Radial parts of Haar measures and probability distributions on the space of rational matrix-valued functions

0.18 MB·

by  Yury A. Neretin

Radial parts of Haar measures
and probability distributions on the space
of rational matrix-valued functions
6 Yu.A.Neretin1
1
0
ConsiderthespaceCofconjugacyclassesofaunitarygroupU(n+m)withrespect
2
to a smaller unitary group U(m). It is known that for any element of the space C we
n
can assign canonically a matrix-valued rational function on the Riemann sphere (a
a
J Livshits characteristic function). In the paper we write an explicit expression for the
natural measure on C obtained as the pushforward of the Haar measure of the group
8
U(n+m) in theterms of characteristic functions.

]
R
1 The statement
G
.
h 1.1. The purpose of the paper. There is a wide literature (see, e.g., [19],
t
a [21], [22], [7] and further references in these works) on Gaussian random func-
m tions, the topic arises at least to the Payley–Wiener book [18], Chapter 10.
[ Relatively recently M. Krishnapur [10] started investigation of random matrix-

valued holomorphic functions.
1
In the present paper we consider measures on the space of rational matrix-
v
1 valued functions on the Riemann sphere. The origin of the question under a
3 discussionisthe following. Tobe definite,consideranunitarygroupU(n). The
9 distributionofeigenvaluesofunitarymatricesisameasureonthesetofn-point
1 subsetsonthecirclewithadensityoftheformC· |z −z |2 (TheHermann
0 k<l k l
. Weylformula,see,e.g.,[8],formula(3.2.2),or[4],TQheorem11.2.1). Thereexists
1
a zoo of similar formulas, a usual corresponding term is ’radial parts of Haar
0
measures on symmetric spaces’. Namely, we consider a Riemannian symmetric
6
1 spaceG/K,thespaceofdoublecosetsK\G/K,andthepushforwardoftheHaar
: measure under the map G→K\G/K (in the example with unitary group,we
v
haveG=U(n)×U(n), andK isthediagonalsubgroupU(n)), generalformulas
i
X are contained in [6], Propositions X.1.17, X.1.19.
r This important topic has numerous applications and continuations. How-
a
ever, its extensions to other pairs of groups and subgroups are almost absent2
G⊃ K. One of obstacles for such extensions are difficulties related to descrip-
tions of double coset and conjugacy classes spaces.
In one case such description is known during a long time. Namely, for con-
jugacy classes of a unitary group U(n+m) with respect to a smaller subgroup
U(m) the solution is given in the terms of Livshits characteristic functions (see
below). In [11] there was proposed a counterpart of characteristic functions for
1TheresearchwascarriedoutattheIITPRASattheexpenseoftheRussianFoundation
forSciences(project 14-50-00150).
2The case when G = SU(2)×···×SU(2), and K = SU(2) is the diagonal subgroup, is
examinedin[12].
1
double cosets of U(n+m) by O(m), see also [17]. In [13], [15] characteristic
functionswereconstructedforawideclasspairsgroup–subgroup. Sincecharac-
teristic functions from [13] were originated from representationtheory, see [14],
the question about radial parts of Haar measures arises naturally.
In the present paper we get an explicit formula for radialpart of Haar mea-
sure in the case of conjugacy classes of U(n+m) by U(m).
α β
1.2. Livshits characteristic function. Let g = be a block
(cid:18)γ δ(cid:19)
matrix of size (n+m)×(n+m). The characteristic function of the matrix g
(another term is a transfer-function) is a function on C defined by the formula
χ(λ):=α+λβ(1−λδ)−1γ. (1.1)
This function takes values in the space of matrices of size n×n. It is easy to
see that the function does not changes under conjugations of the matrix g by
1 0
matrices of the form . I other words, χ is an invariant of conjugacy
(cid:18)0 u(cid:19)
classes of U(n+m) by U(m).
It is known that any rational matrix-valued function that has not a singu-
larity at λ = 0, is a characteristic function of a matrix. On a reconstruction
of the matrix (more precisely, of the conjugacy class) g from its characteristic
function, see, e.g., [3], Chapter 19. This reconstruction is not unique3.
Now let amatrix g be unitary. Then the characteristicfunction satisfies the
following properties:
1) For |λ|=1 values of χ(λ) are unitary matrices4.
2) For |λ| < 1 we have5 kχ(λ)k 6 1; by the Riemann–Schwarz reflection
principle, for |λ|>1 we have kχ(λ)−1k61.
3) Represent detχ(λ) as an irreducible fraction, u(λ). It is easy to show
v(λ)
that degrees of polynomials u(λ), v(λ) are 6 m. Indeed this determinant can
be written in the form6
α −λβ
det
(cid:18)γ 1−λδ(cid:19)
detχ(λ)= . (1.2)
det(1−λδ)
Conversely, any rational function on C taking values in the space of n×n-
matrices, satisfying the above-listed properties, is a characteristic function of a
certainunitarymatrixofsizen+m. DenotethesetofsuchfunctionsbyR (m).
n
3Howeveramatrixofaminimalpossiblesizewithagivencharacteristicfunctionisunique
up to a conjugation, see the textbook Dym [3], Chapter 19. Moreover an element of a cat-
egorical quotient ofGL(n+m,C)by GL(m,C) canbe reconstructed fromthe characteristic
functioninauniqueway(see[16]).
4Aproofofthisandthenextstatement arecontained inRemarkin.2.1.
5Thus characteristic functions are matrix-valued analogs of interior functions, which are
well-knowninclassicaltheoryofanalyticfunctions,see,e.g.,[5]. Thetheoryofmatrix-valued
interiorfunctionswasdevelopedin[20]. Oninteriorfunctionsofmatrixargument,see[13].
6For this, it is sufficient to apply the usual formula for a determinant of a block matrix,
seebelow(2.17).
2
Next, consider block matrices of size n+m = n+(m +m ) having the
1 2
structure
α β 0
g =γ δ 0. (1.3)
0 0 κ
 
It is easy to see that a characteristic function does not depend on the block
κ, on the other hand eigenvalues of the matrix κ are invariants of a conjugacy
class. It is easy to show that any matrix g ∈ U(n+m) by conjugations by
elements of U(m) can be reduced to the form (1.3), where kδk<1.
Finally, spectral data determining a conjugacy class of U(n+m) by U(m)
looks as follows: the characteristic function and the set of eigenvalues of the
matrix δ that are contained in the circle |λ|=1.
Remark. I do not know a source of literature containing simple proofs of
statements about unitary matrices in a (very small) degree of generality that
is necessary for us. For arbitrary matrices a short exposition is contained in
[3], Chapter 19, see also [1]. The unitary case requires additional arguments.
The statements that are necessary for us is an extremely particular case of [2],
Theorem 5.1. A restoring of a unitary matrix from a characteristic function
also can be done by the Potapov method [20], using expansion of a rational
matrix-valued inner function in a Blaschke product. ⊠
Anyway, for elements of U(n + m) in a general position, the block δ is
purely contractive and conjugacy classes are determined by their characteristic
functions. Thus we have the map
U(n+m)→R (m),
n
and we wish to evaluate the image of the Haar measure under this map. The
answer will be presented in terms of the set of λ, where χ(λ) has an eigenvalue
−1, and the corresponding eigenvectors.
1.3. The density of the measure. Consider an element of the space
R (m) in a general position (i.e., an element of a set of full measure, which
n
will be detaches step-by-step). Consider the set of all points t ∈C such that a
k
matrix χ(t ) has an eigenvalue −1. This set is contained in the circle 7 |λ|=1.
k
In a general position the number of such points is m. Let us order the points
t in some way, for instance, let us assume that
k
2π >argt >···>argt >0.
1 n
In a general position there is a unique corresponding eigenvector c ,
k
χ(t )c =−c , c =(c1,...,cn)∈Cn.
k k k k k k
To fix coordinates on R (m), we normalize c from the condition
n k
hχ′(t )c ,c i=−t−1.
k k k k
7For elements of U(n+m) in a general position we have kαk < 1. But α = χ(0). On
the other hand, on the unit circle kχ(λ)k = 1. Applying the maximum principle to linear
functionalsonthespaceofmatricesweeasilygetkχ(λ)k<1insidethecircle.
3
Next, we fix a phase of each vector c by the assumption c1 >0.
k k
Lemma 1.1 The properties mentioned above really hold in a general position.
In a general position, the numbers t , the vectors c , and the unitary matrix
k k
U :=χ(−1)
uniquely determine a characteristic function.
ComposeamatrixC ofsizen×mofvector-columnsc . Composeadiagonal
k
matrix T of numbers t .
j
Theorem 1.2 The image of probabilistic Haar measure under the map
U(n+m)→R (m)
n
has the form
∗ −2n−2m 2m
θ · det(1+T +C (1+U)C) det(1+U) ×
n,m
(cid:12) m (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
× |1+t |2m+2n |t −t |2×
k k l
kY=1 16kY<l6n
m m n n
dt
×dσ (U)· c1dc1 · dRecj dImcj · k, (1.4)
n k k k k it
kY=1 kY=1jY=2 kY=1 k
where dσ (U) is the Haar measure on U(n) and
n
m+n−1j!
θ =2−mπ−mn j=1 . (1.5)
n,m nQ−1j!· m j!
j=1 j=1
Q Q
In view of this statement, we mention two works on random analytic func-
tions.
First, measures on the space of scalar meromorphic functions in terms of
distributions of poles and residues were considered by Wigner [23], 1952. This
is similar to the language of our Theorem 1.2, moreover during a calculation
in Subsect.2.4 we literally use distributions of poles and residues (the main
difference is the following: in Wigner paper distributions of poles and residues
are determined by independent random variables, in our case this is not so).
The class of meromorphic functions, which was considered in [23] can be send
to the class of of (scalar-valued) interior functions by a simple transformation.
Second, ameasureonthe spaceofinteriorfunctions wasconsideredby Kat-
snelsonin [9], 2002,it seems (at leastfromthe firstsight)that his object is not
similar to our considerations.
IamgratefultoM.Sodin,N.Makarov,S.Sodinfordiscussionsofthistopic.
4
2 Proof
Notation:
Mat is the space of complex matrices of size k×l;
k,l
Herm is the space of Hermitian matrices of size k;
k
∗
AHerm is the space of anti-Hermitian matrices (X =−X) of size k;
k
Tk is the torus, i.e. the product of k circles with the corresponding group
structure;
R is the positive semi-axis.
+
2.1. A preparation lemma. Let T be a square matrix. Define its Cayley
transform by the formula
g 7→H =−1+2(1+g)−1 =(1+g)−1(1−g).
This transform is inverse to itself and send unitary matrices to anti-Hermitian
matrices.
Lemma 2.1 Consider a block matrix g of size n+m and perform the following
chain of manipulations:
1) we apply the Cayley transform and get a certain matrix H;
2) we take the characteristic function ϕ(s) of the matrix H;
3) we apply the Cayley transform to ϕ(s) and get a function ψ(s) taking
values in Mat ;
n,n
4) we consider the function ϕ t+1 .
t−1
(cid:16) (cid:17)
Then the resulting function coincides with the characteristic function χ (t)
g
of the matrix g.
First,werecallthatdefinitionsoftheCayleytransformandthecharacteristic
function can be formulated as follows.
• Let g be a square matrix of size n. Let vectors u, v ∈Cn be connected by
the relation
(u−v)=g(u+v). (2.1)
Then v =Hu, where H is the Cayley transform of the matrix g.
• Let g be a block matrix of size n+m. Fix λ ∈ C. Consider the set L of
all pairs q, p∈Cn, for which there exists x∈Cm such that
q α β p
= . (2.2)
(cid:18)x(cid:19) (cid:18)γ δ(cid:19)(cid:18)λx(cid:19)
The subset L⊂Cn⊕Cn obtained in such way is determined by the equation
q =χ (λ)p
g
for all λ except poles of the characteristic function.
5
The first statement is evident, we simply write (2.1) in the form
(1−g)u=(1+g)v.
To verify the second statement, we write (2.2) as
q =αp+λβx, x=γp+λδx.
Eliminating x we get the desired statement.
Remark. Now we can easily derive properties 1)–2) of characteristic func-
tions from Subsect. 1.2. Indeed, if a matrix g is unitary, then
kqk2+kxk2 =kpk2+kλxk2.
Assuming |λ| = 1, we get kqk2 = kpk2, this implies that the matrix χ(λ) is
unitary. If |λ| <1, then kqk2 > kpk2, i.e., kχ(λ)k 61. It is worth noting, that
a proof using directly formula (1.1) is not so easy. ⊠
Proof of the lemma. Letg beablocksquarematrixofsizen+m. Then
v =Hu is equivalent to the equality
u−v =g(u+v).
Next, q =ϕ(s)p, if there exists a vector x such that
p q p q p−q p+q
− =g· + or =g· .
(cid:18)sx(cid:19) (cid:18)x(cid:19) (cid:20)(cid:18)sx(cid:19) (cid:18)x(cid:19)(cid:21) (cid:18)(s−1)x(cid:19) (cid:18)(s+1)x(cid:19)
Now we must apply the Cayley transform, i.e., set q = y−z, p = y+z. It is
more convenient to set q = (y −z)/2, p = (y +z)/2, this does not change a
result. We get
z y
=g· ,
(cid:18)(s−1)x(cid:19) (cid:18)(s+1)x(cid:19)
recallthatweconsiderpairsz,y,forwhichthereexistsxsuchthatthisequality
′ ′
holds. set x = (s−1)x. If we change x by x the condition for q, p does not
change. Thus,
z y
(cid:18)x′(cid:19)=g·(cid:18)ss+−11x′(cid:19).
We get the definition of the characteristicfunction of the matrix g at the point
s+1. (cid:3)
s−1
2.2. Start of proof of the theorem. The first Cayley transform.
Now, we intend to watch step-by-step pushforwards of the Haar measure un-
der the transformations described in Lemma 2.1. It is important that all our
manipulations over matrices commute with the action of the group U(m).
The image of the probabilistic Haar measure on U(k) under the Cayley
transform was evaluated by Hua Loo Keng. It has the form (see [8], §3.1)
τ det(1−X2)−kdX˙, (2.3)
k
6
where X is the Cayley transform of a matrix g,
dX˙ := dImx · dRex , (2.4)
kl kl
16kY6l6n Y16kY<l6n
and τ is a normalizing constant,
k
k−1
τ =2k2−kπ−k(k+1)/2 j!
k
jY=1
Obviously, the density of the measure can be written as
τ det(1+X)−kdet(1−X)−k =τ det(1+X)|−2k.
n n
(cid:12)
In our case, k=n+m. Representing an anti-(cid:12)Hermitian matrix H (i.e., the
A B
Cayley transform of the matrix g) in a block form H = i ∗ , we get a
(cid:18)B D(cid:19)
measure
−2n−2m
A B
τm+n·(cid:12)det(cid:20)1+i(cid:18)B∗ D(cid:19)(cid:21)(cid:12) dA˙dB˙ dD˙,
(cid:12) (cid:12)
where dA˙, dB˙, dD˙ are(cid:12)(cid:12)the naturalLebesgue(cid:12)(cid:12)measureson the space of matrices,
dB˙ := dReb dImb ,
kl kl
16Yk6n16Yl6m
and the notations dA˙, dD˙ are similar (2.4).
2.3. Quotients with respect to the actions of the groups. Reduce
D to the diagonal form, let µ > ··· > µ be the eigenvalues, let M be the
1 m
diagonal form, i.e., M = V−1DV. Denote by Ξ ⊂ Rm the set of all such
m
collections. The distribution of eigenvalues of Hermitian matrices is (see, e.g.,
[8], §3.3, or [4], Theorem 10.1.4)
πm(m−1)/2 m
dwm = |µ −µ |2 dµ . (2.5)
k l k
j!
16j6m 16kY<l6m kY=1
Q
Considerthe actionofthe groupU(m) onthe space ofHermitian matrices 1H,
i
−1
1 0 A B 1 0 A BV
(cid:18)0 V(cid:19)(cid:18)B∗ D(cid:19)(cid:18)0 V(cid:19) =(cid:18)(BV)∗ V−1DV(cid:19). (2.6)
In fact, this group acts on pairs (D,B), the matrix B can be regarded as a
collectionofnrows,denotethembyβ ,..., β . Wegetanactionoftheunitary
1 n
groupU(m)onthecollectionsoftheform: aself-adjointoperatorDandn-tuple
of vectors β ,..., β .
1 n
We can reduce D to a diagonal form. After this we preserve a freedom to
conjugate in (2.6) by elements of the diagonal subgroup Tm. In this way, we
can make all coordinates of the vector
β =(b ,...,b )
1 11 1m
7
being real and positive. After this all coordinates of vectors β , ..., β are
1 n
fixed.
Lemma 2.2 Consider the map
Π:Hermm×Matn,m →Ξm×(R+)m×Matn−1,m, (2.7)
′
which to each pair (D,B) assigns its canonical form (M,B ). Then the image
of the Lebesgue measure under the map Π is
m
dwm·(2π)m· b′ db′ dReb′ dImb′ .
1k 1k kl kl
kY=1 16l6nY,26k6n
Proof. Denote by Fl the space, whose points are ordered collections
m
ℓ = (ℓ ,...,ℓ ) of pairwise orthogonal (complex) lines in Cm. Obviously, Fl
1 m m
is simply a flag space). The group U(m) acts on this space transitively, a
stabilizer of a point is isomorphic Tm, i.e., Fl ≃ U(m)/Tm. Since U(m) is
n
compact,there isaunique U(m)-invariantprobabilisticmeasureonFl ,denote
n
itbydℓ˙. ForanyHermitianmatrixD,weassignthecollectionofitseigenvalues
µ >··· >µ and the collection of its eigenvalues ℓ , ..., ℓ . In this way, we
1 m 1 m
get map
Herm →Ξ ×Fl ,
m m m
which is defined a.s. and bijective a.s. The image of the Haar measure under
this map is dwmdℓ˙. This statement is equivalent to8 formula (2.5).
Thus, the action of the group U(m) on
Herm ×Mat ≃Herm ×(Cm)n
m n,m m
can be regarded as an action on the space
Ξ ×Fl ×Cm×(Cm)n−1,
m m
trivial on the first factor.
ConsiderthequotientofFl ×CmwithrespecttoU(m). Fixapointℓ∈Fl ,
m m
denote by Tm ⊂U(m) its stabilizer. For all ℓ the distribution of Tm-invariants
ℓ ℓ
in a fiber Cm is the same. Therefore, on the space
(Fl ×Cm)/U(m)≃(R )m
m +
we get the same distribution of invariants,
m
′ ′
b db .
1k 1k
kY=1
Now we can identify the following spaces by a U(m)-equivariant measure
preserving transformation
Fl ×Cm ≃(R )m×U(m).
m +
8Actually,proofsofformula(2.5)usuallyusethisfact(directlyorhidden).
8
Thus we come to the action of U(m) on
Ξ ×(R )m×U(m)×(Cm)n−1,
m +
moreover, the action is trivial on the first two factors, on U(m) the group acts
by right shifts, on vector-rows it acts in a natural way. Now a description of
quotient becomes trivial. (cid:3)
In fact, we use a slightly modified version of the lemma. Consider a map
π : Ξm×Matn,m →Ξm×(R+)m×Matn−1,m,
defined as a reduction of the first row to the canonical form using the action of
the torus Tm .
Corollary 2.3 The image of the measure Ξ ×Mat under the map π coin-
m n,m
cides with the image of the measure on Herm ×Mat under the map Π. In
m n,m
other words, the natural identification of the spaces
(Herm ×Mat )/U(m)←→(Ξ ×Mat )/Tm
m n,m m n,m
is measure preserving.
Wecametothefollowingproblem. ConsiderthespaceHerm ×Mat ×Ξ
n n,m m
equipped with a measure
−2n−2m
A B
τm+n(cid:12)det(cid:18)1+i(cid:18)B∗ M(cid:19)(cid:19)(cid:12) dwm(M)dA˙dB˙. (2.8)
(cid:12) (cid:12)
(cid:12) (cid:12)
We mustwatchbe(cid:12)haviorof this measureu(cid:12)nder transformations2)-4)described
in Lemma 2.1.
2.4. The taking of a characteristic function. Next, we write the
A B
characteristic function of the matrix i ∗ ,
(cid:18)B M(cid:19)
m ∗
sb b
ϕ(s)=iA−sB(1−isM)−1B∗ =iA− k k , (2.9)
1−isµ
Xk=1 k
where b are columns of the matrix B.
k
We see thatpolesofϕ(s)arelocatedatpoints s = 1 , andresiduesatthe
k iµk
poles are rank 1 matrices given by
∗
b b
Res ϕ(s)=− k k.
s=1/iµk µ2
k
∗
Notice that ϕ(0) = iA, and the matrix b b remembers the vector b up to a
k k k
phase. Therefore, there is no need to introduce new coordinates, the matrices
A, B, M can be automatically reconstructed from the function ϕ(s).
Notice that in a general position all the vectors b are non-zero.
k
9
2.5. The second Cayley transform. Next, we take the function
ψ(s)=−1+2(1+ϕ(s))−1.
Notice that for a pure imaginarys the matrix ψ(s) is unitary (since the matrix
ϕ(s) is anti-Hermitian). Denote U :=ψ(0). Clearly,
iA=−1+2(1+U)−1.
Next, at points s = 1/iµ a matrix (1+ϕ(s))−1 is degenerate, wherefore (−1)
k
is an eigenvalue of ψ(s). Let us write Taylor expansions of functions 1+ϕ(s)
and (1+ϕ(s))−1 at a point s=1/iµ ,
k
∗
b b 1
1+ϕ(s)=− k k +V+..., (1+ϕ(s))−1 =W+ s− Y+...,
µ2 s− 1 (cid:18) iµk(cid:19)
k(cid:16) iµk(cid:17)
where V, W, Y are certain matrices. Then
∗
Wb b
(1+ϕ(s))−1(1+ϕ(s))=1=− k k +(WV −µ−2Yb b∗)+...
k k k
µ2 s− 1
k(cid:16) iµk(cid:17)
Therefore,
∗
Wb b =0;
k k
WV −µ−2Yb b∗ =1. (2.10)
k k k
We have Wb = 0. Otherwise, multiplying the vector column Wb by a
k k
non-zerorowb we getanonzeromatrix. Next,we write the equality ofmatrix
k
elements h(...)b ,b i for both sides of (2.10),
k k
(WV −µ−2Yb b∗)b ,b =hb ,b i. (2.11)
k k k k k k k
(cid:10) (cid:11)
Notice that
∗
hWVb ,b i=hVb ,W b i=hVb ,0i=0.
k k k k k
∗
Indeed, the matrix W =(ψ(s)+1)/2 is normal, therefore kerW =kerW .
Next,
∗ ∗
(Yb b )b =Yb (b b )=hb ,b iYb .
k k k k k k k k k
Hence equation (2.11) transforms to
hYb ,b i=−2µ2,
k k k
or, equivalently, to
hψ′(1/iµ ),b i=−µ2. (2.12)
k k k
In particular, we see that vectors b can be reconstructed up to a phase from
k
the function ψ(s).
10