, we want to exploit the information in order to derive the arithmeticity of . For this sake, we invoke the following result:

Theorem 5 (Raghunathan-Venkataramana)Assume that is semisimple of defined over , and is -simple. Suppose that and are opposite horospherical subgroups defined over .If is a subgroup such that , resp. , has finite index in , resp. , then has finite index in .

Remark 1As it was kindly pointed out to me by David Fisher, Raghunathan-Venkataramana theorem is due to Margulis in the cases covered by Raghunathan (at least).

Remark 2This result is false when , e.g., (and ).

Roughly speaking, Raghunathan-Venkataramana theorem essentially establishes the desired Theorem 1 provided we know in advance that .

In the sequel, we will treat Raghunathan-Venkataramana theorem as a *blackbox* and we will complete the proof of Theorem 1 in the case reflexive and commutative, and *non-compact* such as and .

Remark 3In some natural situations (e.g., subgroups generated by the matrices of the so-called Kontsevich–Zorich cocycle) we have that . In particular, it is a pity that the lack of time made that Yves Benoist could not explain to me the proof of Raghunathan-Venkataramana theorem. Anyhow, I hope to come back to discuss this point in more details in the future.

*Proof of Theorem 1:* We consider the subgroup of . It is discrete and Zariski dense in . Therefore, the normalizer is Zariski dense in , and it is not hard to check that it is also discrete.

Observe that Proposition 2 says that is a lattice in (because ). Since is non-compact, we have that is infinite. Thus, Margulis’ Theorem 4 implies that for some -form of .

Note also that and are defined over : in fact, if is an algebraic subgroup such that is Zariski dense in , then is defined over .

Hence, we can apply Raghunathan-Venkataramana Theorem 5 to get that is commensurable to . Since , this proves the arithmeticity of .

Remark 4As we noticed above, the arguments presented so far allows to prove Theorem 1 when is reflexive and commutative, and is non-compact.

**3. From infinite to arithmeticity**

In this (final) section (of this post), we discuss some steps in the proof of Margulis Theorem 4 stated above.

We write , i.e., we decompose the Lie algebra of in terms of the Lie algebras of , and .

Our goal is to find a -form of containing . For this sake, let us do some “reverse engineering”: assuming that we found , what are the constraints satisfied by the -structure on its Lie algebra?

First, we note that we dispose of lattices and . Hence, we are “forced” to define and as the -vector spaces spanned by and .

Next, we observe that the choice of above imposes a natural -structure on via the adjoint map . In fact, is defined over because we know that is a lattice (and, hence, Zariski-dense) in and (by definition).

Finally, since , and are already defined over and we want a -structure on , it remains to put a -structure on . In general this is not difficult: for instance, we can take

in the context of the example and .

Once we understood the constraints on a -form of containing , we can work backwards and set

where , , and are the -structures from the previous paragraphs.

At this point, the proof of Theorem 4 is complete once we show the following facts:

- doesn’t depend on the choices (of , etc.);
- for all ;
- is a Lie algebra.

The proof of these statements is described in the proof of Proposition 4.11 of Benoist–Miquel paper. Closing this post, let us just make some comments on the independence of on the choices. For this sake, suppose that is another choice of horospherical subgroup. Denote by its Lie algebra, and let be the associated parabolic subgroup. Our task is to verify that the Lie algebra is defined over . In this direction, the basic strategy is to reduce to the case when is opposite to and in order to get . Finally, during the implementation of this strategy, one relies on the following properties discussed in Lemma 4.8 of Benoist–Miquel article of the action of the *unimodular normalizers* of horospherical subgroups on the space with basepoint :

- If is compact, then is closed;
- If and are closed, then is closed;
- If is compact and is closed, then has finite index in .

In any event, this completes our discussion of Theorem 4. In particular, we gave a (sketch of) proof of Theorem 1 when is commutative and reflexive, and is non-compact (cf. Remark 4).

Next time, we will establish Theorem 1 in the remaining cases of and .

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