Equidistribution of primitive lattices in $\mathbb{R}^n$

We count primitive lattices of rank $d$ inside $\mathbb{Z}^{n}$ as their covolume tends to infinity, with respect to certain parameters of such lattices. These parameters include, for example, the subsapce that a lattice spans, namely its projection to the Grassmannian; its homothety class; and its equivalence class modulo rescaling and rotation, often referred to as a shape. We add to a prior work of Schmidt by allowing sets in the spaces of parameters that are general enough to conclude joint equidistribution of these parameters. In addition to the primitive $d$-lattices themselves, we also consider their orthogonal complements in $\mathbb{Z}^{n}$, and show that the equidistribution occurs jointly for primitive lattices and their orthogonal complements. Finally, our asymptotic formulas for the number of primitive lattices include an explicit error term.


Introduction
The aim of this paper is to extend classical counting and equidistribution results for primitive vectors to their higher rank counterparts: primitive lattices. A primitive vector is an n-tuple of integers (a 1 , . . . , a n ) with gcd(a 1 , . . . , a n ) = 1, and the the set of primitive vectors in R n is denoted by Z n prim . We can associate to each vector 0 = v ∈ R n the discrete subgroup that it spans, Zv; following this logic, a rank d (1 ≤ d ≤ n) analog for a vector is a lattice of rank d in R n , namely Λ = Zv 1 ⊕ · · · ⊕ Zv d where v 1 , . . . , v d ∈ R n are linearly independent. We will refer to it briefly as a d-lattice. We say that a d-lattice Λ is integral if Λ ⊂ Z n , and primitive if Λ = V ∩ Z n , where V is a d-dimensional rational subspace of R n . For example, a primitive 1-lattice is simply all the integral points on a rational line, or, equivalently, it is Zv where v is a primitive vector.
Questions about counting primitive vectors date back to the days of Gauss and Dirichlet, e.g. with the Gauss Class Number problem. In the 20th century, questions about equidistribution of integral vectors began to arise, with the principal example being Linnik-type problems [Lin68,EH99,Duk03,Duk07,ELMV11,BO12,EMV13]. These questions and others generalize naturally to primitive lattices, as we now turn to describe.
The Primitive Circle Problem. The well known Gauss Circle Problem concerns the asymptotic number of integral vectors up to (euclidean) norm X > 0. The analogous question for primitive vectors, namely the asymptotic amount of primitive vectors up to norm X, is often referred to as the primitive circle problem [Now88,ZC99,Wu02]. In lattices, the role of a norm is played by the covolume: the covolume of Λ, denoted covol(Λ), is the volume of a fundamental paralelopiped for Λ in the linear space Thus, the primitive circle problem for lattices is to estimate the asymptotics of #{primitive d-lattices in R n of covolume up to X} (1.1) as X → ∞. Note that for 1-lattices, the notions of norm and covolume coincide: covol(Zv) = v , hence when d = 1 the above recovers the "original" primitive circle problem. Schmidt [Sch68] showed that the amount in (1.1) equals where Leb is the Lebesgue measure, and B i is the unit ball in R i . We remark that the optimal exponent in the error term of the circle problem (primitive or not) is established only in dimensions n ≥ 4. As far as the authors are aware, this (d = 1, n ≥ 4) is the only case where an optimal error exponent is known for the lattices circle problem (primitive or not).
Linnik-type problems. This is a unifying name for questions on the distribution of the projections of integral vectors to the unit sphere, i.e. of v/ v when v ∈ Z n or Z n prim . Viewing the unit sphere as the space of oriented lines in R n , the analogous object when considering d-lattices would be the Grassmannian of oriented d-dimensional subspaces in R n , denoted Gr(d, n) (and defined explicitly in Section 2). Accordingly, we will view our lattices as carrying an orientation, which simplifies our discussion on the technical level but has no effect on the results. In particular, the two-to-one correspondence between primitive vectors and primitive 1-lattices (arising from the fact that v and −v span the same lattice) becomes a one-to-one correspondence between primitive vectors and oriented primitive 1-lattices. The average Linnik problem for primitive lattices is to study the distribution of the (oriented) spaces V Λ in Gr(d, n) as covol(Λ) ≤ X → ∞. Note that V Λ are exactly the rational subspaces in Gr(d, n).
Shapes of orthogonal lattices. More recently, with the rise of dynamical approaches in number theory, another type of equidistribution questions for primitive vectors arose. To a primitive vector v we associate the (n − 1)-lattice v ⊥ ∩ Z n , referred to as as the orthogonal lattice of v, where v ⊥ is the orthogonal hyperplane to v. Several recent papers (by Marklof [Mar10], Aka Einsiedler and Shapira [AES16b,AES16a], Einsiedler, Mozes, Shah and Shapira [EMSS16], Einsiedler Rï¿oehr and Wirth [ERW17]) studied the equidistribution of shapes of the orthogonal lattices to primitive vectors as their norm tends to infinity, where the shape of a lattice is its similarity class modulo rotation and homothety. The space of shapes of d-lattices is a double coset space of SL d (R), denoted X d and defined explicitly in Section 2, and the aforementioned papers show (among other things) that the shapes of the orthogonal lattices to v ∈ Z n prim equidistribute in X n−1 as v → ∞ w.r.t. a uniform measure arriving from the Haar measure on SL n−1 (R). In the works of Einsiedler et al. they in fact show that the shapes of v ⊥ ∩ Z n equidistribute in X n−1 jointly with the directions v/ v in S n−1 .
Just like for (primitive) vectors, orthogonal lattices can be defined for (primitive) lattices as well: for a primitive d-lattice Λ, we let: Λ is the orthogonal complement of V Λ . Note that Λ ⊥ is primitive by definition, and that it has rank n−d. Also note that Λ → Λ ⊥ defines a bijection between primitive lattices of ranks d and n − d. This bijetion extends to a bijection between oriented lattices, with a natural choice of orientation on the orthogonal lattice (Definition 2.2).
One could then ask about the equidistribution of shapes of the orthogonal lattices Λ ⊥ to primitive lattices Λ, where the one dimensional case Λ = Zv recovers the question studied in the aforementioned papers about the equidistribution of shapes of v ⊥ ∩ Z n .
Equidistribution of a sequence in a finite-volume space can be deduced from counting in "sufficiently general" subsets of this space. Indeed, we will count in subsets that have controlled boundary (Definition 3.1).
In the above theorem, vol stands for a uniform Radon measure (that is unique up to multiplication by a positive scalar and will be defined in Section 2), so it implies the joint uniform distribution of the directions, shapes, and shapes of the orthogonal lattices of primitive lattices as their covolume tends to infinity. In particular, it is quite interesting to observe that the shapes of Λ and of Λ ⊥ are independent parameters, meaning that there is no way to know the shape of Λ ⊥ given the shape of Λ, even though the latter lattice determines the first.
We remark that the equidistribution of shapes of primitive d-lattices was shown in a non-quantitative form by Schmidt in [Sch98].
Counting d-lattices: Beyond shapes. There exists a wide body of work on equidistribution problems in the space which is the space of unimodular (i.e. with covolume one and positive orientation) full lattices in R n , as well as in the space X n of their shapes. The restriction to covolume one is due to the fact that the space L n (and therefore X n ) has finite volume, while the space of all lattices, GL n (R)/ GL n (Z), does not. Comparing the spaces L n and X n , the space L n naturally contains "more information" than X n , which is obtained by modding L n by rotations. This brings up the question of whether one can define a space of "unimodular d-lattices in R n " so as to consider the unimodular d-lattices in R n without modding out by rotations. In Section 2 we introduce two such spaces, which are homogeneous spaces of SL n (R), but do not carry an SL n (R)-invariant measure. However, they do support a natural measure which is invariant under a direct product of two subgroups of SL n (R). We will prove a stronger statement than Theorem 1.1, namely Theorem 3.2, where we count primitive d-lattices according to their projections to these more refined spaces, from which we conclude Theorem 1.1, of counting primitive d-lattices according to their projections to X d .
Organization of the paper. The paper is organized as follows: In Section 2 we define all the spaces and notions related to lattices that appear throughout this work. In Section 3 we state Theorem 3.2, and explain how it implies Theorem 1.1; then the rest of the paper is devoted to proving Theorem 3.2. In Section 4 we define coordinates on SL n (R) that are a refinement of the Iwasawa coordinates, and study their properties. In Section 5 we identify subsets of SL n (R) that are sets of representatives for all the quotient spaces that appear in Theorem 3.2. In Section 6, we associate to each primitive d-lattice a unique element in SL n (Z), transferring the problem of counting primitive dlattices to the problem of counting points of SL n (Z) in a family of increasing sets in SL n (R). These sets are not compact, even though the number of lattice points in each of the sets is finite. In Section 7, we reduce to counting in a family of compact subsets, obtaining a "classical" counting lattice point problem in SL n (R). In Section 8 we state the solution of this counting problem (Proposition 8.1), and prove Theorem 3.2 based on it. In Section 9 we prove Proposition 8.1 using a method developed by Gorodnik and Nevo in [GN12], for counting lattice points in semi-simple Lie groups. In the appendix, we expand further on the notions introduced in Section 2, as some of them are new, and others are not new but possibly unfamiliar to the reader.

Spaces of lattices
We begin by defining explicitly the spaces X d and Gr(d, n) appearing in Theorem 1.1. An oriented subspace on R n is a subspace with a sign attached, and the Grassmannian Gr(d, n) is the set of all d-dimensional oriented subspaces of R n . It can also be defined as the following quotients where P d is the group of upper triangular matrices of determinant 1 with positive diagonal entries. A coset of SO n (or SL n ) represents an oriented d-dimensional subspace V if the first d columns of the matrices in this coset span V with the corresponding orientation.
Recall that the shape of (an oriented) lattice is its equivalent class modulo homothety and rotation. The space of shapes of (oriented) d-lattices is Both Gr(d, n) and X d are equipped with natural measures that decent from the Haar measure on ambient group, and are unique up to a positive scalar -for Gr(d, n) it is the measure that is invariant under SO n (R), and for X d it is the measure that is obtained from the Haar measure on SL n (R). The specific normalizations we choose for these measures are given in Section 2.3.
We now proceed to define a space of d-lattices in R n that encodes both their shapes, and their directions (i.e. their projections to the Grassmannian).

Spaces of homothety classes of d-lattices.
A unimodular lattice is an oriented lattice with positive orientation and covolume one. Recall our notation for the space of rank n unimodular lattices where a matrix in SL n (R) lies in the coset that represents a full unimodular lattice in R n if its columns span this lattice. As the space of shapes X n is obtained from L n by modding by SO n , the space L n is more refined, containing not only the information about the shape of a lattice, but also about its position in R n . To define the analogous space for d-lattices in R n , notice that since L n consists of a unique representative from any equivalence class of n-lattices in R n modulo homothety, one can identify L n with the space of such equivalence classes. The space of homothety classes of oriented d-lattices inside R n is where a matrix in SL n (R) lies in the coset that represents an equivalence class of a dlattice in R n if and only if its first d columns span a positive scalar multiplication of this lattice, with the corresponding orientation. The need to mod out by the block-scalar group follows from the fact that the first d columns of a matrix in SL n span a lattice that is hardly ever unimodular, so one needs to mod out by the covolume (hence an element in this quotient space is an equivalence class of d-lattices up to homothety). However, modding out the block-scalar group comes with a price, which is that the space L d,n does not carry an SL n (R)-invariant measure. To fix this flaw, we claim (Remark 4.1) that , and observe that, while the quotiented manifold is not a group, it is diffeomorphic to the group SO n (R) . Therefore, L d,n carries a Radon measure that is invariant under this group, and unique up to a positive scalar. This measure is the natural one in the sense that, as we shall see below, it projects to the uniform measures on L d and X d . With respect to it, L d,n has finite volume (which wouldn't have been the case if we hadn't modded out by the covolume, just like in the case of L n ).

Factor lattices and the space of pairs.
Recall that a lattice Λ is primitive if it is of the form Λ = Z n ∩ V Λ ; this is equivalent to the fact that any basis of Λ can be completed to a basis of Z n . Theorem 1.1 consists of a joint equidistribution result for primitive lattices Λ and their orthogonal complements It is a consequence of the stronger Theorem 3.2 below, in which Λ ⊥ is replaced by another (n − d)-lattice in V ⊥ Λ : Definition 2.1. The factor lattice Λ π of a primitive d-lattice Λ is the projection of Z n to V ⊥ Λ . The factor lattice Λ π is isometric to the quotient Z n /Λ (Proposition B.3), so one should think of Λ π as a realization of Z n /Λ inside R n . Notice that, like Λ ⊥ , Λ π is a full lattice inside V ⊥ Λ , and in particular is of rank n − d. The relation between Λ ⊥ and Λ π is that they are dual to one another (for the definition of dual lattices, see [Cas71, I.5], or Appendix A in the present paper. For the duality of Λ ⊥ and Λ π , see Claim B.5). It holds that Sch68]; see also Proposition B.4 and Corollary B.6 in the Appendix). Since we view d-lattices as carrying an orientation, we need to define an orientation on Λ π and Λ ⊥ , which is done as follows.
Definition 2.2. Let L be a full lattice in V ⊥ Λ . A basis C for L is positively oriented if det(B|C) = 1 for a positively oriented basis B of Λ.
The last space we introduce is the space of pairs of oriented lattices (Λ, L) such that (i) Λ is a d-lattice, (ii) L is a full lattice in V ⊥ Λ (hence is of rank n − d and its orientation is given in Definition 2.2), and (iii) covol(Λ) covol(L) = 1. In fact, it is the space of homothety classes of such pairs, where (Λ, L) is equivalent to (α 1 d Λ, α − 1 n−d L) for every α > 0. This space is given as the quotient Note the similarity to the definition of L d,n above; here also, modding out the blockscalar group results in having to SL n (R)-invariant measure, whih can be fixed by observing (Remark 4.1) that and therefore P d,n carries a Radon measure that is invariant under the group SO n (R) × P d (R) 0 d×n−d 0 n−d×d P n−d , and is unique up to a positive scalar. More details on dual lattices and factor lattices can be found in the appendix.

Relation between the spaces of lattices and their measures
While the spaces L d , X d and Gr(d, n) are well known, the spaces L d,n and P d,n , as far as we are aware, make their first appearance here. It therefore seems appropriate to explain how L d,n and P d,n add to the more "familiar" spaces L d , X d and Gr(d, n). It is not hard to see that P d,n projects canonically to L d,n , which in turn projects canonically to X d and Gr(d, n). In fact, one could also map L d,n → L d and P d,n → L n−d , but these projections are not cannonical, and depend on a choice of coordinates on the ddimensional (resp. n − d dimensional) vector spaces in R n . Such a choice of coordinates will be introduced in Section 4.1, defining the non-canonical identifications The full picture is depicted in the following commutative diagram of projections: Notation 2.3. Given an oriented d-lattice Λ < R n we denote its homothety class by [Λ] ∈ L d,n , and its shape by shape (Λ) ∈ X d . Given a pair (Λ, L) of lattices in orthogonal subspaces where Λ is primitive of rank d, we denote its homothety class by [(Λ, L)] ∈ P d,n . The image of Λ in L d (resp. of L in L n−d ) is denoted Λ (resp. L ).
Remark 2.4. For completeness, we provide an explicit description of the above maps. The projection P d,n → L d,n is the one of modding by \ from the right, and the projection L d,n → Gr(d, n) is modding by SL d (R) 0 d×n−d 0 n−d×d I n−d from the right. The projections L i → X i for i = d, n − d are given by modding from the left by SO i (R). The projections P d,n → L d,n and P d,n → X n−d , as mentioned, depend on a choice of coordinates and will be explicated in Remark 4.2. The projection P d,n → X d is modding from the left by SO n (R) and similarly for P d,n → X n−d . The projections depicted in Diagram (2.2) are critical to understanding why the measures we introduced on L d,n and P d,n are indeed the natural uniform measures on these spaces; as we shall see below (Lemma 2.5), with a consistent choice of normalizations, these measures project to the uniform measures on L d , X d and Gr(d, n).
Normalizations of the measures. As explained above, all the spaces considered are endowed with invariant Radon measures that are unique up to a positive scalar, and have finite volume with respect to these measures. Hence, by fixing the volume of the entire space, a unique measure is determined. This is what we now turn to do, where all the unique measures obtained are denoted by vol (the space in question should be understood from the context).
In accordance with the fact that SO d (R) / SO d−1 (R) is diffeomorphic to the sphere S d−1 , we choose a Haar measure on the special orthogonal group such that Haar(SO d (R)) = Leb(S d-1 ) · Haar(SO d−1 (R)), namely: We refer to [Gar14] for the fact that, with our standard choice (Section 4.3) of Haar Similarly, in accordance with the fact that SO n (R) /(SO d (R)×SO n−d (R)) is diffeomorphic to the Grassmannian Gr(d, n), we choose to normalize the volume of the Grassmannian as Haar(SO n (R))/(Haar(SO d (R)) Haar(SO n−d (R))), namely .
Finally, recall from Theorem 1.1 that where Z denotes the center. Note that the denominator is 1 if d is odd and 2 if it is even.
As we shall now prove (Lemma 2.5), fixing the volumes of L d and Gr(d, n) determines the volumes of L d,n and P d,n in correspondence with the identifications in (2.1). In fact, Interactions between the measures. We claim that the measures we introduced on L d,n and P d,n are the products of uniform measures in a manner that corresponds to the identifications in (2.1). For example, if Ψ ⊆ L d,n projects to the sets E ⊆ L d and Φ ⊆ Gr(d, n), then vol(Ψ) = vol( E) vol(Φ). This (and in particular, (2.3)) is a consequence of the following lemma.

Counting lattices: Main theorem
Theorem (1.1) stated in the Introduction is in fact a consequence of the more general Theorem 3.2, which we now introduce. It our main result, which we prove throughout the rest of the paper. This theorem concerns the equidistribution of [(Λ, Λ π )] ∈ P d,n for primitive d-lattices Λ, implying the equidistribution of the projections of [(Λ, Λ π )] to all the spaces below P d,n , e.g.
Then Theorem (1.1) follows from the fact that the equidistribution still holds when replacing Λ π by its dual, Λ ⊥ , which is the content of Theorem (3.4).
Let us provide more details. First of all, for the counting to hold, we require that [(Λ, Λ π )] and its projections to all the spaces below it fall in sets that satisfy the following regularity condition: is contained in a finite union of embedded C 1 submanifolds of M, whose dimension is strictly smaller than dim M. In particular, B is a BCS if its (topological) boundary consists of finitely many subsets of embedded C 1 submanifolds.
A few more notions before we can state our main theorem. We refer to the left } as the d component, and to the right component in these spaces as the n − d component. We say that the d (resp. n − d) component of a set in one of these spaces is bounded if the image of this set under the projection to X d (resp. X n−d ) is bounded. Notice that this is well defined since the projections to the shape spaces are canonical; they are also proper, so a set in is bounded if and only if both its d and n − d components are bounded.
In particular, the number of primitive Λ with covol(Λ) ≤ X whose homothety class lies inside a BCS Ψ ⊆ L d,n is The 1 2 factor is due to the fact that the lattices we count carry an orientation, so every non-oriented lattice is counted twice. We also note that a set that is BCS is Jordan measurable, and indeed Schmidt (in [Sch98]) provides an example for how the asymptotic formula for number of d-lattices with shapes in E fails when E is not Jordan measurable.
2. The case of d = n−1 was also obtained in [Mar10], using a dynamical approach, as well as in [HK19]. It was also considered in [AES16b, AES16a, EMSS16, ERW17, Ber19] in a more delicate setting.
3. Primitive d-lattices are in one-to-one correspondence with rational subspaces in R n . These spaces are the rational points on the Grassmanian variety: the projective variety consisting of all the d-dimensional spaces in R n . Therefore, the aforementioned result of Schmidt can be read as the counting of rational points up to a bounded height in the Grassmannian variety (the height being the covolume of the unique primitive lattice in the space). As such, it provides yet another example where the Manin conjecture [FMT89,Pey95] on counting rational points in varieties holds. The more refined counting we suggest in Theorem 3.2 (Part 4) plays a key role in the intensive study of rational points on the Grassmannian conducted in [BHW21]. In this paper, the authors confirm a modification to the Manin conjecture suggested by Peyre [Pey17], and obtain an equidistribution result for the integral lattices in the rational tangent bundle of the Grassmannian.
Note that Theorem (1.1) is just Part 1 of Theorem 3.2, only with Λ ⊥ instead of Λ π . As we have already mentioned, these lattices are dual to one another; we denote the dual of a lattice Λ by Λ * . Then Theorem (1.1) is obtained from Theorem 3.2 and the following.

Theorem 3.4. A version of Theorem 3.2 holds when replacing
Indeed, note that in all of the pairs above, the right-hand lattice spans the orthogonal subspace to the one spanned by the left-hand lattice. In other words, their homothety classes are elements in P d,n .
Proof of Theorem 3.4. Note that the three pairs Λ, Λ ⊥ , Λ * , Λ ⊥ and (Λ * , Λ π ) are obtained from the pair in Theorem 3.2 (Λ, Λ π ) by taking the dual of one of Λ, Λ π or of both. According to Proposition A.8 and Proposition A.9, taking the dual of one of the lattices in a pair or of both is a measure preserving auto-diffeomorphism of P d,n .

Refining the Iwasawa decomposition of SL n (R)
Set G = G n := SL n (R) and let G = KAN be the Iwasawa decomposition of G, meaning that K = K n is SO n (R), A = A n is the diagonal subgroup in G, and N = N n is the subgroup of upper unipotent matrices. We also let P n = A n N n . Consider the following isomorphic copy of SL d (R) × SL n−d (R) inside G, .
Let P := A N and Q := KP (note that Q is not a group, but it is a smooth manifold). To complete the definition of the Refined Iwasawa decomposition, we define K , A , N that complete K , A , N to K, A and N respectively. Let . Similarly for K ,P and A .
Fix a transversal K of the diffeomorphism K/K → Gr(d, n), meaning that K = K K ; later (Proposition 5.3) we will restrict to a specific choice of a transversal, with some desirable properties. Then Q is also K G , and we let Then the Refined Iwasawa (or RI, for short) decomposition is given by Remark 4.1. In particular, the two quotients with whom we expressed L d,n and P d,n in the Introduction indeed agree. For L d,n , we have that SL n (R) Similarly, Remark 4.2. The choice of a transversal K determines projections L d,n → L d and P d,n → L n−d : both these maps are induced by k g a n → g a n .
In particular, for every d lattice Λ and every (n − d) lattice L in V ⊥ Λ , the projections Λ ∈ L d and L ∈ L n−d from Notation 2.3 are now determined.

Parameterizations of the RI components
Our use of the RI decomposition of SL n (R) is motivated by the fact that, as we shall see in Section 5, the components appearing in this decomposition or some special subsets of theirs are in a sense isomorphic to the homogeneous spaces of SL n (R) that we mentioned in the Introduction. Specifically, we shall state (Proposition 5.3) that certain subsets of G d , P d , Q d and Q are parameterized by the spaces L d , X d , L d,n and P d,n ; in particular, Also, it is clear that K and N can be parameterized Gr(d, n) and R d(n−d) ; the parameterization for K ⊂ K by the Grassmannian will be the inverse of the restriction of the quotient map, and a choice of an isomorphism for N is arbitrary (but fixed). We let k U denote the element of K that corresponds to an oriented d- When D is a subset of a space that is parameterized by a RI component S (or a subset of it), we let S D denote the image of D in S under this parameterization. For example, D ⊂ R d(n−d) , then N D denotes its image in N . Finally, the groups A, A , A are clearly isomorphic to R n−1 , R d−1 × R n−d−1 and R respectively. We choose the following parameterizations: a = diag(a 1 , . . . a n ) will be written as

RI decomposition of the Haar measure on SL n (R)
It is well known (e.g. [Kna02,Prop. 8.43]) that a Haar measure on SL n (R) can be decomposed according to the Iwasawa components of SL n (R). Let us extend this to a Refined Iwasawa decomposition of the Haar measure on SL n (R). For every S ⊂ G = SL n (R) appearing as a component in the Iwasawa or Refined Iwasawa decompositions of G, we let µ S denote a measure on S as follows: µ K , µ N are Haar measures, and so do N and all three groups are unimodular, µ N = µ N × µ N . We assume that the Haar measures µ K and µ K are normalized such that The measures µ A , µ A , µ A are Radon measures such that Note that these measures are non-Haar. We use the measures on A i , A to determine a (normalization of the) Haar measure on G i : Since Q is diffeomorphic to the group K×P , we endow it with the Haar measure on this group: All in all we have the following: To abbreviate, we will use µ instead of µ G .

Interpretation of the RI components of g ∈ SL n (R)
Like the Iwasawa decomposition, the Refined Iwasawa decomposition induces coordinates on SL n (R). The RI components of an element g ∈ SL n (R) encode certain information regarding the lattices spanned by its columns, as explained in the proposition below. To state it, we extend the definition of primitiveness and of factor lattices from d-lattices in Z n , to d-lattices in any full lattice of R n .
Definition 4.3. Assume that a d-lattice Λ is contained inside a full lattice ∆ < R n . We say that Λ is primitive inside (or with respect to) ∆ if Λ = ∆∩V Λ . In other words, if there is no subgroup of ∆ that lies inside V Λ and properly contains Λ. Given a d-lattice Λ that is primitive inside ∆, we define the factor lattice of Λ (w.r.t. ∆), denoted Λ π,∆ , as the orthogonal projection of ∆ into the space (V Λ ) ⊥ . When Λ is primitive inside Z n , we omit the explicit mentioning of Z n , and say just that Λ is primitive. Accordingly, we denote Λ π , and say that it is the factor lattice of Λ.
Let us also introduce the following notation.
Notation 4.4. For g ∈ SL n (R) and d ∈ {1, . . . , n − 1}, let Λ g denote the lattice spanned by the columns of g, and let Λ d g denote the lattice spanned by the first d columns of g. Let Λ d ← − g denote the lattice spanned by the last d columns of Λ g . Finally, given a lattice Λ in R n , recall that V Λ denotes the linear space spanned by Λ.
and similarly for p . The RI components of g represent parameters related to Λ in the following way: Since the columns of k are obtained by performing the Gram-Schmidt orthogonalization procedure on the columns of g, we have that the first d columns of k span , then the first d columns of k span (and are in fact an orthonormal basis to) the same space as the first d columns of k, which is V Λ . This proves (i) and (i) , by definition of orientation on V ⊥ Λ . Write g(n ) −1 (a ) −1 = k g = q; right multiplication by an element of N does not change the first d columns of g, and right multiplication by (a ) −1 multiplies each of these columns by e − t d . This means that e − t d Λ = Λ d q , proving (iv), and (v), (vi) directly follow. Also, notice that Λ d q has covolume one, since it is a rotation of the unimodular Λ d g d ; then, by considering the covolumes of the lattices on both sides, we obtain e −t covol(Λ) = 1, proving (ii).
Considering g , it is clear that the lattice Λ g n−d is the factor lattice of Λ g d w.r.t.
Λ g . Rotating it by left multiplication by k , we have that the lattice Λ is the factor lattice of Λ d q w.r.t. Λ q . But since q = g(n ) −1 (a ) −1 , we may also say that Λ Noticing that this projection kills the contribution of (n ) −1 , as well as the first d columns of g(n ) −1 (a ) −1 , we remain only with the projection of the lattice spanned by the last n − d columns of g, on which (a ) −1 acts as multiplication by e − t n−d . In other words, Λ This proves (iv) , from which (v) and (vi) follow, and then similarly to how we proved (ii) we also obtain (ii) 1 . Since span orthogonal subspaces and are both unimodular, then (iv) and (iv) imply (vii).
It is well known that if g = kan and a = diag(α 1 , . . . , α r ), then i 1 α j = covol(Λ i g ). Since the lattice Λ d g d is a rotation of e −t/d Λ, it has the same covolume and partial covolumes; writing g d = k d a d n d , we have that the product of the first 1 ), while recalling that Λ

Sets of representatives for the spaces in Theorem 3.2
In this section we find "isomorphic" copies of the spaces X d ×X n−d , L d ×L n−d , L d,n , P d,n and Gr(d, n) inside the RI components of SL n (R), namely subsets of the RI components that are parameterized by these spaces. These will be in fact sets of representatives for the cosets that are the elements of these spaces, that are carefully chosen so that they imitate the geometry of the spaces as much as possible; for example, they will have the property that under the quotient map (that is one to one on a set of representatives), a BCS in the space will correspond to a BCS in the set. In [HK20, Section 6] we have defined such "good" sets of representatives, and named them spread models. For the convenience of the reader, we omit the definition, and instead list here the properties of spread models that will be used in this work.
The spread model we use for the space of shapes X i (i = d, n − d) is a fundamental domain for the action of SL i (Z) on SO i (R) \ SL i (R) ∼ = P i , whose construction is essentially due to Siegel and is made explicit in [Gre93, Sch98, HK20, Section 7]. In the case of i = 2, where P 2 is diffeomorphic with the hyperbolic upper half plane, this fundamental domain is the well known set depicted in Figure 1. Notation 5.2. We denote this fundamental domain in P i by F i . Also, we denote by F i the fundamental domain in SL i (R) for SL i (Z) that is obtained as ([HK20, Theorem 7.10 and Proposition 7.13]) where for every z ∈ F i , K z ⊂ SO i (R) is a fundamental domain for the finite group of rotations in SO i (R) that preserve Λ z .
Indeed, we will use F i as a spread model for X i , and F i as a spread model for L i . Here is the full list of spread models representing the spaces that appear in Theorem 3.2: Proposition 5.3 ([HK20, Prop. 8.1]). There exists a spread model K ⊂ K for the space Gr(d, n) ∼ = K \K = Q/G . Moreover, the following subsets of SL n (R) are spread models for the corresponding spaces: The measures we have defined on this spaces in the Introduction are the unique measures from part 2 of Proposition 5.1.

Correspondence between integral matrices and primitive lattices
The goal of this section is to translate Theorem 3.2 into a counting problem of integral matrices. The first step is to establish a correspondence between primitive d-lattices and integral matrices in a fundamental domain of the following discrete group of SL n (R): Proposition 6.1. There exists a bijection Λ ↔ γ Λ between oriented primitive d-lattices and integral matrices in a fundamental domain of SL n (R) Γ, that sends a d-lattice Λ to γ Λ , the unique integral matrix in the fundamental domain whose first d columns span Λ.
Proof. The direction ⇐ is simple: given γ ∈ Ω ∩ SL n (Z), its columns span Z n hence by definition its first d columns span a primitive d-lattice. In the opposite direction, let B be basis for Λ. Since Λ is primitive, B can be completed to a basis of Z n ; let γ ∈ SL n (Z) be a matrix having this basis in its columns, with B in the first d columns. The orbit γ · Γ, whose elements consist of integral matrices having a basis for Λ in their first d columns, meets Ω in a single point, γ Λ .
Let us construct an explicit fundamental domain for Γ. Let := the unit cube (−1/2, 1/2] d(n−d) and, recalling Notation 5.2, set (6.1) It is easy to see that Ω is a fundamental domain for the right action of Γ on SL n (R). The next goal is to reduce the proof of Theorem 3.2 into a problem of counting integral matrices in subsets of SL n (R), and specifically of Ω. We begin by defining these subsets.
Ψ ⊆ L d,n and Ξ ⊆ P d,n , consider Ω T (Φ, E, F) = Ω ∩ g = k k p a n : and Now the following is immediate from Proposition 6.1 and Proposition 4.5: Corollary 6.3. Consider the correspondence Λ ↔ γ Λ between primitive d-lattices and matrices in Ω ∩ SL n (Z), and let T > 0.

For Φ ⊆ Gr(d, n) and E × E
3. For Ψ ⊆ L d,n and F ⊆ L n−d ,

Neglecting lattices up the cusp: reduction to counting in compact sets
We are now at the point where we have reduced the proof of Theorem 3.2 to a problem of counting integral matrices inside the subsets of SL n (R) that are defined in Notation 6.2. The main obstacle in handling this counting problem, is that these sets are not compact: while their A component is bounded in [0, T ], their A component is not bounded from above. This section is devoted to tackling this issue, by reducing to counting in compact subsets of Ω. These subsets will be obtained by truncating the A coordinates of the sets from Notation 6.2; truncating the coordinates in A d will result in bounding the d component, and truncating the coordinates in A n−d will result in bounding the n − d component: where π A is the projection to the A component. If π A n−d (B) = e, then we denote B S , and similarly if π A d (B) = e then we denote B W .
Recall the fundamental domain Ω ⊂ G for Γ appearing in Formula 6.1. Accordingly with Notation 7.1, we let Ω S,W T Proof. Let γ Λ = ka s,w a t n ∈ Ω T ∩ SL n (Z). In what follows, Λ i is the lattice spanned by the first i columns of γ Λ , (Λ π ) j < Λ π is the lattice spanned by the first j columns of the n × (n − d) matrix obtained by projecting the columns of γ Λ to V ⊥ Λ , and Λ ⊥ k < Λ ⊥ is the lattice spanned by the first k columns of the matrix that represents Λ ⊥ inside F n−d .
Recall that γ Λ ∈ Ω T − Ω By where C > 0. Hence, up to an additive constant that becomes negligible when t is large, w j ≥ ω j T implies .

Almost a proof of the main theorem
In this section, we prove Theorem 3.2 based on a proposition for counting lattice points in SL n (R) that we state here and prove in Section 9. The statement includes a parameter τ of lattices in Lie groups, that we now define.

When only S is bounded the condition on S (T ) + W (T ) becomes W (T ) ≤ nδλ n T + O Ξ (1), and when only W is bounded the condition becomes S (T ) ≤ nδλ n T + O Ξ (1).
Proof of Theorem 3.2 assuming Proposition 8.1. By Corollary 6.3, when setting T = log X, the quantities we seek to estimate in parts 1,2,3 and 4 of the theorem is in one to one correspondence with the integral matrices in the following subsets of SL n (R), respectively: 1) Ω T (Φ, E, F), 2) Ω T (Φ, E, F), 3) Ω T (Ψ, F), or 4) Ω T (Ξ). Observe that, according to Section 4.3 on the measures on the RI components, to Proposition 5.3 on the spread models for X i , L i , L d,n and P d,n , to 5.1(2) for the agreement of the measures on the lattice spaces and the measures on their spread models, and to our choice of measure on SO i (R), indeed the main terms in Theorem 3.2 are the volumes of these sets, divided by the measure of SL n (R) / SL n (Z) which is vol(L n ) = n i=2 ζ (i). First we claim that it is sufficient to prove part 4 of the theorem, and then the other parts will follow. Indeed, the family Ω T (Φ, E, F) appearing in part 1 is a special case of the family Ω T (Φ, E, F) appearing in part 2, by taking E, F to be the lifts of E, F. According to Lemma 2.5(1), E, F are BCS's when E, F are, and vol( E) = vol(E)Υ(d) (similarly for F and n − d). In particular, the main term provided in part 1 of the theorem for Φ, E, F coincides with the one provided in part 2 for Φ and the lifts of E, F. Hence, part 1 is a consequence of part 2. Similarly, due to Lemma 2.5(3) and (4), parts 2 and 3 respectively are a consequence of part 4. We conclude that it is sufficient to prove part 4 of the theorem.
Let 0 < < τ n and 0 < δ < τ n − . Suppose first that the d component of Ξ ⊆ P d,n is not bounded, and n − d component is bounded. Recall λ n = n 2 2(n 2 −1) and let According to Remark 7.2, we may replace the measure of (Ω) σ d T,W Ξ T (Ξ) in the expression above with the measure of Ω T (Ξ), since the difference in measures is O(e nT (1− δλn d−1 ) ), which is swallowed in the error term. Then we choose δ that will balance the two error terms above, i.e. δ that satisfies: The family {B T } is Lipschitz well-rounded (LWR) with (positive) parameters (C, T 0 ) if for every 0 < < 1/C and T > T 0 : The parameter C is called the Lipschitz constant of the family {B T }.
The definition above allows any family {O } >0 of identity neighborhoods; in this paper we shall restrict to the following: is a parameter that depends on the rate of decay of the matrix coefficients of the Grepresentation on L 2 0 (G/Γ). The parameter T 1 is such that T 1 ≥ T 0 and for every (9.2) Proof of Proposition 8.1. To prove part 1 using Theorem 9.4, we must show that the family {Ω (S,W ) T (Ξ)} T >0 is Lipschitz well rounded. Define the map a , a , n , n ) .

Appendix: Dual lattices, factor Lattices
This appendix is completely independent from the paper. It aims to be a source for facts that are not necessarily unknown or hard to prove, but can be hard to find in the literature. For example, while the concept of dual lattices (sometimes referred to as polar lattices) is well known [Cas71], and we have also found the notes [Reg04] very useful), the concept of factor lattices, a term coined by Schmidt in [Sch68], is much less known, and in fact we have not found any reference for it. Another aim of this appendix is to add the differential perspective of smooth maps between spaces of lattices. For example, we will see that the map sending a d-lattice to its dual is an auto-diffeomorphism of the space of d-lattices inside R n .
In what follows, if the columns of a matrix B = B n×d span a d-lattice Λ < R n (resp. a subspace V < R n ), we say that B is a basis for the lattice Λ (resp. the subspace V ). Recall that the covolume of a d-lattice Λ, which is the volume of the fundamental parallelepiped for Λ in the linear subspace that it spans, equals (| det(B t B)|) 1/2 where B is (any) a basis for Λ. We will use an underscore to denote that an object is being spanned by a set, so that Λ B is the lattice spanned (over Z) by B, V Λ is the linear space spanned (over R) by Λ, etc.

A Dual lattices
Given a d-lattice Λ < R n , we define the dual lattice of Λ as Note that it is contained in V Λ , and that a priori, it is unclear that Λ * is indeed a lattice. This proposition motivates the notation D = B * for D as above, as well as the name the dual basis of B. The proof will make use of the following lemma: Lemma A.2. B and D span the same d-dimensional linear space in R n , and their columns satisfy the orthonormality relation Proof of Proposition A.1. In order to show Λ D = Λ * B , we will prove two inclusions. For Λ D ⊇ Λ * B , we first note that by Lemma A.2, V D ⊇ Λ * B . As a result, an element y ∈ Λ * B is of the form be the columns of B. It remains to show that α j ∈ Z for every j, which is the case since where in the second equality we applied Lemma A.2, and in the final inclusion we used b i ∈ Λ B and y ∈ Λ * B . For the reverse inclusion Λ D ⊆ Λ * B , it is sufficient to check that D ⊆ Λ * B , namely that x, d j ∈ Z for every x ∈ Λ B and j = 1, . . . , d.
Then again by Lemma A.2, Corollary A.3. The dual lattice Λ * is a d-lattice, that spans the same linear subspace as Λ.
Proof. Check that D(D t D) −1 = B and that (| det(D t D)|) 1/2 = 1/(| det(B t B)|) 1/2 . Example A.5. The dual of Z n is Z n , but in general the dual of an integral (and even primitive) lattice must not be integral. For example, the dual of Z(1, 1) is Z( 1 2 , 1 2 ). More generally, when Λ < Z n , then the entries of Λ * are in the ring Z[ 1 covol(Λ) 2 ]. To see this, recall that (Proposition A.1) if B is a basis for Λ, then B(B t B) −1 is a basis for Λ * . Since (B t B) −1 = adj (B) / det(B t B) = adj (B) / covol(Λ) 2 where adj (B) is the adjucate matrix of B, and adj (B) is integral since B is, we get that the entries of (B t B) −1 (and therefore also the entries of B(B t B) −1 ) are in Z[ 1 covol(Λ) 2 ]. Consider the following two spaces: Let us justify why these quotients are indeed the spaces of lattices we claim they are. A coset of a group element g inside L d,n corresponds to the d-lattice spanned by the first d columns of g. A coset of a group element g inside P d,n corresponds to the pair (Λ, L), where Λ is the d-lattice spanned by the first d columns of g, and L is the lattice spanned by the orthogonal projections of the last n − d columns of g to V ⊥ Λ . Note that since the columns of g are independent, and the first d columns span V Λ , then the projections of the last n − d columns to V ⊥ Λ must be independent; in particular, L is an (n − d)-lattice. Proving that these two identifications are well defined and bijective is an easy exercise.
Proposition A.6. The map Λ → Λ * is a measure preserving diffeomorphism from L d,n to itself.
Proof. Write a matrix g ∈ GL n (R), as two rectangular matrices g = [B|C], where B n×d and C n−d×n . Define a map from GL n (R) to itself by , and so it descends to a map from L d,n to itself . The latter maps the lattice represented by the coset of g, Λ = Λ B , to Λ B(B t B) −1 , which is Λ * , according to Proposition A.1. Hence, the descended map from L d,n to itself is Λ → Λ * . Being an involution by Corollary A.4, this map is bijective and measure preserving. It is a diffeomorphism (into its image, which is the whole space), since it is obtained from a map on GL n (R) which is a diffeomorphism, because it is algebraic and therefore smooth, and since it is its own inverse, hence also its inverse is smooth.
Following Proposition A.6, one expects that (Λ, L) → (Λ * , L * ) is a measure preserving diffeomorphism of P d,n . Proving this fact will require the following lemma.
To show Λ D ⊆ Λ π(C) , it suffices to show that D ⊆ Λ π(C) . Namely, that for every c ∈ C and y ∈ D it holds that π (c) , y ∈ Z. Indeed, π (c) , y = c, y ∈ Z where the equality is due to the fact that y ∈ V ⊥ B and the inclusion holds since c ∈ B|C and y ∈ (B|C) * . This shows that Λ D ⊆ (Λ π(C) ) * . For the reverse inclusion, let y ∈ (Λ π(C) ) * . Since in particular y ∈ V D , so y = α i d i . We need to show that α i are integers. indeed, Proof. Consider first the map GL n (R) → GL n (R) defined in the proof of Proposition A.6. This map is well defined modulo the subgroup GL d (Z) R d,n−d 0 n−d×d GL n−d (Z) , and so it descends to a map from P d,n to itself. As such, according to the explanation about the identification between cosets of this subgroup in SL n (R) and elements (Λ, L) in P d,n , the descended map from P d,n to itself is (Λ * , L) → (Λ * , L). Similarly to the proof of Proposition A.6, the map is a measure preserving diffeomorphism. For

Spaces of unimodular lattices
The space Indeed, the dual of a unimodular lattice is also unimodular, by Corollary A.4; so the map Λ → Λ * is an involution of L d,n , and similarly the maps in Proposition A.8 are involutions of P d,n . The proofs for the adaptations of Propositions A.6 and A.8 to unimodular lattices are obtained by adjusting the proofs of these propositions: the appearances of GL are replaced by SL, and the ambient group should be quotiented by A as well.
Remark A.10. Since the spaces L d,n and P d,n are unbounded but of finite volume, it is desireble to have a criterion for determining when a set is compact, or alternatively, when does a sequence of elements in the space diverge to infinity. For the more familiar space of full unimodular lattices in R d , SL d (R) / SL d (Z), which is also non-compact but of finite measure, the answer is provided by Mahler's Compactness Criterion [BM00,Chapter V.3]. The latter states that a set is compact if and only if there exists δ > 0 such that all the lattices in this set have shortest vector of length > δ. Equivalently, a sequence of lattices {Λ m } diverges if and only if the lengths of the shortest vectors in {Λ m } is a squence of positive real numbers that converges to zero. The purpose of the following is to state an analogous criterion for compactness in the spaces L d,n and P d,n . For every equivalence class [Λ 0 ] ∈ L d,n (resp. [(Λ 0 , L 0 )] ∈ P d,n ), a unimodular representative is the unique representative Λ ∈ [Λ 0 ] of covolume one (resp. (Λ, L) ∈ [(Λ 0 , L 0 )] such that Λ and L are of covolume one).
Proposition A.11. A subset Ψ of L d,n is bounded iff there is some δ > 0 such that for every [Λ 0 ] ∈ Ψ, its unimodular representative Λ ∈ [Λ 0 ] has the property that the shortest vector of Λ is of length ≥ δ.
Proof. We will prove the claim for P d,n , since case of L d,n is similar. It is not hard to see (e.g., Remark 4.1 in the present paper) that P d,n = Q/G (Z) where , is a fundamental domain representing P d,n with the property that the restriction of the natural projection π : Q → P d,n to Since K is bounded, and by definition of the Siegel sets (or the construction of is unbounded if and only if its projection to the diagonal subgroup in G is unbounded. Again by the construction of the Siegel sets, an element in the projection of ) is bounded from above. As a consequence, Ξ is bounded if and only if the entries a 1 and b 1 in the diagonal components of π| −1 (Ξ) are bounded from below by some > 0.
Let Λ 1 and L 1 be the rank one lattices spanned by the shortest elements in Λ and L respectively. According to Proposition 4.5 parts (ii), (ii) # , (iii), (iii) # , and using the construction of a Siegel set (that relies on the notion of a Siegel basis for a lattice, in which the first element is a shortest vector in a lattice), the boundedness from below of a 1 and b 1 is equivaent to boundedness from below of the covolmes of Λ 1 and L 1 , which are clearly the lengths of the shortest vectors in Λ and L respectively.

Relation between symmetries of a lattice and of its dual
A shape of a lattice (sometimes referred to as a type of a lattice) is its similarity class modulo rotation and rescaling. It is a very common parameter to study in the context of lattices, and appears e.g. in crystallography and the theory of periodic tilings. If two lattices in R n have the same shape, then in particular they are invariant under the same symmetries, meaning the same orthogonal transformations of R n . The opposite does not hold, e.g. the lattices Ze 1 ⊕Zαe 2 and Ze 1 ⊕Zβe 2 for 1 < α < β have different shapes, but both are invariant only under the orthogonal transformations ± [ 1 0 . Below we observe the surprising fact that even though a lattice and its dual in general do not have the same shape (e.g., Ze 1 ⊕ Zαe 2 and Ze 1 ⊕ Ze 2 /α are dual), they do share the same group of symmetries.
We conclude our introduction to dual lattices with a short discussion on the successive minima of the dual lattice.
Proof. From Theorem A.14 and Formula A.1 it follows that for every 1 ≤ j ≤ k, which proves the left-hand side inequality. The right-hand side inequality is proved similarly.

B Factor lattices
The term was coined by Schmidt in [Sch68], for primitive integral lattices. We extend it here to general lattices, that are not necessarily integral; for this, we begin by extending the definition of primitiveness to lattices that are not necessarily integral.
Definition B.1. Assume ∆ is a full lattice in R n , and Λ is a d-lattice that is contained in ∆. We say that Λ is primitive inside (or with respect to) ∆ if Λ = ∆ ∩ V Λ .
Note that when ∆ = Z n , this definition agrees with the standard definition of a primitive lattice. Indeed, in this case, we will call Λ primitive (and omit the "w.r.t. Z n ").
Definition B.2. Given a d-lattice Λ that is primitive inside ∆, define the factor lattice of Λ (w.r.t. ∆), denoted Λ π,∆ , as the orthogonal projection of ∆ into the space (span R (Λ)) ⊥ . When Λ is primitive inside Z n , we omit the "w.r.t. Z n " from the name and the notation: we denote Λ π and refer to it as the factor lattice of Λ.
For example, the factor lattice of Z(1, −1) is Z( 1 2 , 1 2 ). A priori, it is not clear that Λ π,∆ is indeed a lattice; the following proposition assures us that this is the case. Proposition B.3. For a d-lattice Λ that is primitive inside a full lattice ∆, consider the inner product on the quotient R n /V Λ : where π : R n → V ⊥ Λ is the orthogonal projection and ·, · is the standard inner product on R n . Then the quotient lattice ∆/Λ with ·, · R n /V Λ is isometric to the factor lattice Λ π,∆ with ·, · .
Then Λ π,∆ is a lattice, because it is isometric to one. The above also demonstrates that it is necessary to require that Λ is primitive inside ∆; otherwise, ∆/Λ (and therefore Λ π,∆ ) is a finite group.
In the case where ∆ = Z n , namely of primitive integral lattices, the concepts of a dual, a factor, and the orthogonal lattice are related in the following way. Proof. Let B n×d be a basis for Λ, and use primitiveness to complete it to a basis B n×n of Z n ; so B ∈ SL n (Z). Let D n×n be the dual basis of B, namely D = B(B t B) −1 = B −t , which is also in SL n (Z). Let D n×n−d denote the last n − d columns of D. Since B t D = I n , then B is orthogonal to D ; in particular, Λ D ⊆ V ⊥ B ∩ Z n = V ⊥ Λ ∩ Z n , so by definition of orthogonal lattice Λ D ⊆ Λ ⊥ . But this inclusion is in fact an equality, since on both sides there are primitive lattices in the same space V Λ and of the same rank.
Proof. this is a direct consequence of Corollary A.4, Proposition B.4 Proposition B.5, while noticing that covol(Z n ) = 1.
Remark B.7. One could wonder if the result of Proposition B.5 can be extended to general lattices, that are not necessarily primitive, or even integral. For example, it is quite natural to extend the definition of Λ ⊥ for general lattices in a similar fashion to what we did with Λ π,∆ , namely for a lattice Λ that is primitive inside a full lattice ∆, define the orthogonal lattice of Λ w.r.t. ∆, as ∆ ∩ (span R (Λ)) ⊥ . Say we denote it Λ ⊥,∆ . Now the question becomes, is it true that the dual of Λ π,∆ is Λ ⊥,∆ . The answer to this is no! Let us consider two counter examples.
The first example shows that the definition of Λ ⊥,∆ is in a sense meaningless; the second example shows us that Λ ⊥,∆ being the dual of Λ π,∆ fails even when ∆ is integral (which happens when we take α ∈ Z in the second example). What could be said in general about the dual of Λ π,∆ then? (Here Λ < ∆ < R n , with rank (Λ) < rank (∆) ≤ n, and ∆ must not necessarily be a full lattice). According to Lemma A.7, we know that a basis for (Λ π,∆ ) * is obtained in the last n − d columns of the matrix [B|C] * , where B n×d is a basis for Λ such that [B|C] is a basis for ∆. Now, even though (Λ π,∆ ) * needs not be integral when ∆ is not Z n (i.e. we are not in the case of Proposition B.5), it is "almost integral" when ∆ < Z n . Indeed, in this case (Λ π,∆ ) * is defined over the ring Z[ 1 covol(∆) 2 ], by a similar argument to the one in Example A.5.
The space of factor lattices of d-lattices in R n is the following: L d,n π := GL n (R) / GL d (R) R d,n−d 0 n−d×d GL n−d (Z) , since, given a coset of g ∈ GL n (R), the projection of the last n − d columns to the space that is orthogonal to the span of the first d columns, is the factor lattice of the lattice spanned by the first d columns, that is clearly an element in L d,n . It is easy to see that there is a natural bijection between the spaces L d,n π and L n−d,n : the factor lattice of a d-lattice in R n (w.r.t. any full lattice) is an (n − d)-lattice in R n ; conversely, if L ∈ L n−d,n , then by taking any d-lattice Λ in the space V ⊥ L , we have that L is the factor lattice of Λ w.r.t. the full lattice Λ ⊕ L. In fact, this bijection is a diffeomorphism between the two spaces: Proposition B.8. This bijection is a measure preserving diffeomorphism between L d,n π and L n−d,n .
Proof. Write a matrix g ∈ GL n (R), as two rectangular matrices g = [B|C], where B n×d and C n−d×n . Denote P = I n −B(B t B) −1 B t ∈ GL n (R), the orthogonal projection to V ⊥ B , and define a map from GL n (R) to itself by [B|C] → [PC|B]. This map is algebraic and therefore smooth, and since it is defined modulo the group GL d (R) R d,n−d 0 n−d×d GL n−d (Z) , it descends to a map from L d,n π to L n−d,n . Note that the lattice spanned by the first n − d columns of the resulting matrix is the factor lattice of Λ B w.r.t. Λ g . In particular, the resulting matrix lies in the coset of GL n (R) representing (in L n−d,n ) the (n − d)-lattice Λ π,Λg B , so it is indeed the direction L d,n π → L n−d,n in the bijection we described. Conversely, consider g = [C|B] ∈ GL n (R) with B n×d and C n−d×n and define the map [C|B] → [B|PC]. Again, it is smooth since it is algebraic, and it is defined modulo the group GL n−d (Z) R n−d,d 0 d×n−d GL d (R) so it descends to a map from L n−d,n to L d,n π . Again we note that the lattice spanned by the last n − d columns of the resulting matrix is the factor lattice for the lattice spanned by the first d columns w.r.t. Λ g , so this is indeed the direction L d,n π ← L n−d,n in the bijection we described. Finally, let us check that these two maps are the inverses of one another: where the equality is because P 2 = P. Now it is only left to observe that