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ABSTRACT
We parametrize the space of double cosets of the group with respect to two subgroups T−, T+ of block strictly triangular matrices. In the appendix, we consider the quasi-regular representation of
in L2 on
, observe that it admits an additional group of symmetries, find the joint spectrum, and observe that it is multiplicity free.
1. The statement
1.1. Double cosets
Let G be a group, K, L subgroups. A double coset of G with respect to K, L is a set of the type K · g · L, i.e. the set of all elements of G that can be represented in the form kgl, where g is fixed, k ranges in K, l ranges in L. We denote the set of all double cosets by .
A description of this set is equivalent to a description of orbits of K on the homogeneous space G/L, and to a description of orbits of L on the homogeneous space . If G is finite, then a description of double cosets is equivalent to a description of intertwining operators between quasi-regular representations of G in ℓ2(G/K) and ℓ2(G/L) (see, e.g. [Citation1, Sect. 13.1]). For Lie groups and locally compact groups the picture is more complicate, in any case understanding double cosets seems necessary for understanding analysis on the corresponding homogeneous spaces.
In any case a problem of the description of double cosets arises quite often, but not quite often it admits a tame solution.
1.2. The problem
Let be a field, V be a finite dimensional linear space over
. Denote by GL[V] the group of all invertible linear operators in V. We also use the notation
for
. Split V into a direct sum
Denote
the group of all block strictly upper triangular matrices of the size
, i.e. matrices of the form
(1)
(1) where 1m denotes the unit matrix of size m. By
we denote the group of all block triangular matrices, i.e. we allow arbitrary invertible matrices on places of units. Clearly, T + is normal in P +,
By
and
we denote the corresponding groups of lower triangular matrices.
In this paper we describe double coset spaces
(2)
(2)
1.3. The statement
Recall some definitions. Let X, Y be linear spaces over . A linear relation (see, e.g. [Citation2, Sect. 2.5]) L:X ↔ Y is a linear subspace in
. A graph of a linear map X → Y is a linear relation but not vice versa. For a linear relation we define:
the kernel
is the intersection
;
the image
is the projection of L to Y along X;
the domain
is the projection of L to X along Y;
the indefiniteness
is the intersection
.
If L is a graph of a linear operator A:X → Y, then kernel and image are the usual kernel and image. In this case also ,
.
Any linear relation L determines a canonical invertible operator
Moreover, a linear relation L:X ↔ Y is determined by subspaces
in X, subspaces
in Y and the operator Θ (L).
Now consider a linear space , consider its copy W ≃ V and take two decompositions
(3)
(3)
(4)
(4) Consider the double cosets (Equation2
(2)
(2) ). For each element A ∈ GL[V] we assign a canonical collection of linear relations
defined in the following way. We say that (ξ, η) ∈ χij(A) if there exist x1 ∈ V1, …, xi−1 ∈ Vi−1 and yj+1 ∈ Wj+1, …, yq ∈ Wq such that
Proposition 1.1
(a) | The relations χij depend only on the double coset containing A; | ||||
(b) | The relations χij = χij(A) satisfy the conditions:
|
Proof is contained in Sections 2.1–2.3.
We say that a collection of linear relations ξij:Vi ↔ Wj is a bi-hinge Footnote1 if it satisfies conditions (Equation5(5)
(5) )–(Equation7
(7)
(7) ). We denote the set of all bi-hinges by
Theorem 1.2
The map determines a one-to-one correspondence between the double coset space (Equation2
(2)
(2) ) and the set of bi-hinges.
1.4. Some known cases of classification of double cosets
We discuss shortly such cases related to classical (also, semisimple, reductive) groups. There are three big important series of solvable problems with tame solutions related to symmetric subgroups. Recall that a subgroup H ⊂ G is symmetric if it is the set of fixed points of some involution σ:G → G (i.e. σ(g1)σ(g2) = σ(g2g1), σ(σ(g)) = g).
Let G be a real semisimple group, H, L are symmetric subgroups. In particular, this problem includes the Jordan normal form (more generally, description of conjugacy classes in all semisimple Lie groups Footnote2 Q), reduction of pairs of nondegenerate quadratic (or symplectic) forms, canonical forms of pairs of subspaces in a Euclidean space, etc. A formal reference to a ‘general case’ is [Citation3].
We consider
, where G is a semisimple group, H is a symmetric subgroup and P is a parabolic subgroup (a block triangular subgroup). A formal reference to a ‘general case’ is [Citation4].
For p-adic groups, the most important case is related to the Iwahori subgroups, see [Citation5].
There is a big family of minor variations of these series (we can slightly enlarge G or slightly reduce subgroups).
Next, there are different ways to assign spectral data to several matrices (this also can be regarded as a classification of double cosets): a spectral curve with a bundle, see [Citation6–8], or a spectral surface with a sheaf, see [Citation9].
On the other hand, for infinite-dimensional groups quite often a double coset space has a structure of a semigroup. There arise questions about spectral data visualizing such multiplications. This also leads to objects of algebraic-geometric nature as spaces of holomorphic maps of the Riemann sphere to Grassmannians (see [Citation2, Sect.X.3]) or rational maps of Grassmannians to Grassmannians (see [Citation10]).
Our case arose as a by-product of a construction of the latter type in [Citation11, proof of Theorem 1.6], it is quite elementary. However, I could not find it in the literature. Apparently, a natural generality here are spaces , where G is a classical (or semisimple) group, H is a symmetric subgroup, and T is the maximal unipotent subgroup in a parabolic subgroup.
A possibility to describe double coset space implies a question about harmonic analysis for L2 on . Such an analysis is possible, see the appendix to this paper (but it is not directly related to the description of double cosets).
2. Proof of Theorem 1.2
2.1. Proof of Proposition 1.1.a.
We must show that χij does not depend on the choice of a representative of a double coset.
Reformulate the definition of χij(A) in the following way. Recall that W is a copy of , the spaces V and W are decomposed as (Equation3
(3)
(3) )–(Equation4
(4)
(4) ). We consider the intersection
and send
by the formula
(8)
(8)
Next, we send Z to the quotient
Clearly, the relation χij(A) is the image of H(A) in this space.
Now, let C ∈ T +, consider AC instead of A. Then H(AC) = C−1 H(A). The set of vectors (h, Ah), see (Equation8(8)
(8) ) changes to (C−1h, Ah). But C−1 acts trivially in
Therefore χij(AC) = χij(A).
2.2. Proof of (5)
For the sake of concreteness, let us verify that . Write the equation
A vector xi is contained in
if there are x1, …, xi−1, yj+1, …, yq such that this equation is satisfied. This implies that (xi, yj+1) ∈ χi(j+1). In particular,
.
Conversely, let . Then there are xi and x1, …, xi−1, yj+2, …, yq satisfying the equation. This implies that
.
2.3. Verification of (6)–(7)
To be concrete, let us prove the statements from the first row (Equation6(6)
(6) ).
Let us show that . Let
. Then there yj+1, …, yq such that
But the right-hand side must be zero and η = 0.
The statement follows from the subjectivity of A.
2.4. The action of ∏GL[Wj] × ∏GL[Vi] on the double cosets space
Let G be a group, K, L its subgroups, and
the normalizers of K and L. Then the group
acts on
. Indeed, for
,
we have
So this transformation sends double cosets to double cosets. Clearly, orbits of
on
are in one-to-one correspondence with double cosets
.
In our case the normalizers of and
are the groups
and
, the quotients are ∏GL[Wj] and ∏GL[Vi].
So let us describe the double coset spaces
(9)
(9) Our subgroups contain the usual subgroups of lower and upper triangle matrices. Applying the usual Gauss reduction we observe that any double coset contains a permutation matrix (a permutation matrix is a matrix consisting of zeros and units and containing only one unit in each column and each row), i.e. an element of the symmetric group S(n). After this reduction we can permute basis elements in each Vi and in each Wj, so the double coset space (Equation9
(9)
(9) ) is in one-to-one correspondence with
Our matrix has a natural decomposition into pq blocks, it is important only the number of units in each block. We formulate our observation in the following complicate form.
Lemma 2.1
For any double coset (Equation9(9)
(9) ) there are canonical decomposition of each Vj and each Wi into a direct sum of coordinate subspaces
(10)
(10)
(11)
(11) such that
A representative of the double coset is the map sending each
to the corresponding
coordinate-wise.
Less formally, we get a matrix of the form
(12)
(12) Here p = 4, q = 3. We present a decomposition of a matrix into blocks corresponding to the decompositions
and
, and refined blocks corresponding to decompositions
and
. Units are put in bold to make them visible among zeros.
Lemma 2.2
For the matrix (Equation12(12)
(12) ) the corresponding linear relations χij are the following:
(13)
(13)
(14)
(14) The operator
is the identical map
.
We say that such a bi-hinge is standard and denote it by
Proof.
The statement is semi-obvious. First, let us write χ32 for the matrix (Equation12(12)
(12) ) (the general case differs from considerations below only by longer notation, see below). We apply this matrix to a vector
(15)
(15) and get
(16)
(16) On the other hand, this must be equal to
(17)
(17) Recall that the linear relation χ32 consists of vectors
for which there are
,
such that (Equation16
(16)
(16) ) equals to (Equation17
(17)
(17) ). We get
(18)
(18)
(19)
(19) The remaining equations contain no information:
(20)
(20)
(21)
(21) The sets of variables in (Equation18
(18)
(18) )–(Equation19
(19)
(19) ) and (Equation20
(20)
(20) )–(Equation21
(21)
(21) ) do not intersect (since we start with a permutation matrix), On the other hand, the second system (Equation20
(20)
(20) )–(Equation21
(21)
(21) ) has a solution, since each variable is present in it only one time (again, this is a priori clear, because we start with a permutation matrix).
Finally, we get the linear relation χ23 consisting of vectors
For a general case we must write
(22)
(22) instead of (Equation15
(15)
(15) ) and
instead of (Equation17
(17)
(17) ). We apply the permutation matrix to (Equation22
(22)
(22) ), repeat the same steps and come to the linear relation consisting of all vectors of the form
2.5. The action of ∏GL[Vi] × ∏GL[Wj] on the set of bi-hinges
Clearly, the group ∏GL[Vi] × ∏GL[Wj] acts on the space
therefore it acts on the set of bi-hinges.
Lemma 2.3
(a) | Any orbit of the group ∏GL[Vi] × ∏GL[Wj] on the set
| ||||
(b) | The stabilizer |
Proof.
(a) In a fixed Vi we have a flag
We choose an element of GL[Vi] sending this flag to the flag of decreasing coordinate subspaces of the form
(25)
(25) this flag is canonically determined by dimensions of
.
Similarly, we fix Wj, consider the flag
and choose an element of GL[Wj] sending this flag to a flag consisting of an increasing sequence of subspaces of the form
.
So for any element of our bi-hinge we fixed positions of its domain, kernel, image and indefinity. After this fixing, it remains a possibility to choose coordinates in subquotient of the flag (Equation25(25)
(25) ).
We get the desired canonical form. The statement (b) also becomes obvious, since the stabilizer must regard the flags in each Vi and Wj and the maps Θ ( · ).
2.6. Coincidence of stabilizers
Thus, we have a map
(26)
(26) which is ∏GL[Vi] × ∏GL[Wj]-equivariant and establishes a bijection of the sets of orbits. For a proof of Theorem 1.2 it is sufficient to show that this map establishes a bijection for each pair of corresponding orbits. So we must check the coincidence of stabilizers of canonical representatives of orbits. So it suffices to prove the following lemma.
Lemma 2.4
Consider a representative of a double coset in the canonical permutation form , see Lemma 2.2, and the corresponding standard bi-hinge
. Then any element of ∏GL[Vi] × ∏GL[Wj] stabilizing the bi-hinge stabilizes J.
Remark 2.1
Notice that the inverse inclusion of stabilizers follows from the ∏GL[Vi] × ∏GL[Wj]-equivariance of the map (Equation26(26)
(26) ).
Proof.
The stabilizer of a standard bi-hinge is described in Lemma 2.3. It is a product of subgroups (Equation23(23)
(23) ) and (Equation24
(24)
(24) ). For the reductive factor (Equation23
(23)
(23) ) the statement is clear. The unipotent factor itself is a product, and it is sufficient to prove the statement for any factor in (Equation24
(24)
(24) ), say
.
We first study the matrix J given by (Equation12(12)
(12) ) and a matrix
(27)
(27) Multiplying (Equation12
(12)
(12) ) by this element we get the matrix
(28)
(28) We put nonzero symbols in bold to make them visible on the field of zeros. Clearly, this matrix can be reduced to the initial form (Equation12
(12)
(12) ) by a left multiplication by an element of
. The boxed units allow to delete x, y, z.
More precisely, consider the second block column Ξ of our big matrix (Equation28(28)
(28) ). It contains several zero rows. Removing such rows from Ξ we get precisely the matrix (Equation27
(27)
(27) ). On the other hand, consider a row of (Equation28
(28)
(28) ) whose part contained in Ξ is non-zero. Then the remaining part of the row is zero. Elements x, y, z of Ξ are located precisely under units, this allows to ‘kill’ them applying a multiplication from the left by a lower triangular matrix.
For a general case, we multiply the corresponding permutation matrix by an element of and get the same structure in the ith block column Ξ of the matrix.
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Notes
1 Cf. similar objects in [Citation13].
2 Namely, we set G = Q × Q, H = L = diag Q is the diagonal subgroup.
References
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Appendix 1
The spaces L2 on ![](//:0)
![](//:0)
A.1. The principal series of unitary representations of the groups ![](//:0)
![](//:0)
Denote by Λ the set of pairs λ|λ′ of complex numbers of the type
For λ|λ′ ∈ Λ we have a well-defined ‘generalized power’ of any nonzero
:
Consider the subgroup
consisting of all upper triangular matrices,
(A1)
(A1) A signature λ is a collection of the form
For such λ denote by χλ(A) the character of B+(n) defined by
(A2)
(A2) By ρλ we denote the representation of
unitary induced in the sense of Mackey (see, e.g. [Citation1, Subsect. 13.2], [Citation12, Sect. 16, Sect.19.1]) from a one-dimensional representation χλ of the subgroup B+(n). Such unitary representations are called representations of the nondegenerate principal series, see, e.g. [Citation12, Sect. 19.3]. Representations ρλ and ρμ are equivalent if and only if a collection
can be obtained from a collection
by a permutation. Denote by Σn the set of all signatures defined up to a permutation.
Next, consider the action of on
by left and right multiplications,
. This determines the left–right regular representation of
in
. According to Gelfand and Naimark, see, e.g. [Citation12, Sect. 14.4.A], this representation decomposes as a direct integral
(A3)
(A3) of representations of
.
A.2. The space ![](//:0)
.
Let us decompose the quasiregular representation of in
. This problem has an additional symmetry. Indeed, the subgroup
normalizes
. Therefore, the quotient group
acts on
by left multiplications. So we get a unitary representation of the group
(A4)
(A4) in our L2.
For elements denote by
the row obtained by concatenation of rows λj.
Theorem A.1
The decomposition of under the action of the group
is multiplicity-free and has the form
Proof.
We have a space homogeneous with respect to group G. The stabilizer G0 of the initial point consists of tuples
By definition our representation is induced from the trivial representation of the stabilizer G0. Consider a larger group
defined by
We apply induction in stages (see, e.g. [Citation1, Subsect.13.1], [Citation12, Sect. 16.2]), first from G0 to
, second from
to G.
On the first step we have the same normal subgroup in both G0 and
. Since the initial representation of G0 is trivial, the induced representation is trivial on
. In fact we have the induction from
The second group is the double
, the first group is
embedded to the double as the diagonal. So the induced representation is the left–right representation of the double. Clearly, it is equivalent to the tensor product of the left–right regular representations of the factors
,
We decompose spaces
according (EquationA3
(A3)
(A3) ).
In this way, we come to a direct integral of irreducible representations of having the form
(A5)
(A5) where
is the representation of
, which is trivial on
and is defined by
Notice that
itself is an induced representation. It is induced from a one-dimensional representation of B+(n), see (A.1), namely from the character given by the formula
(A6)
(A6) where
is determined by (EquationA1
(A1)
(A1) ).
Next, we consider the representation of G induced from an irreducible representation (EquationA5(A5)
(A5) ) of
. Since the factor ∏GL(αj) is present in both groups
and G, actually we have the induction from
to GL(n) (formally, we can refer to [Citation12, Sect.16.2.D, Theorem 3]). But the representation
itself is induced from the character (EquationA6
(A6)
(A6) ) of B+(n). Applying induction in stages we get that our representation of GL(n) is induced from the character (EquationA6
(A6)
(A6) ) of B+(n). But this is a representation
of the principal series.