Invariant measures on p-adic Lie groups: the p-adic quaternion algebra and the Haar integral on the p-adic rotation groups

We provide a general expression of the Haar measure $-$ that is, the essentially unique translation-invariant measure $-$ on a $p$-adic Lie group. We then argue that this measure can be regarded as the measure naturally induced by the invariant volume form on the group, as it happens for a standard Lie group over the reals. As an important application, we next consider the problem of determining the Haar measure on the $p$-adic special orthogonal groups in dimension two, three and four (for every prime number $p$). In particular, the Haar measure on $\mathrm{SO}(2,\mathbb{Q}_p)$ is obtained by a direct application of our general formula. As for $\mathrm{SO}(3,\mathbb{Q}_p)$ and $\mathrm{SO}(4,\mathbb{Q}_p)$, instead, we show that Haar integrals on these two groups can conveniently be lifted to Haar integrals on certain $p$-adic Lie groups from which the special orthogonal groups are obtained as quotients. This construction involves a suitable quaternion algebra over the field $\mathbb{Q}_p$ and is reminiscent of the quaternionic realization of the real rotation groups. Our results should pave the way to the development of harmonic analysis on the $p$-adic special orthogonal groups, with potential applications in $p$-adic quantum mechanics and in the recently proposed $p$-adic quantum information theory.


Introduction
During the last decades of the XX century, a new branch of mathematical physics, the socalled p-adic mathematical physics, has been developed as an effort to find a non-Archimedean approach to space-time and string dynamics at the Planck scale [1][2][3][4][5].Since then, various p-adic quantum mechanical models have been considered and studied [6][7][8][9][10][11][12][13][14][15][16][17][18][19], and several applications to quantum field theory and string theory have been proposed [20][21][22][23][24].Although the original focus of these theories was on the foundational aspects, further investigation has revealed new surprising applications, especially in the context of statistical and condensed matter physics.For instance, it has been observed that the natural ultrametric hierarchical structure of p-adic numbers makes them suitable for the description of the dynamics of chaotic and disordered systems.From this observation, Mézard, Parisi and their collaborators have shown, in the first half of the 1980s, that the ground state of the spin glasses exhibits a natural (non-Archimedean) ultrametric structure [25][26][27].
In more recent years, further intriguing applications of p-adic numbers have emerged, well beyond their original mathematical and physical context.Indeed, p-adic numbers have proved to be a valuable tool in solving issues related to algebraic dynamical systems, image analysis, compression of information, image recognition, cryptography and computer science (see [28], and references therein).Even more recently, there has been an increasing interest in the potential applications of the field of p-adic numbers to quantum information theory, as well [29][30][31].This interest stems from the unique properties of p-adic numbers, that may provide new solutions to challenging problems in quantum information science.E.g., it has been observed that the p-adic numbers can be profitably used in the construction of mutually unbiased bases (MUBs), for any Hilbert space dimension [32].
As a first step in establishing the foundations of a p-adic theory of quantum information, it has been argued [31] that a suitable model of a p-adic qubit can be obtained by resorting to twodimensional irreducible projective representations of the group of rotations on the configuration space Q 3 p (for an alternative 'purely p-adic' approach to the qubit also see [33]).The special orthogonal groups over the p-adic fields can be defined through quadratic forms over Q p .Unlike the real case, however, definite (i.e., representing the zero trivially) quadratic forms over Q p exist only in dimension two, three and four [34].The resulting symmetry groups SO(2, Q p ), SO(3, Q p ) and SO(4, Q p ) are the only compact p-adic special orthogonal groups.In particular, SO(3, Q p ) can be thought of as the group of rotations on Q 3  p , and its geometrical features have been explored in [35].The compactness of the aforementioned groups entails that all their irreducible unitary representations occur (and can be studied) as subrepresentations of the regular representation, according to the celebrated Peter-Weyl theorem [36].
Now, the study of the regular representation of compact groups -in particular, the formulation and the application of Schur's orthogonality relations -as well as several other fundamental issues of abstract harmonic analysis, involve the Haar measure on such groups, namely, the essentially unique (say, left) invariant Radon measure, or, regarding such a measure as a functional [36], the Haar integral.More generally, the irreducible -in general, projectiverepresentations of compact groups are square integrable (see [37][38][39][40], and references therein), and thus satisfy suitable orthogonality relations, where, once again, the Haar measure is involved.Still another class of problems where this measure plays a central role, is related to the 'phase-space' formulation of quantum mechanics [41].Here, the phase space appears in quotes for a two-fold reason: first, because a p-adic model of phase space is what we have in mind; second, because the usual group of translations on phase space (with its genuinely projective representations) is replaced with a locally compact group -e.g., with a compact p-adic Lie group -admitting square integrable representations.Such representations allow one to define generalized Wigner transforms mapping quantum-mechanical operators into complex functions on the relevant group [37][38][39][40].
In the present work -as a first point of an ideal program devoted to the study of harmonic analysis on the compact p-adic special orthogonal groups, and to its applications to quantum information science -we face the problem of describing the Haar integral on the p-adic Lie groups SO(2, Q p ), SO(3, Q p ) and SO(4, Q p ), for every prime p. Whereas the Haar measure on standard special orthogonal groups over the reals has been extensively studied using different approaches, the corresponding p-adic problem seems to be (to the best of our knowledge) still unexplored.
Our strategy to deal with such a problem articulates in two main steps: 1. On the one hand, we derive a general formula for the invariant measure on a generic p-adic Lie group.Our construction relies on the existence of a suitable atlas of mutually disjoint charts on such a group, which allows one to express its Haar measure in the local Q n pcoordinates by exploiting the change-of-variables formula for integrals on Q n p .Precisely, we first obtain a quasi -invariant measure on the group.At this point, we observe that every quasi-invariant measure on a (in general, locally compact) group immediately yields a Haar measure.This method is tailored on the peculiar properties of a p-adic Lie group, but we next show that our result can be interpreted within the invariant volume form approach to the Haar integral usually adopted for standard Lie groups over the reals.general approach may not be very practical or convenient, depending on the (more or less) manageable parametrization of the group one has to work with.In concrete applications, it is often more convenient to exploit a realization of the group one is interested in as a suitable quotient group X = G/H, where G is some suitable p-adic Lie group and H is a closed subgroup of G. Now, if the approach outlined in the previous step provides us with a convenient expression of the Haar integral on G, then we can simply lift the Haar integral living 'downstairs' on X to an integral 'upstairs' on G.This nice lifting strategy for computing the Haar integrals relies on the so-called Weil-Mackey-Bruhat formula [36,42,43].
A direct application of the general formula mentioned in the first point above easily yields the Haar measure on SO(2, Q p ).In dimensions three and four, instead, we find it convenient to introduce a quaternion algebra over Q p first.We can then realize SO(3, Q p ) and SO(4, Q p ) as suitable quotient groups and next apply the lifting strategy of the Haar integrals outlined in the second point above.This approach is reminiscent of the quaternionic realization of the standard rotation group SO(3) the reader may be familiar with.
The structure of the paper is as follows.In Section 2, we collect the basic notions and tools which will be used throughout the remaining sections of the paper.Specifically, in Subsection 2.1 we recall some basic facts concerning the Haar measure on locally compact groups and the lifts of Haar integrals on quotient groups.In Subsection 2.2, we discuss p-adic manifolds and introduce the notion of a p-adic Lie group, before delving into the specific class of p-adic special orthogonal groups, in Subsection 2.3.Section 3 deals with a general construction of the Haar measure on a p-adic Lie group, eventually showing that it naturally coincides with the measure associated with the (maximal-rank) invariant differential form defined on the group.Section 4 is devoted to the applications of the (previously constructed) theory to the p-adic special orthogonal groups in dimension two, three and four.Specifically, in Subsection 4.1, we derive the Haar measure on SO(2, Q p ).In Subsection 4.2, we explicitly construct, for any prime number p, the p-adic quaternion algebra and, in Subsection 4.3, we highlight its relation with the elements of SO(3, Q p ) and SO(4, Q p ).Then, in Subsections 4.4 and 4.5 we construct the Haar integrals on SO(3, Q p ) and SO(4, Q p ) by exploiting the suitable 'lifting strategy' and, hence, by realizing them as Haar integrals on specific subgroups of the p-adic quaternion algebra.Finally, in Section 5, conclusions are drawn, with a quick glance at future prospects.

Basic notions and tools
In this section, we collect some basic results and tools which will be relevant for all our later derivations.We begin by recalling the notion of Haar measure on a locally compact group.Then, we introduce the p-adic Lie groups and we consider, in particular, the class of p-adic special orthogonal groups.Finally, we provide a brief outline of integration theory on a p-adic manifold.

The Haar measure on a locally compact group and the lifts of Haar integrals
Let G be a locally compact (Hausdorff) topological group; in short, a LC group.By a left (resp.right) Haar measure µ on G we mean a non-zero Radon measure for which the following condition holds: for every Borel set E ⊂ G, and g ∈ G [36,44].We refer to (1) as to the left-invariance (resp.right-invariance) property of the measure.
It is worth recalling a remarkable characterization of the left (resp.right) Haar measure provided by a suitable left-(right-)invariance condition for a class of functionals on C c (G)the algebra of compactly-supported continuous complex-valued functions on G [36,45].
Remark 2.1.We are adopting the convention that the support, supp(f ), of a continuous function Let µ be a fixed Radon measure on a LC group G.The map defined as is easily seen to be a positive linear functional on C c (G).On the other hand, the celebrated Riesz Representation Theorem (cf.Theorem 7.2 in [45]) assures that for every positive linear functional on C c (G), there is a unique Radon measure µ on G such that I is represented as in (2).Exploiting this correspondence, a Radon measure µ is a left Haar measure iff the associated functional is left-invariant, i.e., iff the condition holds for every f ∈ C c (G).Here, the map By defining the right translation via (R h f )(g) = f (gh), we capture analogously right-invariance of the measure.In what follows, whenever µ is a Haar measure on G, we will refer to the integral in the r.h.s. of (2) as to the Haar integral associated with µ.
It is a well known result (see, e.g., Theorem 2.10 and 2.20 in [36]) that any LC group admits an essentially uniquely defined Haar measure.In particular, if µ and ν are left Haar measures on G, then there exists c ∈ R + * such that µ = cν.If G is a LC group, its left and right Haar measures are related via the so-called modular function ∆ : G → R + * [36].In the case where ∆ ≡ 1 (as it happens for abelian and compact groups), G is called unimodular, meaning that left and right Haar measures coincide.
Remark 2.2.A locally compact group G has finite left (and right) Haar measure µ if and only if it is compact [36,46]; in this case, it is possible (and customary) to normalize the Haar measure in such a way that µ(G) = 1.
Example 2.3 (Haar measure on Q p ).The (additive) group of the field of p-adic numbers Q p (p ∈ N prime) is a LC group once endowed with its standard ultrametric topology (namely, the topology induced by the non-trivially valued, non-Archimedean absolute value | • | p on Q p ).Therefore, it admits a left Haar measure λ.Since (Q p , +) is abelian (hence, unimodular), λ is right-invariant as well, i.e., holds for every Borel subset E in B Qp , and any x ∈ Q p .Since the subring Z p of p-adic integers is a compact subset of Q p , we can normalize λ by setting It is now not difficult to explicitly construct the measure λ.Indeed, let B(r, x 0 owing to the invariance condition (4) and the normalization (5), we get λ B(1, x) = 1 for every x ∈ Q p .Moreover, the topological features of Q p -i.e., any ball of radius p k , k > 0, is a disjoint union of p k balls of radius 1 -also entail that λ B(p k , x) = p k for every k ∈ Z, x ∈ Q p .Hence, we get to the conclusion that the measure of every Borel set E of Q p is given by analogous to the formula for the Lebesgue measure on the real line.
Example 2.4.The group Q n p = Q p × . . .× Q p (n-times), endowed with the product topology, has a natural structure of (additive) LC group; hence, it admits a left (and right) Haar measure.To find it explicitly, it is enough to observe that, being Q p a second countable LC group, there is no distinction between the standard product of measures and the Radon product (see §2.2 in [36]).Therefore, the Haar measure on Q n p is provided by the n-times product of the Haar measure on Q p , i.e., With a slight abuse of notation, we will denote by λ the Haar measure on Q n p for every n ∈ N, as the dimension n will be clear from the context.
Let G be a LC group, and let X be a LC Hausdorff space.We call X a (transitive) G-space whenever it is equipped with a (transitive) continuous left action ( If G is a locally compact second countable Hausdorff (in short, LCSC) group, and H a closed normal subgroup of G (e.g., the centre of G), let X ≡ G/H denote the quotient (LCSC) group.Furthermore, let q : G → X be the quotient map (i.e., the projection homomorphism) which is an open continuous map.We can then define a natural continuous action ( This action is transitive and, hence, turns X ≡ G/H into a transitive G-space.In the literature, one refers to such a G-space as to a homogeneous space [36, 42-44, 46, 47]. Let now µ G , µ H , µ X denote the (left) Haar measures on G, H, X ≡ G/H respectively, and let ∆ G , ∆ H be the modular functions on G and H.It is a standard fact that (since X admits a X-invariant, hence G-invariant, measure µ X ; see Theorem 2.51 of [36]) i.e., ∆ H = ∆ G | H . Therefore, if G is unimodular, then H shares the same property.Let (X, B X ), (Y, B Y ) be (Borel) measurable spaces.We recall that a map ϕ : is a quotient group, we also denote by s : X → G a Borel (cross) section of X into G, i.e., a Borel map satisfying the condition q(s(x)) = x, for every x ∈ X. Proposition 2.5 (Lemma 6 of [38]).For every Borel section s : X → G, the mapping is a Borel isomorphism (X × H being endowed with the product topology).
For every f ∈ C c (G), we put Remark 2.6.It is worth observing that the function is a compact subset of G and, hence, of H).Therefore, the integral on the r.h.s. of ( 11) is well-defined.
Remark 2.7.Note that, by the left-invariance of µ H , the integral H dµ H (h)f (gh) is constant w.r.t.g varying in q −1 ({x}), for every x ∈ X.Hence, (P f )(x) ∈ C does not depend on the choice of the cross section s.
Theorem 2.8.For every f ∈ C c (G), the function belongs to C c (X), and the mapping where the Haar measures µ G , µ H , µ X are supposed to be suitably normalized and s : X → G is any Borel cross section.
(Note: Since X, H are LCSC groups, in the first line of (13) it is not necessary to make a distinction between the standard product of measures and the Radon product [36].) Proof.See Section 2.6 of [36]; in particular, Proposition 2.50 and Theorem 2.51.
For every φ ∈ C c (X) and ψ ∈ C c (G), we set It is easy to see that L ψ φ ∈ C c (G); in particular, we have that is a compact subset of G.
Lemma 2.9.For every compact subset K of X, there exists a function ψ ∈ C + c (G) such that Here and in the following, we set Proof.Use Lemma 2.49 of [36].
By Lemma 2.9, for every nonempty compact subset K of X, we can define the following (nonempty) subset of C + c (G) By convention, we put Ψ ∅ = {ψ ≡ 0}.
The notion of Weil-Mackey-Bruhat lift in Definition 2.10 is strictly related to the WMB formula (13).Indeed, given a ψ-lift of a function φ ∈ C c (X), exploiting the WMB formula, it is not difficult to prove the following results: Lemma 2.11.For every φ ∈ C c (X), and every ψ ∈ Ψ supp(φ) , we have that Proof.In fact, by Lemma 2.9, we have: where s : X → G is any Borel cross section (q(s(x)h) = x).
We are now able to express any Haar integral on X as a Haar integral on G: Theorem 2.12.Let φ be a function in C c (X).Then, for every WMB lift where a suitable (mutual) normalization of µ X and µ G is assumed.
Proof.In fact, by the second assertion of Theorem 2.8, where, for the second equality, we have used Lemma 2.11.
We will call the Haar integral on the r.h.s. of (20) a lift of the Haar integral on the l.h.s. of the same formula.
In our specific applications, X = G/H will be a compact group.In this case, some of the previously discussed results admit a remarkable generalization.To start with, let us notice that, when X is compact, C c (X) coincides with the set C(X) of all continuous functions on X.Let us put Ψ ≡ Ψ X .From Theorem 2.12, we can immediately prove the following: Corollary 2.13.Let X = G/H be compact.Then, for every ψ ∈ Ψ, we have that where a suitable (mutual) normalization of µ X and µ G is assumed.
Remark 2.15.Without any assumption of compactness of X, the map P : C c (G) → C c (X) can be extended to a (surjective) map P : L 1 (G) → L 1 (X), defined by see Lemma 7 of [38] and Theorem 3.4.6 of [42] (L 1 (G) ≡ L 1 (G, µ G ) denotes the set of complexvalued functions on G whose absolute value is integrable w.r.t.µ G ).Moreover, the extended WMB formula holds: for a suitable (mutual) normalization of the Haar measures µ X , µ G .
The forthcoming Theorem will provide us with a suitable generalization of the results in Lemma 2.11 and in Theorem 2.12, tailored to the case where X = G/H is a compact group.To this end, we find useful to preliminary recall the notion of pushforward measure.Definition 2.16.Let (X, B X ) and (Y, B Y ) be (Borel) measurable spaces.Let µ be a Borel measure on X.
for every Borel set E in B Y .
Remark 2.17.If (X, B X ) and (Y, B Y ) are Borel measurable spaces, and if f : Y → R is a Borel function on Y , the following (abstract) change-of-variables formula (C.O.V.F., in short) holds [48]: Moreover, from (26), it is not difficult to prove the following relation [48]: for every Borel function g : X → R. We shall constantly resort to this formula in our description of integration theory on Q p -manifolds.
We are now ready to prove the following result Theorem 2.18.Let X = G/H be compact.Then, for every ψ ∈ Ψ, the map L ψ : C(X) → C c (G) admits an extension -a so-called extended WMB lift - that is a right inverse of P : Moreover, for every φ ∈ L 1 (X), we have that for a suitable (mutual) normalization of µ X , µ G .
In fact, by Lemma 7 of [38], for any Borel section s : X → G, the pushforward measure (γ s ) * (µ X × µ H ) coincides (up to normalization) with µ G .Hence, we have that where the last equality is obtained by Tonelli's theorem.Therefore, we find that and L ψ φ ∈ L 1 (G).At this point, one easily proves (30) and (31).
To conclude this section, we state the following remarkable consequence of Theorem 2.8 Theorem 2.20.Let us suppose that H is compact.Then, for a suitable normalization of µ G and µ X , q * (µ G ) = µ X .
Proof.Since H is compact, q * (µ G ) is a Radon measure on X (for every compact E ⊂ X, q −1 (E) = KH, with K ⊂ G compact by Lemma 2.48 of [36]; hence, KH is compact too).Then, for every φ ∈ C c (X), by the WMB formula we have that where we have assumed that µ H (H) = 1 and µ G ,µ H are suitably normalized.Hence, q * (µ G ) = µ X .

p-Adic Lie groups
In this subsection, we discuss the main features of p-adic manifolds and p-adic Lie groups [49][50][51].
As in the standard real setting, the starting point is to introduce a suitable notion of chart on a Hausdorff space.Definition 2.21.Let X be a Hausdorff space.A chart on X is a triple (U , ϕ, Q n p ), where U ⊂ X is an open subset and ϕ : U → Q n p is a map such that ϕ : U → ϕ(U ) is a homeomorphism.We refer to U as the domain of the chart, and to n ∈ N as its dimension.
If x ∈ U ⊂ X , we say that (U , ϕ, Q n p ) is a chart around x.In the following, we will set ↼ ϕ : ϕ(U ) → U to be the inverse map of ϕ on its range.
Definition 2.24.An atlas A for X is a family {(U α , ϕ α , Q nα p )} α∈A of pairwise compatible charts which cover X , i.e., X = α∈A U α .An atlas A for X is called n-dimensional if all the charts in A have dimension n.
Similarly to the standard real case, it is now natural to set the following: Definition 2.25.A Hausdorff space, X , together with a maximal (w.r.t.inclusion) atlas A is called a Q p -analytic manifold.The manifold is called n-dimensional if the atlas A is ndimensional.
For notational convenience, in what follows we shall denote an n-dimensional atlas on X as A = {(U α , ϕ α )} α∈A ; moreover, we will refer to 'Q p -analytic manifold' simply as 'Q p -manifold'.If X , Y are two Q p -manifolds of dimension m and n respectively, we shall say that a map f from X to Y is Q p -analytic if, for every x ∈ X , there exist a chart (U , ϕ, Q m p ) on X around x, and a chart (V , ψ, Remark 2.26.Every Q p -manifold X is both totally disconnected and locally compact (TDLC in short).In particular, the latter condition entails that for every point x of X , the set T x of all compact open subsets in X containing x forms a base at x (see Lemma 7.1.1 in [52]).Therefore, the set T (X ) = x∈X T x of all the compact open subsets of X forms a basis for the topology of X .
Analytic differential forms on Q p -manifolds are defined in a similar fashion to the standard real setting (see Chapter 2 in [52] for a thorough discussion).Indeed, let X be a Q p -manifold of dimension n, and let A = {(U α , ϕ α )} α∈A be an atlas on X .If Θ is a differential form of degree k < n on X , its restriction Θ α := Θ| Uα -in the local coordinates of (U α , ϕ α ) -is given by where θ α j 1 ...j k are Q p -valued functions on U α , and where we set ϕ α (u) = (x 1 , . . ., x n ) to denote the local coordinates of u in U α .If, for every α ∈ A, the maps θ α j 1 ,...,j k are all Q p -analytic functions on U α , we say that Θ is a Q p -analytic differential k-form on X .If Ω is a Q p -analytic differential n-form on X (i.e., of maximal degree equal to the dimension n of X ), its local expression Ω α := Ω| Uα can be written as for where (x i ) n i=1 and (y j ) n j=1 denote the systems of local coordinates of U α and V β respectively, and where DF is the Jacobian matrix of the transformation F .
To conclude this subsection, we now discuss the principal object of our investigations: Definition 2.27.A p-adic Lie group G is a Q p -manifold which is also a group, and such that the multiplication map From Definition 2.27, it follows that the inverse map, G ∋ g → g −1 ∈ G, is a Q p -analytic map.Moreover, it is clear that every p-adic Lie group is a TDLC Hausdorff space (see Definition 2.25 and Remark 2.26).
Remark 2.28.Let G be a p-adic Lie group.For h ∈ G, the map ℓ h of left translation by h is defined as: This map is the composition of the map G ∋ g → (h, g) ∈ G × G, and the multiplication map defined in (38); hence, it is Q p -analytic (the composition of Q p -analytic maps is a Q p -analytic map; see Lemma 8.4 in [49]).Similarly, one can define the map of right translation, r h , on G, which is Q p -analytic as well.
Remark 2.29.A classical result by van Dantzig (see Theorem 7.7 in [44]) states that a TDLC group admits a base at the identity consisting of compact open subgroups (and vice versa).This result provides a peculiar characterization of the topology of p-adic Lie groups.
Since a p-adic Lie group, G, is a Q p -manifold, we can clearly define differential k-forms on it.In particular, we say that a differential k-form holds for every g and h in G. Right-invariant differential n-forms are defined similarly with ℓ h replaced by r h .By taking h ≡ g −1 and g ≡ e in (40), we also see that that is, if Θ is left-invariant on G, its value at every point on G is determined by the value Θ assumes at the identity e in G.In the next subsection, we shall prove that a left-invariant n-form on G can always be constructed, and that it naturally induces the left-invariant Haar measure on G.For the moment, we want to stress a relevant topological feature of p-adic Lie groups which will turn out to be central in our later derivations.We recall that a Hausdorff space X is called paracompact, if every open covering of X can be refined into a locally finite open covering.We say that X is strictly paracompact if every open cover of X admits a refinement consisting of pairwise disjoint open sets.
Proposition 2.30.Let G be a second countable p-adic Lie group.Then, G is a strictly paracompact space.
Proof.By assumption, G is locally compact, second countable and Hausdorff, hence σ-compact (i.e.union of countably many compact subspaces).Every σ-compact space is Lindelöf, and, therefore, paracompact (cf.Theorem 5.1.11. in [54]).Then, the proposition follows by the equivalence of the points i and ii of Proposition 8.7 in [49].
Remark 2.31.From Proposition 2.30 it follows that any second countable p-adic Lie group G can always be endowed with an atlas consisting of pairwise disjoint charts.Indeed, since every atlas is an open covering, it admits a refinement consisting of pairwise disjoint open sets.Then, the restriction of the coordinate maps of the initial atlas to the sets in the refinement provides a new system of charts for G.
Remark 2.32.It is well known that a LC group G is Polish iff it admits a second countable topology (see Theorem 5.3 in [53]).Therefore, a second countable p-adic Lie group is also a Polish group.
In this work, p-adic Lie groups will always be assumed to be second countable (as so are the most important examples); hence, they are LCSC Polish groups.

p-Adic rotation groups
A noteworthy class of p-adic Lie groups is given by the special orthogonal groups over the p-adic fields.We devote this subsection to recall some of their basic properties [31,35].
The general definition of special orthogonal group is given in terms of quadratic forms Q : V → F, for V a vector space over a field F (see [34,55] for a thorough discussion).Quadratic forms, up to linear equivalence and scaling, lead to isomorphic special orthogonal groups [34].In this work, we always assume that the quadratic forms Q are non-degenerate (i.e. they have maximum rank).
If the characteristic of F is different from 2 (as it is for R and Q p ), a bilinear form b(x, y), x, y ∈ V, induces a quadratic form Q(x) = b(x, x) and, vice versa, a quadratic form induces a bilinear form, i.e.,

b(x, y)
Therefore, we are allowed to use interchangeably quadratic forms and bilinear forms.The unique non-degenerate definite quadratic form on R n , for every n ≥ 2, is given (up to linear equivalence and scaling) by Q R (x) = n−1 i=0 x 2 i .This is represented in the canonical basis by the n-dimensional identity matrix I n .Thus, the (compact) special orthogonal group over R of degree n is where • , • : R n × R n → R is the Euclidean scalar product on R n , and M n (R) denotes the associative algebra of n × n matrices over the field of real numbers R.
The following theorem characterizes the definite quadratic forms over Q p in every dimension, as explicitly derived in [35] (see also [34,55]).
Theorem 2.33.For every prime p > 2, let u ∈ U p be a non-square -with U p denoting the group of p-adic units, i.e., the group of all invertible elements of Z p -and let v ∈ U p be defined by In the case where p > 2, there are (precisely) three definite quadratic forms on Q 2 p , up to linear equivalence and scaling, and there are seven on Q 2 2 , namely, There is a unique definite quadratic form on Q 3 p (depending on p), up to linear equivalence and scaling, i.e., as well as on Q 4 p , i.e., No quadratic form on Q n p is definite for n ≥ 5.
Remark 2.34.Note that all restrictions of the definite quadratic form Q (4) on Q 4 p to any three variables, and indeed to any three-dimensional subspace, are equivalent to We can now characterize a relevant class of special orthogonal groups over Q p .
Corollary 2.35.The p-adic special orthogonal groups associated with the definite quadratic forms on Q 2 p are (up to isomorphism) where A κ are the matrix representations, in the canonical basis of Q 2 p , of the quadratic forms in (46) and (47).Index κ ranges in {−v, p, p u } whenever p > 2, while κ ∈ {1, ±2, ±5, ±10} when p = 2.For every p ≥ 2, the special orthogonal group associated with the definite quadratic form on while the one on A and A ′ are the matrix representations in the canonical basis of Q  [6,56].SO(n, Q p ), n = 2, 3, 4, are closed, as they are groups of solutions of a system of continuous (polynomial) equations.On the other hand, every matrix in the groups of Corollary 2.35 has bounded entries (see Theorem 5 and Remark 14 in [35] for the details); specifically, we have: This entails that M p ≤ p for every Remark 2.36.We have used definite quadratic forms to define the p-adic special orthogonal groups.It turns out that those groups defined on indefinite quadratic forms are not bounded, whence, not compact.
In the light of the discussion above, the following result is now clear: Proposition 2.37.The groups SO(n, Q p ), n = 2, 3, 4 of Corollary 2.35 are all and the only compact p-adic special orthogonal groups.
The next theorem provides a parameterization of the compact p-adic special orthogonal groups in dimension two [35].

Integration on p-adic manifolds
This subsection deals with integration theory on p-adic manifolds [52].For our purposes, we will need a p-adic counterpart of the well known change-of-variables formula for multiple integrals on R n .Therefore, we start with the following: between an open neighborhood U of a, and an open neighborhood V of ξ(a), such that Then, for every integrable function f on V , the following formula holds: where λ is the Haar measure on Q n p .
Proof.Formula ( 58) is actually a special case of the abstract C.O.V.F.(see (26) in Remark 2.17) specialized to the case where the pushforward of the measure on U is realized via a Q p -analytic map.See Proposition 7.4.1 in [52] for the technical details.
Let X be a second countable n-dimensional Q p -manifold, and Ω a differential n-form on X .If A = {(U α , ϕ α )} α∈A is an atlas on X , Ω is expressed as in (36) in the local coordinates of each chart in A. Then, we can associate a Radon measure µ Ω with Ω by setting for every compact (open) subset C ⊂ U α of X , and where d(( that is, µ Ω (C ) does not depend on the considered chart containing C .We shall give the proof of this result in Remark 2.42 below.
Remark 2.41.Since a second countable Q p -manifold is σ-compact, the measure (59) is regular (cf.Theorem 7.8 in [45]).Then, the measure of a Borel set E ⊂ U α of X is given, by inner regularity, by the supremum of the measures of the compact (open) sets contained in E.
µ Ω is also well-defined, and is given by Let now C be an arbitrary compact (open) subset of X .Its measure w.r.t.µ Ω can be defined as follows.First, we can decompose C as i.e., as a disjoint union of compact (open) subsets C i , each contained in some U α .Then, the measure of C is given by µ Similarly, we can then extend the measure (63) to arbitrary Borel sets E in X (see Remark 2.41).Exploiting (61), it is then not difficult to define the integral of an arbitrary function f ∈ C c (X ) w.r.t.µ Ω , as well.We can consider the pushforward of the measure µ Ω via ϕ α to a measure on Q n p .This allows us to treat the integration theory on a manifold X via integrals on Q n p .Indeed, using formula (27), we have: where ϕ α (u) = (x 1 , . . ., x n ) =: x denotes the coordinate representation of the point u ∈ X .Hence, using the (abstract) C.O.V.F.(cf.relation (26) in Remark 2.17) with f = χ C , we obtain With the above discussion, we get to the following two conclusions.First, from (59), we see that The latter condition means that the pushforward of the measure µ Ω on X via the maps ϕ α , for every α ∈ A, provides an absolutely continuous measure w.r.t. the Haar measure λ on Q n p .Their Radon-Nikodym derivative -which, for notational convenience, hereafter we will simply denote by η -is globally defined, and it is uniquely defined up to a set of points of null measure (any other Radon-Nikodym derivative is equal to η almost everywhere).Accordingly, we shall denote by p , for every U α in the covering atlas A of X .Exploiting this notation, and recalling condition (65), we can then write Remark 2.42.Using the local representation (68), it is now not difficult to prove the equality of integrals in (60).Indeed, let C ⊂ U α ∩ U β be a compact (open) set in X .Then, we want to show that We first consider the change of variable y = (ϕ β • ↼ ϕ α )(x) in the r.h.s. of (70).Then, using Theorem 2.40, we see that (70) holds iff where det D(ϕ β • ↼ ϕ α )(x) denotes the Jacobian of the transformation ϕ β • ↼ ϕ α .On the other hand, the pullback formula (37) also shows that Therefore, taking the p-adic absolute value of the l.h.s. and of the last equality in (72) entails that (71) (and, hence, (70)) holds.
To conclude this subsection, we prove that it is always possible to construct an essentially unique -i.e., uniquely defined up to a multiplicative constant -(left-)invariant differential nform on every n-dimensional p-adic Lie group.We will then show that it is naturally associated with the (left) Haar measure on the group.This will draw a parallel with the standard theory of (real) Lie groups [57,58].
Let us first note that, also in the p-adic setting the tangent space T e G to G at e ∈ G has a natural structure of Lie algebra g, whenever the elements X ∈ T e G are identified with the corresponding left-invariant vector fields X on G [50].Let X 1 , . . ., X n be a basis of T e G, and let X 1 , . . ., X n be the corresponding left-invariant vector fields in g.We can now define, for all g in G, the 1-forms ω 1 , . . ., ω n on G via the condition By construction, ω 1 , . . ., ω n are left-invariant 1-forms on G, as follows by observing that In particular, this also entails that ω 1 , . . ., ω n form a basis of the dual space of T g G for every g ∈ G. Therefore, the differential form Ω inv defined as is a (nowhere vanishing) left-invariant n-form on G. Indeed, since the pullback ℓ * g commutes with ∧, we have: that is, Ω inv is left-invariant.It is clear that any constant multiple of Ω inv is a left-invariant nform as well.Conversely, if Ω is another left-invariant n-form on G, there must exist c ∈ Q p such that Ω(e) = cΩ inv (e).But then, the left-invariance condition (41) entails that Ω(g) = cΩ inv (g) for every g in G.
We want now to show that if Ω inv is the left-invariant differential n-form on G, its induced measure, µ Ω inv , is the left Haar measure on G (up to multiplicative constants).Indeed, we already know that µ Ω inv is a Radon measure.To conclude that it is a Haar measure, we have to show that it is left-invariant.Let C be a compact (open) set in G. From the left-invariance of Ω inv we see that for every g ∈ G.This entails that µ Ω inv (gC ) = µ Ω inv (C ), for every compact (open) set C ⊂ G, and g in G (see (65)).Moreover, since G is second countable, µ Ω inv is regular.In particular, inner regularity entails that for every Borel set E in G, and every g ∈ G. Concluding, we proved that µ Ω inv is a left-invariant Radon measure on G, and since the Haar measure is essentially uniquely defined, it must coincide with the Haar measure on G up to a multiplicative constant.
3 The Haar measure on p-adic Lie groups In this section, we show how to construct a left Haar measure µ on a (second countable) p-adic Lie group G. Our approach exploits the peculiar topological features of p-adic Lie groups, and relies on the possibility to construct a quasi-invariant measure for G.Eventually, we will prove that the measure thus constructed coincides with the measure induced by the left-invariant differential n-form Ω inv on G (see Subsection 2.4).We begin by recalling the notion of a quasi-invariant measure [36].Let G be a p-adic Lie group, and let ν be a Radon measure on it.For h ∈ G, we can define the left translation ν h , of ν by h, as for every Borel set E ∈ B G .We say that ν is quasi-invariant if the measures ν h are all equivalent, i.e., mutually absolutely continuous [45].In such a case, we have: where η : The function η is the Radon-Nikodym derivative dν h /dν.For h, h ′ ∈ G, since ν hh ′ = (ν h ) h ′ , the chain rule for the Radon-Nikodym derivative entails the following cocycle formula: for every g ∈ G.In particular, using (81) it is not difficult to prove the following result.
Lemma 3.1.Let G be a p-adic Lie group, and let ν be a quasi-invariant measure on it.The measure defined as dµ(g) := η(g, e) −1 dν(g) -where e denotes the identity element in G -is a left Haar measure on G.
Proof.Let µ h be the left translation, by h in G, of the measure µ, as defined in (79).For every Borel set E in B G , we have: where in the last equality we have used the change of variable h −1 g → g.Then, taking into account condition (80) for quasi-invariant measures, and exploiting the cocycle formula (81), we have: which yield Therefore, the first equality in (83) and the last one in (85) give the desired result.
From Lemma 3.1 we see that it is always possible to construct a left Haar measure on a p-adic Lie group G once known a quasi-invariant measure on it.Hence, our next step is to show how to explicitly construct a quasi-invariant measure on G.
Let A = {(U α , ϕ α )} α∈A be a disjoint atlas on G (cf. Proposition 2.30).We can construct a (regular) Radon measure ν on G as follows.First, in every chart (U α , ϕ α ) in A, we define a measure ν α on U α by setting that is, ν α is the pushforward measure, via ↼ ϕ α : ϕ α (U α ) → U α , of the restricted Haar measure λ| ϕα(Uα) on Q n p .Note that since ν α is finite on compact sets, it is a Radon measure.In this way, we have constructed a Radon measure on every chart (U α , ϕ α ) in A. To obtain a Radon measure ν on the whole group G, we can then act as follows.Given any Borel set E in B G , we express it as the disjoint union E = α∈A E α , where E α := E ∩ U α , and set Since A is countable, the series in (87) contains a countable number of non-null terms.It is now easily proved that the measure defined in (87) is a (regular) Radon measure on G. Indeed, ν takes values in [0, +∞] as so do all the ν α s, and , as follows by observing that the ν α s are σ-additive, and that the summation order can be exchanged in the double series α i ν α (E i ∩ U α ) by positivity of the ν α s.Moreover, ν is clearly finite on compact sets, and since G is second countable, we can conclude that ν is a regular, and hence Radon, measure on G (cf. Theorem 7.8 in [45]).Our next step is to show that this measure is quasi-invariant.To this end, let h ∈ G be some fixed point, and let us set, for any α, β ∈ A, Note that U h α,β ⊂ U α is an open set and Assuming that U h α,β = ∅, for every j = 1, . . ., n, and at given h ∈ G, we put where ϕ β,j is the j-th vector component of ) by varying β in A. In this way, we obtain a map ζ β,j (h; • ) : ϕ α (U α ) → Q p (for suitable labels β depending on the charts as in (90)), for any given h ∈ G.We can then define a function Eventually, we obtain a function η : G × G → R + * , defined as follows: Let us now define a (regular) Radon measure µ h on G by setting dµ h (g) = η(h, g)dν(g).
We want to prove that ν is quasi-invariant and, moreover, µ h = ν h , so that Since ν is a regular measure, then ν h and µ h are regular measures.Hence, by outer regularity, it is sufficient to show that for every open set O ⊂ G. Actually, since it is enough to prove (95) on every open subset O ≡ O h α,β of U h α,β .Assuming that O h α,β = ∅ (otherwise there is nothing to prove), we have: η(h, g)dν(g) (by ( 86)-( 87) and ( 92) In conclusion, we have ν h = µ h .Therefore, ν h and ν are mutually absolutely continuous, for every h ∈ G -namely, ν is quasi-invariant -and dν h dν (g) = η(h, g).As a direct consequence of Lemma 3.1, the left Haar measure µ on G is of the form dµ(g) = η(g, e) −1 dν(g). (99) With the above construction, we have proved the following result: Theorem 3.2.Let G be a p-adic Lie group, and let A = {(U α , ϕ α )} α∈A be a disjoint atlas on G.If µ is the left Haar measure on G then, for every Borel set E in B G , and any U α in A, the following equality holds: where (U 0 , ϕ 0 ) is the chart around the identity e ∈ G, (x k ) n k=1 denotes a system of local coordinates w.r.t.(U 0 , ϕ 0 ), and ζ j is the map defined in (90).Remark 3.3.In (100), the functions ζ α,j are correctly labelled by α.In fact, their derivatives are performed in a neighborhood of x = ϕ 0 (e), and, thus, Now, we prove that Theorem 3.2 still holds in the case of an atlas including possibly overlapping charts.Indeed, let A = {(U α , ϕ α )} α∈A be an arbitrary atlas on G. Since G is strictly paracompact (see Proposition 2.30), we can always find a refinement A ′ of A consisting of pairwise disjoint charts.Then, Theorem 3.2 provides us with a left Haar measure on (every chart of) A ′ .To show that this measure is well-defined on A as well, we have to prove that for every Borel set E in B G contained in the intersection of two charts in A, the value of the integral in (100) is the same w.r.t. the local coordinates of the two charts; that is, we want to prove To start with, the r.h.s. of (101) explicitly is where we have denoted by y the local coordinates in the chart (U β , ϕ β ).Then, the change of where, for notational convenience, we have omitted 1 ≤ j, k ≤ n in the Jacobians.Note that in the second equality of (103), we have used the C.O.V.F. for multiple integrals in Q n p (cf. Theorem 2.40).Moreover, in the third equality, we have used the fact that ζ β,j and ζ α,j are related via the condition ζ α,j = ϕ α,j • ↼ ϕ β,j • ζ β,j , and then we have exploited the usual chain rule for the Jacobian of a composite function.Therefore, (101) shows that the (left) Haar measure in Theorem 3.2 is well-defined over overlapping charts; that is, it does not depend on the particular chosen chart in A. Concluding, we have the following Corollary of Theorem 3.2.Corollary 3.4.Let G be a p-adic Lie group, and let A = {(U α , ϕ α )} α∈A be a (not necessarily disjoint) atlas on G.The left Haar measure µ on G is expressed, in the local coordinates of any given chart (U α , ϕ α ) in A, as for every Borel set E ∈ B G , where (U 0 , ϕ 0 ) is the chart around e ∈ G, and (x k ) n k=1 denotes a system of local coordinates w.r.t.(U 0 , ϕ 0 ).
To conclude this section, we now show that the Haar measure (104) coincides with the measure on G associated with the left-invariant differential n-form Ω inv on G, as constructed in Subsection 2.4.Indeed, let us denote with Ω the differential n-form on G whose local expression Ωα , in every chart U α in A, is given by where, as usual, Dζ α denotes the Jacobian matrix of ζ α = (ζ α,j ) n j=1 , and where we set ϕ α (g) = (x 1 , . . ., x n ).It is clear that the measure µ Ω, associated with Ω via relation (65), coincides with the Haar measure in (104).(It is worth noting that, from Corollary 3.4, it follows that Ω does not depend on the particular chosen chart on G, i.e. it is a well-defined differential n-form on G).To prove that the form (105) coincides with the left-invariant differential n-form Ω inv on G, it is enough to show that condition (40) holds, i.e., ℓ * h Ω(hg) = Ω(g), for every h, g in G. Indeed, this will prove that Ω is a left-invariant differential n-form on G, and due to its essential uniqueness, we can then conclude that it coincides with Ω inv (up to a multiplicative constant).In fact, we have: where we set ϕ α (hg) = (y 1 , . . ., y n ).Note that, in the second equality we have used the pullback formula (37) for differential forms, while in the third equality we have used the formula for the Jacobian of a composite function, taking into account the relation ζ α,j = ϕ α,j • ↼ ϕ β,j • ζ β,j between ζ α,j and ζ β,j .Hence, since the Haar measure is essentially uniquely defined, we get to the conclusion that the left Haar measure (104) must coincide (up to a multiplicative constant) with the measure µ Ω inv induced by the left-invariant differential n-form Ω inv on G. Remark 3.5.Let us clarify how the local formula (104) for the Haar measure µ on G allows us to globally integrate a function on G. Given f ∈ C c (G), its Haar integral G f (g)dµ(g) can be computed by splitting f as a sum of its components on local supports contained in the domains of the charts in an atlas for G.This is done by making use of a partition of unity {χ α } α∈A under an atlas {(U α , ϕ α )} α∈A of G.Then, the following relations hold: Each integral in the summation can be computed by using the local formulas (104).

Applications
As previously observed (Proposition 2.37), the groups SO(n, Q p ), n = 2, 3, 4, are compact.Hence, they admit a (left and right) Haar measure, which is essentially uniquely defined, i.e., unique up to a normalization constant factor.The construction of the Haar measure on SO(2, Q p ) κ immediately follows by formula (104).On the other hand, we will explicitly construct the Haar integrals on SO(3, Q p ) and SO(4, Q p ).A fruitful approach is to introduce a suitable p-adic quaternion algebra, H p , and exploit its relations with the p-adic special orthogonal groups in dimension three and four.In particular, we will prove that the latter groups can be realized as suitable quotients of the quaternion groups H × p and P(H × p ) respectively (c.f.Theorems 4.14 and 4.16), whose Haar measures are determined, once again, by means of a direct application of (104).Then, exploiting the Weil-Mackey-Bruhat formula introduced in Subsection 2.1, we will express the Haar integrals on SO(3, Q p ) and SO(4, Q p ) as lifts to the Haar integrals on the covering quaternion groups (see the forthcoming Theorems 4.21 and 4.26).

4.1
The Haar measure on SO(2, Q p ) κ In this subsection, we explicitly construct a left and right Haar measure on every SO(2, Q p ) κ , as in Corollary 2. 35.
According to parameterization (54), SO(2, Q p ) κ is homeomorphic to the p-adic projective line, and it is covered by two disjoint charts.One coordinate map, say ϕ (κ) , is defined on SO(2, Q p ) κ \{−I} to Q p , and it is such that Theorem 2.38); the other one maps −I ∈ SO(2, Q p ) κ to ∞.Since the groups SO(2, Q p ) κ are compact and infinite (uncountable), the singleton {−I} has zero Haar measure.The Jacobian in ( 104) is now easily computed: by recalling the composition law (55), we find (Note that −κ is never a square [35], i.e., 1 + κα 2 = 0 for every α ∈ Q p ).Therefore, an application of (104) -with β = ϕ (κ) (I) = 0 -immediately yields the Haar measure of every Borel subset E in SO(2, Q p ) κ : with dλ(α) the Haar measure on Q p .
Remark 4.1.One can directly verify that the measure in (109) is a Haar measure, i.e., leftand right-invariant.Indeed, let us consider the functional invariance condition in (3): , whenever X is compact), and where g = R κ (β), for some β ∈ Q p .In the last integral, we have also used the fact that Recalling formula (55), we have: Let us now set ̟ = (α − β)/(1 + καβ).We have: and, by inserting (112) into (111), we obtain This shows the left-invariance of the measure.On the other hand, since the group is compact, this also entails the right-invariance of the measure (109).
Remark 4.2.The Haar measure of any Borel subset F of SO(2, R) is given by where λ denotes the Haar measure on R, and the coordinate map on SO(2, R) is given by On the other hand, with κ = 1 and α = tan θ 2 , an element of SO(2, Q p ) 1 becomes formally identical to an element of SO(2, R) (cf.Remark 2.39).Therefore, one may expect that such a 'reduction' applies also for the Haar measure.Indeed, using the C.O.V.F. for p-adic integrals (see Theorem 2.40) we have: i.e., the Haar measure on SO(2, Q p ) κ reduces to that on SO(2, R), up to the normalization constant factor.

The quaternion algebra H p
The study of real quaternions was originally motivated by their property to model Euclidean orthogonal transformations of R 3 and R 4 [59].It turns out that this familiar picture keeps some of its main futures -but also requires some essential modifications -when switching from the real to the p-adic setting.In what follows, we will describe the quaternion algebra H p over the field Q p of p-adic numbers [60], in a way that closely mimics its real counterpart (briefly reminded in Appendix A.1); later, (cf.Subsection 4.3), we shall clarify its relations with the p-adic special orthogonal groups in dimension three and four.The cases where p > 2 and p = 2 will be discussed separately.

Case p > 2
In the standard real case, the quaternion algebra H is the vector space R 4 ∼ = R × R 3 equipped with a suitable standard basis, namely, the one consisting of the vectors 1, i, j, k in R 4 satisfying the commutation rules (200) of Appendix A.1.From this, one can then define an isomorphism which realizes H as a subalgebra of M 2 (C).Switching to the p-adic setting, it is then natural to set the following Definition 4.3.Let p > 2 be an odd prime.By a p-adic quaternion algebra we mean a fourdimensional vector space H p ∼ = Q p ×Q 3 p over Q p which is a Q p -algebra, and satisfies the following conditions: (a) There exist i, j in H p such that, denoting by 1 the multiplicative identity in H p , the set {1, i, j, k := ji} is a Q p -basis in H p .
(b) The basis vectors i, j, k in H p satisfy the following commutation rules: for v ∈ Q p a non-quadratic p-adic unit.
Remark 4.4.By means of a direct calculation, one verifies that the centre of the quaternion algebra H p coincides with the base field Q p .This is reminiscent, to some extent, of the standard real case where, similarly, one shows that the field of real numbers R is the centre of the real quaternion algebra H.
On the quaternion algebra H p , we can define a natural involutive anti-automorphism by setting Then, it is easily checked that, for every ξ ∈ H p , the product of ξ and ξ results into that is, the unique (up to linear equivalence and scaling) four-dimensional definite quadratic form over Q p , for p > 2 (cf.( 49) in Theorem 2.33).Therefore, we can express the inverse ξ −1 of every (non-null) p-adic quaternion as In what follows, we shall denote by the multiplicative group of invertible quaternions.
In what follows, we shall denote by ξ/nrd(ξ) the inverse element (119) of a quaternion ξ ∈ H × p .
In the group of invertible quaternions H × p , it is possible to single out the subgroup of the so-called unit quaternions, namely, the group: We want now to show that, as in the standard real case, H p can be realized as a suitable matrix algebra.To begin with, we recall that in the quadratic form and let H p be the subalgebra of the matrices where where the inverse of every non-null element M ∈ H p is and where det(M ) = Q (4) (x 0 , x 1 , x 2 , x 3 ).Let us now introduce the matrices i, j, k in M 2 (Q p ( √ v)) It is clear that every M in H p can be expressed as follows: (here, we are omitting the identity matrix I 2 multiplying x 0 ); that is, H p coincides with the Q p -linear span of the set {I 2 , i, j, k}.Moreover, i, j, k satisfies the following commutation rules: from which we can argue that H p is a non-commutative Q p -division algebra.
Remark 4.6.As in the complex case, the subset of invertible elements in H p forms a group where 0 2 denotes the null 2 × 2 matrix on Q p ( √ v).Moreover, we can single out the subgroup which provides the p-adic counterpart of (206) in Appendix A.1.
In the light of the discussion above, it is now not difficult to prove the following result: Proposition 4.7.For every prime p > 2, the p-adic quaternion algebra Proof.Let us consider the map It is clear that θ p is one-one, onto and linear, i.e., it is an isomorphism of vector spaces.Also, θ p is a ring homomorphism, since θ p (ξη) = θ p (ξ)θ p (η) for every ξ, η ∈ H p .Hence, it defines an algebra isomorphism from H p to H p .
The algebra isomorphism θ p identifies the basis vectors 1, i, j, k of H p with I 2 and the matrices (125) in the spanning set of H p , respectively.This then also justifies our abuse of notation in using the same symbols for the basis elements of both H p and H p .Remark 4.8.Exploiting the algebra isomorphism θ p , one can easily check that nrd(ξ) = det(θ p (ξ)) = Q (4) (q 0 , q 1 , q 2 , q 3 ). (131) Therefore, we can interchangeably use nrd(ξ), det(θ p (ξ)) and Q (4) (q 0 , q 1 , q 2 , q 3 ) to denote the reduced norm of ξ = q 0 + iq 1 + jq 2 + kq 3 in H p .
Remark 4.9.Using the isomorphism θ p , it is clear that the subgroups U(H p ) and H × p of H p are isomorphic, respectively, to the subgroups U(H p ) and H × p of H p (cf. Remark 4.6).

Case p = 2
As for the p > 2 case, we start by giving the following Definition 4.10.Let p = 2.By a 2-adic quaternion algebra we mean a four-dimensional vector space

and satisfies the following conditions:
(c) There exist i, j in H 2 such that, denoting by 1 the multiplicative identity in H 2 , the set (d) The basis vectors i, j, k satisfy the following commutation rules: We can endow H 2 with the involution (117), thus turning it into an involutive algebra.Then, the inverse ξ −1 of every non-null 2-adic quaternion ξ can be expressed as Moreover, we can single out the subgroup H × 2 ≤ H 2 of invertible 2-adic quaternions by putting as well as the subgroup U(H 2 ) ≤ H × 2 of unit quaternions defined as We want now prove that H 2 can be made in a one to one correspondence with a suitable matrix algebra.To this end, we recall that the definite quadratic form of Q 4  2 is now given by (49); moreover, since −1 is not a square in Q 2 , we can consider the quadratic extension )) denote the algebra of two-dimensional matrices on Q 2 ( √ −1), and let )) be the subalgebra of matrices M defined by By construction, we have that det . Hence, every non-zero M ∈ H 2 is invertible, with inverse given by i.e., H 2 is an associative (unital Every M in H 2 can be expressed as M = x 0 + ix 1 + jx 2 + kx 3 (we have omitted the identity I 2 multiplying x 0 ); that is, H 2 can be realized as the Q 2 -linear span of {I 2 , i, j, k}.By further noting that i, j, k satisfy the commutation rules we also see that H 2 is a non-commutative Q 2 -algebra.
Remark 4.11.As in the p > 2 case, we can introduce the group of the invertible matrices in H 2 , as well as the subgroup The following result is a straightforward adaptation of Proposition 4.7 Proposition 4.12.Let p = 2.Then, the 2-adic quaternion algebra H 2 is isomorphic with the subalgebra Proof.It suffices to consider the map and observe that it provides the desired algebra isomorphism.
Remark 4.13.The quaternion algebra H 2 shares some analogies with the standard real quaternion algebra H.In particular, the matrix representation of a 2-adic quaternion is 'essentially the same' as in the standard case (just set √ −1 := i for the square root of the non quadratic element −1 ∈ Q 2 ).This is what one expects upon considering the 'formal equivalence' of the real four-dimensional quadratic form Q R with the four-dimensional quadratic form Q (4) on Q 2 .However, the analogies between standard and p-adic quaternion algebras cannot be pursued too far.Indeed, a fundamental difference between H p , for every prime p ≥ 2, and H is the following.For the latter, we have that Q R (q 0 , q 1 , q 2 , q 3 ) = (q 0 , q 1 , q 2 , q 3 ) 2 R 4 , i.e., the definite quadratic form Q R on R 4 coincides with the squared Euclidean norm of R 4 .(This also entails that the reduced norm of H is equivalent to the (square of) the Euclidean norm of R 4 .See Remarks 4.5 and A.1).On the other hand, in the p-adic setting, we only have the equivalence Q (4) ≡ nrd, i.e., the reduced norm of H p does not coincide with the square of the p-adic norm of Q 4 p .

Relation between p-adic quaternions and special orthogonal groups
This subsection clarifies the relations between p-adic quaternions and the p-adic groups of rotations SO(3, Q p ) and SO(4, Q p ), for every p ≥ 2. We begin with SO(3, Q p ).Let us consider the action by conjugation of the group H × p of invertible quaternions on H p ; namely, the map where ξ ∈ H × p , and p ≥ 2. This map is an isometric linear transformation of H p , since it preserves the reduced norm of every quaternion η in H p : equivalently, the action by conjugation of H × p preserves the definite quadratic form of Q 4 p .Moreover, the operation η → ξηξ −1 leaves the centre Q p of H p pointwise fixed and, hence, also leaves the orthogonal subspace Q 3 p invariant.
The above discussion shows that κ p (H × p ) ⊆ SO(3, Q p ).We are now going to prove that, actually, also the reverse inclusion SO(3, Q p ) ⊆ κ p (H × p ) holds.Indeed, let us first introduce the map τ ρ : H 0 p → H 0 p defined, for every ρ ∈ H × p ∩ H 0 p , as where b denotes the bilinear form associated with the quadratic form Q + in Q 3 p (cf. Section 2).It is easily shown that this map satisfies the conditions τ ρ (ρ) = −ρ and defines a hyperplane reflection (w.r.t.ρ) in H 0 p .Moreover, by taking into account the defining properties of b, nrd and H 0 p , and recalling that, for a pure imaginary quaternion ν in H 0 p , one has ν = −ν, we see that the reflection (150) is explicitly given by τ ρ (ν) = −ρνρ −1 , i.e. τ ρ ≡ −κ p (ρ).On the other hand, by a classical theorem of Cartan and Dieudonné (cf.Theorem 4.5.7. in [59]), every special orthogonal transformation in SO(3, Q p ) can be written as the composition of two such reflections, i.e. g = τ ρ 1 τ ρ 2 , for all g ∈ SO(3, Q p ), and suitable ρ 1 , ρ 2 ∈ H × p ∩ H 0 p .Therefore, every p-adic rotation in SO(3, Q p ) is expressed by for ξ := ρ 1 ρ 2 ∈ H × p .This then shows that κ p (H × p ) = SO(3, Q p ), i.e. that κ p is surjective.The following result is now straightforward and crucial for our purposes.
Theorem 4.14.The group SO(3, Q p ) is isomorphic to the quotient of the group H × p of invertible quaternions, and the multiplicative group Proof.To prove the group isomorphism (152), we can equivalently show that the following is a short exact sequence.We already know that κ p is surjective.Furthermore, the kernel of κ p , ker(κ p ), coincides with the image Q × p of the embedding in the short sequence: as The exact sequence (153) is reminiscent, to some extent, of the exact sequence of the standard real case (cf. the isomorphism (210) in Appendix A.2). Here, the main difference with the sequence (153) is provided by the fact that the groups U(H) and F 2 = {±1} are replaced, in the p-adic setting, by the groups H × p and Q × p respectively.The reason for this discrepancy is related to the peculiar features of the base field Q p .Indeed, it is possible to prove [59,61] that a sequence as in (155) is exact if and only if nrd(H × ) ⊂ (F × ) 2 , namely, iff the reduced norm of every invertible quaternion is a quadratic element of the field.In the case where F = R, this is certainly true.Instead, in the p-adic setting, nrd(H × p ) ⊂ (Q × p ) 2 is never true.
We want now to show that SO(3, Q p ) and H × p /Q × p are homeomorphic.This fact will indeed play a fundamental role in our construction of the lift of the Haar integrals on SO(3, Q p ) to H × p .Let us preliminary recall that every LCSC Hausdorff space is a standard Borel space once endowed with its Borel σ-algebra.Accordingly, one calls a space X a standard Borel G-space if X is a G-space (cf.Subsection 2.1), its Borel structure is standard, and if the action of G on X is a Borel map.If X is a standard Borel G-space, and x ∈ X is a fixed point, let = x} be the stability subgroup at x.One can show (cf.Corollary 5.8 in [47]) that G x is a closed subgroup of G.Moreover, denoting by q : G → G/G x the projection homomorphism, the map is a Borel isomorphism, and it is a homeomorphism whenever X is LCSC (cf.Theorem 5.11 in [47]).Therefore, in such a case, X ∼ = G/G x are homeomorphic spaces in a natural way.We are now ready to prove the following result.
Proposition 4.15.The group isomorphism (152) between SO(3, Q p ) and H × p /Q × p is also an isomorphism of topological groups.
Proof.The proof we give here is based on general measure-theoretical arguments on G-spaces; for a more specific proof, involving the reduced norm of p-adic quaternions, see Appendix B.
As a vector space, It is easily shown that the map (157) provides a continuous left action of H × p on SO(3, Q p ). Indeed, continuity follows from that of κ p and of the matrix multiplication in SO(3, Q p ). Next, we have that ξ p , as follows by observing that R → ξ[R] is surjective (since the multiplication in SO(3, Q p ) by the matrix κ p (ξ) is so), and injective (since if κ p (ξ)R 1 = κ p (ξ)R 2 , then R 1 = R 2 by the invertibility of κ p (ξ) ∈ SO(3, Q p )), and both the map and its inverse are continuous (as they are just matrix multiplications and inverses).This shows that (157) is a continuous (actually, Borel) left action of H × p on SO(3, Q p ).This action is also transitive, since it exists an element R ∈ SO(3, Q p ) such that its orbit {κ p (ξ)R | ξ ∈ H × p } is the whole space SO(3, Q p ) (it is enough to consider R = I, and the surjectivity of κ p ).Therefore, we can argue that SO(3, Q p ) is a standard Borel (transitive) H × p -space.On the other hand, the stability subgroup at every R ∈ SO(3, hence, we can conclude that SO(3, Q p ) and H × p /Q × p are homeomorphic spaces.In particular, the homeomorphism is as in (156) with, for instance, the stability subgroup at I ∈ SO(3, Q p ). Explicitly, the homeomorphism is . This is, indeed, the same map providing the isomorphism in Theorem 4.14.Proposition 4.15 concludes our discussion on the relations between p-adic quaternions and rotations in SO(3, Q p ). Now, we carry out a similar analysis to clarify the relation between quaternions and the elements in SO(4, Q p ).To begin with, let us introduce the left action of H × p × H × p on H p defined by This action is by similarities, as follows by noting that nrd(ξη̺ −1 ) = nrd(ξ)nrd(η)nrd(̺ −1 ) = nrd(ξ) nrd(̺) nrd(η).
In particular, the action is by isometries whenever nrd(ξ) = nrd(̺).Hence, let us introduce the group The restriction of the action (158) to a pair (ξ, ̺) ∈ P(H Since this action is by isometries, and Proof.Since char(Q p ) = 2, the isomorphism (162) follows from Proposition 4.5.17. in [59].In particular, to prove (162), it suffices to show that the following is a short exact sequence.This is done similarly to the proof of Theorem 4.14: Surjectivity of the map κ ′ p : P(H × p ) → SO(4, Q p ) again follows by the Cartan-Dieudonné Theorem (cf.Theorem 4.5.7. in [59]), and its kernel is In particular, the last condition must hold for η = 1 ∈ H p , providing the necessary condition ξ = ̺; hence, That is, the kernel of κ ′ p is the diagonally embedded Remark 4.17.The short exact sequences (163) is the p-adic counterpart of the following sequence for the standard real setting: from which we argue that We can now compute the function , where the vectors of coordinates of e and ξ are (1, 0, 0, 0) and (q 0 , q 1 , q 2 , q 3 ) := q respectively: This yields which, as anticipated, is globally defined on H × p .Then, using (104), we can conclude that the Haar measure of any Borel subset E of where we recall that dλ(q) denotes the Haar measure on Q 4 p (cf. Example 2.4).For the p = 2 case, a similar discussion to the one leading from (168) to (171) shows that det ∂ζ i ∂x j ↼ ϕ(q); ϕ(e) = det     q 0 −q 1 −q 2 −q 3 q 1 q 0 −q 3 q 2 q 2 q 3 q 0 −q 1 q 3 −q 2 q 1 q 0 Then, using (104), the Haar measure of any Borel subset We summarize the above results with the following Proposition 4.20.Let p ≥ 2 be a prime number, and let H × p be the group of invertible p-adic quaternions.The Haar measure µ H × p on H × p is given by for every Borel set E ⊂ H × p , where ξ = q 0 + iq 1 + jq 2 + kq 3 , ϕ(ξ) = (q 0 , q 1 , q 2 , q 3 ), dλ(q) = dq 0 dq 1 dq 2 dq 3 is the Haar measure on Q 4 p , and Q (4) is the definite quadratic form of Q 4 p (see (49) in Theorem 2.33).
where c 1 , c 2 , c 3 , d 0 , d 1 , d 2 , d 3 can be assumed as independent variables.Here, we are referring to the coordinate map on such a neighborhood of e ∈ P(H × p ) as where γ and δ are as in (180).The same argument can be repeated for the condition nrd(ζ 1 ) = nrd(ζ 2 ), to express z 0 in terms of the other independent coordinates z ′ 0 , z i , z ′ i , i = 1, 2, 3.In conclusion, we are left with the following 7 × 7 Jacobian matrix where, in the l.h.s., ϕ 0 (e) = (0, 0, 0, 1, 0, 0, 0) and (x k ) 7 k=1 = (c 1 , c 2 , c 3 , d 0 , d 1 , d 2 , d 3 ).By using (182), the partial derivatives of the dependent variable c 0 w.r.t. the independent ones are ∂c and ∂c 0 ∂d 0 Hence, using the expressions in (181), the Jacobian matrix (184) is straightforwardly computed and reads: The p-adic absolute value of the determinant of such a matrix is For the last equality, we used again the condition nrd(ξ) = nrd(ρ) in a suitable neighborhood for the coordinates of the identity of P(H × p ), where is well-defined.
Let us now switch to the case where p = 2.The components of ζ 1 and ζ 2 are now given by: .Then, analogously to the case where p > 2, it is not difficult to prove that, in a suitable neighborhood of (1, 0, 0, 0, 1, 0, 0, 0) in Q 8 p , c 0 can be expressed in terms of the independent variables c 1 , c 2 , c 3 , d 0 , d 1 , d 2 , d 3 , as the forthcoming Remark will clarify.Remark 4. 24.Consider an open ball in Q 8  2 of centre (1, 0, 0, 0, 1, 0, 0, 0) and radius 1/2, in which d 0 = 1 + 4y 0 , d i = 4y i , c i = 4x i , with y 0 , y i , x i ∈ Z 2 , i = 1, 2, 3.In this case, Q (4) (d 0 , d 1 , d 2 , d 3 ) = (1 + 4y 0 ) 2 + (4y 1 ) 2 + (4y 2 ) 2 + (4y 3 ) 2 = 1 + 8 y 0 + 2Q (4) (y 0 , y 1 , y 2 , y 3 ) and Q As a consequence, the coordinate map on a suitable neighborhood of e ∈ P(H × 2 ) to Q 7 2 is as in ( 183).An analogous discussion can be carried out for the condition nrd(ζ 1 ) = nrd(ζ 2 ), to show that z 0 can be expressed as a function of the (independent) variables z ′ 0 , z i , z ′ i , i = 1, 2, 3, in a suitable neighborhood of the identity.It follows that the Jacobian matrix for p = 2 is of the same form (184) for p > 2 and, as one can easily check, the partial derivatives of the dependent variable c 0 are again given by (185) and (186).Thus, we obtain A The real quaternion algebra and its relations with SO(3, R) and SO(4, R) We devote this appendix to a brief account on the real quaternion algebra H, along with a discussion of the quaternionic realization of the elements in SO(3, R) and SO(4, R).This will also give us the opportunity to highlight analogies and differences with the p-adic case of Section 4.2.

A.1 The real quaternion algebra H
There are several ways of describing the real quaternion algebra H [36,62].As a real vector space H ∼ = R × R 3 , and any element in H is written as ξ = (a, x), with a ∈ R and x ∈ R 3 .The multiplication law is given by: (a, x)(b, y) = (ab − x • y, bx + ay + x × y), where x • y and x × y are respectively the usual inner product and vector product between vectors in R 3 .It is easily verified that the product (198) is associative.The centre of H is given by the subspace R × {0} ∼ = R.Likewise, we identify {0} × R 3 with R 3 , in such a way that every element in H can be expressed as ξ = a + x, a ∈ R, x ∈ R 3 .Denoting by i, j, k the canonical basis vectors of R 3 , ξ can be expressed as where a, x 1 , x 2 , x 3 ∈ R.Then, the multiplication law between quaternions is given by specifying the products between the basis vectors i, j, k [36,63]: It is straightforward to realize that H is a non-abelian algebra.
H is an involutive algebra, as the map H ∋ ξ = a + x → ξ := a − x ∈ H is an involutive anti-automorphism.Moreover, ξξ = |ξ| 2 = a 2 + |x| 2 = Q R (a, x 1 , x 2 , x 3 ), where Q R denotes the definite quadratic form of R 4 .Thus, every non-zero element in H is invertible, with ξ −1 = ξ/|ξ| 2 , and so H is a division algebra.Those elements in H for which |ξ| = 1 are called unit quaternions.They form a group in H, denoted by U(H): Remark A.1.In the literature (e.g., [59]), the quantity ξξ is referred to as the reduced norm of ξ in H, and denoted by nrd(ξ) (see Remark 4.5).From the definition, it is clear that the reduced norm on the real quaternion algebra H is equivalent to the square of the Euclidean norm on R 4 (since the definite quadratic form on R 4 induces the Euclidean inner product, and vice versa).However, this is not the case when one defines a quaternion algebra over a generic field w.r.t.some quadratic form (see, for instance [59,61]): The latter does not necessarily induce the considered norm on the vector-space structure of that algebra.
There is another (yet equivalent) way in which H can be described [62].Let us consider the subset H of the algebra M 2 (C) of complex 2 × 2 matrices of the form M = a b −b a = q 0 + iq 1 q 2 + iq 3 −q 2 + iq 3 q 0 − iq 1 , q j ∈ R, for every j = 0, 1, 2, 3, where i denotes the imaginary unit.One can easily verify that H is a subalgebra (actually, a division algebra) in M 2 (C).In particular, that every non-null element M ∈ H is invertible easily follows by observing that det(M ) = det q 0 + iq 1 q 2 + iq 3 −q 2 + iq 3 q 0 − iq 1 = q 2 0 + q 2 1 + q 2 2 + q 2 3 (203) is the (non-degenerate) four-dimensional definite quadratic form over R. From (202), we also see that every element M ∈ H can be written as M = q 0 + iq 1 + jq 2 + kq 3 , where and where we omitted the identity matrix I 2 multiplying q 0 .Moreover, it can be easily checked that i, j, k obey commutation relations which are analogous to the quaternion commutation relations (200).Indeed, the correspondence θ := H ∋ ξ = (q 0 , q 1 , q 2 , q 3 ) → θ(ξ) := q 0 + iq 1 q 2 + iq 3 −q 2 + iq 3 q 0 − iq 1 ∈ H defines an algebra isomorphism from the quaternions H to the algebra of complex matrices H [36].In particular, unit quaternions are identified in H by

U
(H) = {M ∈ H | det(M ) = 1}.(206) It is not difficult to prove that the special orthogonal groups SO(n, Q p ), n = 2, 3, 4, in Corollary 2.35 are compact as subsets in Q n 2 p .Indeed, we can introduce a p-adic (non-Archimedean) norm on SO(n, Q p ) by settingM p = (M ij ) ij p := max i,j=1,...,n |M ij | p .Clearly, SO(n, Q p ), n = 2,3, 4, turn into topological groups, whenever they are endowed with the natural topology generated by the open balls of the p-adic norm.We recall that a set K ⊂ Q m p is compact if and only if it is closed and bounded w.r.t. the ultrametric topology generated by (the open balls of) the p-adic norm N p (x) := max i=1,...,m |x i | p of Q m p and we can provide H p with the product topology (on Q p , we consider the natural (ultra-)metric topology generated by the p-adic absolute value).Similarly, H × p and Q 4 p − {0} are homeomorphic topological spaces whenever they are equipped with the induced topology as subspaces of H p and Q 4 p respectively.The continuity of the group operations (multiplication and inverse) of H × p is inherited from the continuity of the addition, inner multiplication (according to the commutation relations among the basis elements) and multiplication by scalars of Q 4 p − {0}; therefore H × p is a topological group.Also, H × p is LCSC, as Q 4 p − {0} is so (being an open subspace of the locally compact Hausdorff space Q 4 p ).We have already observed that SO(3, Q p ) is a compact second countable Hausdorff group, once endowed with the topology introduced in Section 2. Hence, SO(3, Q p ), supplied with its Borel σ-algebra is a standard Borel space.We want now to show that SO(3, Q p ) is a standard Borel H × p -space.To this end, we have to find a Borel action of H × p on SO(3, Q p ).Let us introduce the map from H × p × SO(3, Q p ) to SO(3, Q p ) defined as for every x ∈ Q 4 p }.It can be easily checked that, as done for the maps κ p (ξ) in the three-dimensional case,κ ′ p (ξ, ̺) ∈ SO(4, Q p ),for every (ξ, ̺) ∈ P(H × p ). Also, κ ′ p : P(H × p ) → SO(4, Q p ) is a group homomorphism, and we get to the following result: Theorem 4.16.The group SO(4, Q p ) is isomorphic to the quotient of the group P(H × p ) and the multiplicative group Q × p of non-null p-adic numbers: Similarly to what we did for SO(3, Q p ), we are interested in proving that SO(4, Q p ) and P(H × p )/Q × p are homeomorphic; this will allow us to consider the lift of the Haar integrals on SO(4, Q p ) to that on P(H × p ).Let us assume p > 2 first.If ξ = q 0 + iq 1 + jq 2 + kq 3 , and χ = x 0 + ix 1 + jx 2 + kx 3 are two quaternions in H × p , their composition will result in another quaternion ζ = z 0 + iz 1 + jz 2 + kz 3 ; namely ζ