On eigenvector statistics in the spherical and truncated unitary ensembles

We study the overlaps between right and left eigenvectors for random matrices of the spherical and truncated unitary ensembles. Conditionally on all eigenvalues, diagonal overlaps are shown to be distributed as a product of independent random variables. This enables us to prove that the scaled diagonal overlaps, conditionally on one eigenvalue, converge in distribution to a heavy-tail limit, namely, the inverse of a $\gamma_2$ distribution. These results are analogous to what is known for the complex Ginibre ensemble. We also provide formulae for the conditional expectation of diagonal and off-diagonal overlaps, with respect to all eigenvalues.


Spherical and Truncated Unitary Ensembles
This work considers two ensembles of random matrices defined as follows.
(i) The spherical ensemble consists of products G 1 G −1 2 , where G 1 , G 2 are i.i.d. complex Ginibre matrices. We denote the N ×N complex Ginibre ensemble by CGE(N ) and the corresponding spherical ensemble by Sph(N ). The name spherical comes from a geometric description of the eigenvalues, stated as Proposition 2.5 and illustrated on Figure 1.  The reason for treating these two ensembles in the same paper is the strong analogy between them, underlined and exemplified by [9], that extends to the overlap distribution. All results are presented in details for the spherical case in Section 2, while the corresponding results in the truncated unitary case are found in Section 3 -with less detail whenever the two computations are exactly the same.

Results
The matrix of overlaps associated to the bi-orthogonal family of left and right eigenvectors of a non-Hermitian random matrix has been introduced and studied by Chalker & Mehlig in [4,5], then more recently in a series of paper involving a variety of methods [1-3, 6, 8, 10, 13, 15]. It is defined as follows: for a given matrix G ∈ M N (C) with simple spectrum {λ 1 , . . . , λ N } (note that the random spectra we consider are almost surely simple), if R j = |R j is the right eigenvector associated to λ j and L j = L j | the left eigenvector associated to the same eigenvalue, chosen such that for every j, we define the matrix of overlaps O by Chalker & Mehlig computed the conditional expectation of the overlaps in the complex Ginibre ensemble and conjectured several of their properties. For a more detailed presentation, see the introduction of [3] and the appendix of [8].
The results we obtain in the spherical and truncated unitary cases are analogous to some of the results obtained in [3] for the complex Ginibre ensemble CGE(N ). We recall these results, and point out which statement of the present paper corresponds to each one.
(i) A decomposition of the distribution of diagonal overlaps. The first notable fact is that, conditionally on the spectrum Λ ∈ C N , diagonal overlaps can be decomposed as a product of independent variables. In the complex Ginibre ensemble, Theorem 2.2 from [3] states that, conditionally on the event {Λ = (λ 1 , . . . , λ N )}, the distribution of diagonal overlaps is given by where (Z i ) N i=2 are i.i.d. standard complex Gaussian. Instead of Gaussian variables, the analogous statements in the spherical and truncated unitary ensembles involve i.i.d. variables whose distribution is specific to each case. Namely, in Sph(N ), we have where the X where the Y (k) M are i.i.d. variables whose distribution is defined in (3.5). These decompositions are stated as Theorem 2.6 and 3.5 respectively.
(ii) A limit theorem for diagonal overlaps. In the complex Ginibre ensemble, Theorem 1.1 from [3] states that conditionally on the event {λ 1 = z} with z ∈ D, the scaled diagonal overlap O 1,1 converges to the inverse of a γ 2 distribution: This heavy-tail limit 1 appears to be universal.
In particular, the exact same convergence holds at the origin for the spherical and truncated unitary ensembles, which is stated as Proposition 2.8 and 3.7 respectively. Unlike the complex Ginibre case, where O −1 1,1 follows a beta distribution when {λ 1 = 0}, the distribution of the overlap for fixed N does not take an especially simple form here; nevertheless, the asymptotical result can be worked out in an analogous way. The specific structure of the spherical ensemble allows one to extend this result to the whole complex plane, yielding the following Theorem.
It is to be expected that a similar statement holds for TUE(N, M ) in the bulk of its limit density of eigenvalues, with a scaling parameter coherent with the expressions of [2].
(iii) Conditional expectations of overlaps. In the complex Ginibre ensemble, it follows from (1.3) that the expectation of diagonal overlaps of CGE(N ) takes the following form: (see [10] and [8]).
which had been obtained earlier by Chalker & Mehlig [4,5] by a direct computation.
Analogous identities derive from equations (1.4) and (1.5); they are stated in Theorem 2.6 and 3.5 respectively. Moreover, expressions of the same kind can be obtained for off-diagonal overlaps, although no decomposition in independent variables is known in that case. In the Ginibre ensemble, this yields . (1.9) The analogous results for Sph(N ) and TUE(N, M ) are stated as Theorem 2.11 and 3.8 respectively.
(iv) Conditional expectation of mixed moments. The conditional expectation of Tr G * G with respect to Λ also exhibits a remarkable decomposition in all three ensembles. One reason for considering this particular quantity, which is the simplest 'mixed moment', is that it is obtained from the eigenvalues and the overlaps by the identity: More general mixed moments are linked to the generalized overlaps considered in [6] by similar relations.
In the complex Ginibre case, it suffices to write The conditional expectation follows immediately, using the fact that the upper-diagonal entries of the Schur transform are Gaussian and independent of the eigenvalues.
The spherical and truncated unitary ensembles yield slightly more intricate expressions, stated as Proposition 2.12 and 3.9 respectively.
We summarize all results relative to (iii) and (iv) in the table below, Section 1.3. It follows from (1.10) that the third column is related to the first two by elementary linear relations -a fact which is not directly seen from the quenched expressions.

Synoptic table of conditional expectations
Chalker & Mehlig [4,5] Chalker & Mehlig [4,5] From eq. (1.11)  We first note that the conditions (1.1) can be achieved by choosing R i as the columns of P and L i as the rows of P −1 for a given diagonalization G = P ∆P −1 ; the overlaps are independent of this choice. Moreover, overlaps are unchanged by an unitary change of basis, and therefore one can study directly the overlaps of the Schur form

Sph(N )
(1.12) By exchangeability of the eigenvalues, we can also limit ourselves to studying the variables O 1,1 and O 1,2 , whose definitions only involve the first two left and right eigenvectors of T , chosen such that The recurrence formula is (1.13) The first overlaps, according to (1.2), are then given by the expressions (1.14) The reason why the recurrence (1.13) leads to a decomposition in distribution (resp. a decomposition of the conditional expectation with respect to all eigenvalues) of the overlaps in different ensembles is that the distribution of the Schur form is known and allows to perform such a computation explicitly. For instance, in the complex Ginibre case, the upper-triangular entries (T i,j ) i<j are i.i.d. complex Gaussian variables with variance 1/N , so that u k+1 is a k-dimensional Gaussian vector with independent coordinates, and independent of u 2 , . . . , u k . The Schur forms of Sph(N ) and TUE(N, M ) have explicit densities expressed in the form of a determinant; a structure which allows an analogous analysis.

Notations and conventions.
Throughout the paper, N is the size of the system (i.e. the number of eigenvalues); the spectrum is Λ = Λ N = (λ 1 , . . . , λ N ). For any n N , we denote by T n the n × n top-left submatrix of the Schur form T , and by u n the first n − 1 coordinates of the last column vector of T n , so that E A denotes the conditional expectation with respect to A (if A is a random variable or a sigma algebra), or the expectation for the conditional probability (if A is an event); the context should prevent any ambiguity to arise. In particular, E Λ is the conditional expectation with respect to the spectrum Λ. When conditioning on Λ, we will also use the following filtration, adapted to the nested structure of the Schur transform: (This convention differs from the one chosen in [3]. In particular, b 2 = T 12 λ 1 −λ 2 ∈ F 2 , and F 1 is trivial.) With any suitable function V , the generalized Gamma and Meijer functions are defined as We also define the partial sums e (m) , and the generalized Gamma distributions γ V (α), with density with respect to the Lebesgue measure. We will use the fact, established for instance in [7,11,12], that a point process in C with joint density given by , is such that the following identity in distribution holds: where the latter variables are independent, and γ V (k) is distributed according to (1.15) with α = k.
What we need here is a specific form of Kostlan's property, obtained by applying Proposition 1.2 to the conditioned measure.
where the latter variables are independent, and γ V (k) is distributed according to (1.15) with α = k.
In the complex Ginibre ensemble, the scaled variables N γ V follow usual gamma distributions.
Other notations or conventions relatives specifically to the spherical or truncated unitary case are mentioned in the corresponding section.

Spherical ensemble
This section contains the proof of all claims related to the spherical ensemble Sph(N ). These proofs rely on a few estimates that are found in Subsection 2.3.

Schur form and eigenvalues
We first present a few general results in order to illustrate the method; the tools and definitions that follow are specific to the spherical case. We recall that the Schur transfom T of a matrix from Sph(N ) is distributed with density proportional to with respect to the Lebesgue measure on all complex matrix elements, diagonal (dΛ = dλ 1 · · · dλ N ) and upper-triangular (du 2 · · · du n ). We introduce the Hermitian, definite-positive matrices The following lemma is the essential tool used in [9].
Lemma 2.1. The determinant of H n = I n + T n T * n can be reccursively decomposed as Proof. We first write Elementary operations on columns brings this matrix to an upper-triangular form, so that The claim follows by Sylvester's identity, det(I + AB) = det(I + BA).
For any p > n, we denote by V (n) p a random vector with density with respect to the Lebesgue measure on C n ; the value of C n,p is given by (2.21). For any m 0, we denote by X m a real random variable with density with respect to the Lebesgue measure. In particular Note that the i.i.d. variables that appear in Theorem 2.6 follow the distribution of X m with m = N .
Lemma 2.2. Identity holds between the following expressions, for p n and f, g integrable functions of the matrix elements: Proof. Lemma 2.1 and the change of variable u n = S n−1 v n bring the left hand side to the form Recall that u n , and therefore v n , are column vectors of size n − 1. The claim follows by definition of the random vector V A first relevant fact that can be deduced from the above Lemma is the distribution of every top-left submatrix of the Schur form T .
Proposition 2.3. Conditionally on Λ and for 2 n N , the submatrix T n of the Schur transform is distributed with density proportional to with respect to the Lebesgue measure on upper-triangular matrix elements (du 2 · · · du n ).
Proof. The claim is known for n = N . We deduce it for all n by a backward recurrence; indeed, as long as n − 1 2, the claim for n − 1 follows from the claim for n by Lemma 2.2 with g = 1 and generic f .
We can also derive the joint eigenvalue density of the spherical ensemble from the density of its Schur form, as was done in [9].
Theorem 2.4. The joint density of eigenvalues for the spherical ensemble is proportional to with respect to the Lebesgue measure on C N .
Proof. Let h be a bounded and continuous function of the spectrum Λ n . We use Lemma 2.1 with p = 2N , g = 1 and which yields and so on; this recurrence leads to the expression which is equivalent to the claim.
Theorem 2.4 can be rephrased by saying that the eigenvalues of Sph(N ) are distributed according to (1.16) with potential V (t) = (N + 1) ln(1 + t). A straightforward computation shows that Origin of the name spherical. The stereographic projection from S 2 to C is defined by and its inverse map from C to S 2 is given by : where λ ∈ C. (2.10) The reason for the name spherical is that the following identity in distribution holds. In other terms, the eigenvalues of Sph(N ) can be described as the stereographic projection of a one-component plasma on S 2 , with respect to a uniform potential 2 .
2 Note that the appropriate convention for the stereographic projection here is such that the unit circle is mapped to the equator of S 2 . In particular, the average proportion of eigenvalues of Sph(N ) falling in the unit disk is 1 2 .
Proof. This is obtained by a change of variable applied to the density (2.7), noting that and that the Jacobian of p at λ is 4 (1+|λ| 2 ) 2 .

Distribution and conditional expectation of overlaps
We now give the proof of the claims concerning diagonal and off-diagonal overlaps in the spherical ensemble. Some results hold conditionally on the whole spectrum Λ N , whereas others only imply a condition on one eigenvalue.
where the X (k) N are i.i.d. distributed according to (2.5) with m = N . In particular, the quenched expectation is given by (2.14) Proof. For 1 d N , we define the partial sums It follows from the general facts presented in Section 1.2 that O 1,1 = 1, and for any d 1, In order to characterize the distribution of this factor, we use our preliminary results in the following order: • Proposition 2.3 gives the distribution of T d+1 , so that p = N + d + 1 in the following steps.
• Lemma 2.2 with n = d + 1, generic f and g(u d+1 ) := h(|B d u d+1 | 2 ) with generic h gives that and is independent of F d .
• Lemma 2.16 with a = b = B * d and S = S d yields where X N is distributed according to (2.5) with parameter m = N , and independent of F d .
We notice that, as T is triangular and T d , B d are obtained from T and L 1 , where X N is independent of F d ; we denote this variable by X Diagonal overlap are (deterministically) larger than one, and typically of order N . The following proposition states that in the spherical ensemble the expectation of the diagonal overlap for an eigenvalue conditioned to be at the origin is exactly N , as is also the case in the complex Ginibre and truncated unitary ensembles.
Proof. We know from Proposition 1.2 that the squared radii are distributed like independent variables with distributions γ V,k with V (x) = (N + 1) log(1 + x) and 2 k N . We have and according to Lemma 2.13, so that the expectation is given by the telescopic product as was claimed.
Proposition 2.8. Conditionally on {λ 1 = 0}, the following convergence in distribution takes place: The proof relies on the following elementary Lemma. Then there exists a sequence (k n ) n 1 such that 1 k n n, k n −−−→ n→∞ ∞, and for any m ∈ N, Proof of Lemma 2.9. We first prove the statement for one double-indexed sequence (u k,n ) 1 k n . We define, for 1 k n, the partial sums S k,n = k i=1 u i,n , and the following sequence, iteratively: n 1 := 1, n j+1 := min l | ∀n l, S j+1,n 1 2 S j,n j .
By assumption on u k,n , the sequence (n j ) j 1 is well defined, increasing, and goes to infinity. Moreover, by construction we see that S j,n j converges to zero. It is straightforward to check that the sequence k n := max{j ∈ [[1, n]] | n j n} is such that 1 k n n, k n −−−→ n→∞ ∞, and ∀n ∈ [[n j , n j+1 − 1]], S kn,n = S j,n 1 2 S j−1,n j−1 , so that S kn,n converges to 0; thus, the Lemma is established for one sequence. We extend this to a countable family of double-indexed sequences u Indeed, every term being positive, as k n → ∞, the latter sum can be bounded by the first one as soon as k n m. This concludes the proof of Lemma 2.9.
The following argument uses the multiplicative version of Lemma 2.9; namely, if a countable family of double-indexed sequences p Note that this existential statement does not give any estimate on the growth rate of (k n ).
Proof of Proposition 2.8. We first recall how convergence to γ −1 2 arises for the complex Ginibre ensemble; part of the argument then relies on comparison with this case, treated in [3]. The reason why this situation is more tractable is that the distribution of the diagonal overlap yields an exact expression: using a few classical identities of the beta and gamma distributions, we see that Now, for any sequence of integers (k n ) n 1 such that the same product can be decomposed as It is straightforward to check that In other words, the limit distribution γ 2 essentially depends on the first k N factors, provided k N goes to infinity. Similarly in the spherical case, using Theorem 2.6 and Proposition 1.3, we write: We will prove that the first factor F (2, k N ) converges to γ 2 for a suitable sequence k N that allows comparison with the complex Ginibre case, whereas the second factor F (k N + 1, N ) converges to 1. By the identity (2.8), the independent variables involved are distributed as follows: where X N is defined by (2.5).
Convergence of F (2, k N ) to γ 2 , for a suitable sequence (k N ). For fixed k, each term F N,k converges to its analog in the complex Ginibre case. Indeed, The function x → x −m being smooth and bounded on (1, ∞) for any integer m, we have that and so, by the multiplicative version of Lemma 2.9 applied to the appropriate fraction of moments, there exists a sequence k n verifying (2.17), such that for every m, which implies, by comparison with the product arising in the complex Ginibre case, Convergence of F (k N + 1, N ) to the constant 1. Let k n be the sequence of integers used in the first part of the argument; in particular, it satisfies (2.17). We check that this is enough to ensure the convergence of F (k N + 1, N ) to 1. A straightforward computation, similar to the one performed in Proposition 2.7, yields so that, thanks to telescopic products, we obtain the following expressions As k N verifies condition (2.17), the second half of the proof. The claim of the Theorem follows by Slutsky's theorem.
The following proposition relies on the spherical structure of Sph(N ) and has no analog in Section 3. Proof. Recall that the Jacobian of p at λ ∈ C is 4 (1+|λ| 2 ) 2 and that, for any λ, µ ∈ C, identity (2.12) holds. For any continuous and bounded function F of N − 1 variables, evaluated in we have for any z ∈ C, by a straightforward change of variables, where (w 1 , . . . , w N ) is a point process on the sphere with density proportional to (2.11). As the expectation on the right hand side does not depend on z (by invariance under orthogonal transformations), neither does the one on the left hand side. The claim follows by noting that for any continuous and bounded function G, by the tower property of conditional expectation, where F (l 2 , . . . , l N ) := E Λ G (O 1,1 ) is indeed a function of the variables l 2 , . . . , l N .
Clearly, Propositions 2.7, 2.8 and 2.10 provide together a full proof of Theorem 1.1.
Theorem 2.11. The quenched expectation of off-diagonal overlaps in the spherical ensemble is given by the formula Proof. Similarly to the diagonal case, we define the partial sums It follows from the facts presented in Section 1.2 that One can check, following the proof of Theorem 2.6, that |u 2 | 2 d = X N , so that which initiates the recurrence. We now compute the conditional expectation of b n+1 d n+1 by integrating out the vector u n+1 . We use Proposition 2.3 and (2.25) from Lemma 2.15 with a = B * n , b = D * n and S = S n such that S 2 n = (1 + |λ n | 2 )(I n−1 + T n−1 T * n−1 ). It follows that We notice that, as T is triangular and B n , D n are subvectors of L 1 and L 2 , The factorization follows.
Proposition 2.12. The conditional expectation of 1 N Tr G * G with G distributed according to Sph(N ) is given by the formula: Proof. It is clear that Tr G * G = Tr T * N T N , and that for any n N , Tr T n T * n = |λ n | 2 + u n 2 + Tr T n−1 T * n−1 , so that defining v N,n = v N,n (λ 1 , . . . , λ n ) := E N,Λ Tr T n T * n , yields a recursion with v N,1 = |λ 1 | 2 and, using Proposition 2.3 and (2.26) from Lemma 2.15, This suggests the introduction of w N,n = v N,n + N + n, for which we see that so that for every n N , which is equivalent to the statement, when n = N .

Constants and integrals
Lemma 2.13. The normalization constant for generalized gamma variables γ V,k with potential V (x) = M log(1 + x) and 1 k M − 1 is Proof. Let us compute, for any suitable function f , which implies the first claim. As which is the second claim.
Lemma 2.14. For any p > n, C n,p :=

21)
and for p > n + 1, Proof. We first compute C n,p by induction on n. For n = 1, p > 1, and one can note that for any α > 0, For general n, using the above equalities with α n = 1

Equation (2.21) follows. A similar induction can be performed on C
(1) n,p . The only difference is that the last step involves the following identity: for any p > 2, which, in general, yields the extra factor 1 p−(n+1) in (2.22).
Note that when we begin the recursion from [9] with n = N − 1, p = 2N , the extra factor is 1 N at every step. Lemma 2.15. For any p > n, a, b ∈ C n and any Hermitian positive-definite matrix S,

26)
where the constant C n,p is explicitly computed in Lemma 2.14.
Proof. Integral (2.23) was computed in [9]. (2.24) is zero by symmetry. For (2.25), the change of variables u = Sv yields We notice that where A = Sba * S is a matrix of rank 1. If we express v = v i e i in a unitary basis such that the vectors (e 2 , . . . , e n ) form a basis of ker(A) and denote Therefore, after a unitary change of basis the integral becomes, using Lemma 2.14 and the fact that cross-terms v 1 v i vanish by symmetry, The value of λ 1 (A) can be obtained by writing from which the claim (2.25) follows. The same technique applied to (2.26) yields and a unitary change of variable to a basis that diagonalizes S, together with Lemma 2.14, gives dv λ 1 (S 2 )v 2 1 + · · · + λ n (S 2 )v 2 n (1 + v 2 1 + · · · + v 2 n ) p = (λ 1 (S 2 ) + · · · + λ n (S 2 )) 1 p − (n + 1) C n,p , concluding the proof of the last claim.
Lemma 2.16. For any p > n, a ∈ C n and any Hermitian positive-definite matrix S, if u ∈ C n is distributed with density 1 C n,p | det S| 2 1 (1 + u * S −2 u) p with respect to the Lebesgue measure on C n , then the following identity in distribution holds: Proof. By a direct change of variable, it is clear that u d = SV n p . We note that |a * Sv| 2 = v * Av where A = Saa * S is a Hermitian matrix of rank one. A unitary change of variable brings it to the form λ 1 (A)v 2 1 with λ 1 (A) = Tr A = a * S 2 a = Sa 2 . Successive integration of the other coordinates v 2 , . . . , v n yields the result.

Truncated unitary ensemble
This section contains the proof of all claims concerning the truncated unitary ensembles TUE(N, M ) when N M . Almost every step in this study is analogous to what was done in the spherical case; we therefore refer constantly to the corresponding parts of Section 2.

Schur form and eigenvalues
As in Section 2, we first present a few general results in order to illustrate the method, as well as a few tools and definitions that are specific to the truncated unitary case. We first recall that the Schur transfom T is distributed with density proportional to with respect to the Lebesgue measure on all complex matrix elements, diagonal (dΛ = dλ 1 · · · dλ N ) and upper-triangular (du 2 · · · du n ).
Provided T T * < 1 (which implies the same condition on every submatrix T n ), we introduce the Hermitian, definite-positive matrices Note that the only differences with the matrices H n , S n−1 used in the spherical case are the minus sign and the condition on the eigenvalues of T T * .
Lemma 3.1. The determinant of H n = I n − T n T * n can be reccursively decomposed as The proof is analogous to the proof of Lemma 2.1. For any p 0, we denote by W (n) p a random vector with density with respect to the Lebesgue measure on C n ; the value of C n,p is given by (3.14). For any m 2, we denote by Y m a real random variable with density with respect to the Lebesgue measure, i.e. it follows a β 1,m−1 distribution; in particular EY m = 1 m . If w i is a coordinate of W (n) p , it follows from Lemma 3.12 that Note that the i.i.d. variables that appear in Theorem 3.5 follow the above distribution with m = M .
Lemma 3.2. Identity holds between the following expressions, for p n and f, g integrable functions of the matrix elements: where H n , S n−1 , W (n) p are defined in (2.2) and (2.4).
We deduce from the above Lemma the distribution of every top-left submatrix of the Schur form, analogously to Proposition 2.3. with respect to the Lebesgue measure on upper-triangular matrix elements (du 2 · · · du n ).
We also derive the joint eigenvalue density of the truncated unitary ensemble from the density of its Schur form, as was done in [9]. The result itself was first proven in [14].
with respect to the Lebesgue measure on C N .
The proof is analogous to the one of Theorem 2.4.  where the Y (k) M are i.i.d. distributed according to (3.5) with m = M . In particular, the quenched expectation is given by the formula

Distribution and conditional expectation of overlaps
Proof. It is similar to the one of Theorem 2.6; we sketch it again to see where the differences lie. We first write In order to characterize the distribution of this factor, we use Proposition 3.3, then Lemma 3.2 and Lemma 3.12 with a = b = B d and S = S d+1 such that S 2 d+1 = (1 − |λ d+1 | 2 )(I d − T d T * d ). This yields where Y N is distributed according to (3.5) with m = M , and independent of F d ; we denote this variable by Y Note that the same statement, which is an exact identity for any N , holds in the complex Ginibre ensemble and spherical ensemble respectively.
Proof. We know from Proposition 1.3 that the squared radii, conditionally on the event {λ 1 = 0}, are distributed like independent variables with distributions γ V,k with V (x) = −(M − 1) log(1 − x)1 (0,1) and 2 k N . We already noticed that γ V,k d = β k,M . A straightforward computation follows: For any k 2, so that the expectation is given by the telescopic product as was claimed.
Proposition 3.7. Conditionally on {λ 1 = 0}, the following convergence in distribution takes place: Note that N → ∞ implies M → ∞, as we study the truncated unitary ensemble in the regime where N M . The rate at which N, M go to infinity does not have any impact on the following proof (although it is expected to play a role when conditioning on a generic z in the bulk).
Proof. The technique is similar to the proof of Proposition 2.8. We decompose the distribution obtained by Theorem 3.5 in two factors Lemma 3.12. For any p > n, a ∈ C n and any Hermitian positive-definite matrix S, if u ∈ C n is distributed with density 1 C n,p | det S| 2 (1 − u * S −2 u) p with respect to the Lebesgue measure on C n , then the following identity in distribution holds: The proofs of Lemmata 3.11 and 3.12 are exactly analogous to the proofs of their spherical counterpart, Lemmata 2.15 and 2.16.