**Applied Mathematics** Vol.4 No.11(2013), Article ID:38983,6 pages DOI:10.4236/am.2013.411205

Canonical and Boundary Representations on Rank One Para-Hermitian Spaces

G. R. Derzhavin Tambov State University, Tambov, Russia

Email: tria@tsu.tmb.ru

Copyright © 2013 Anatoli A. Artemov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received July 24, 2013; revised August 24, 2013; accepted September 2, 2013

**Keywords:** Para-Hermitian Symmetric Spaces; Overgroups; Canonical Representations; Boundary Representations; Poisson and Fourier Transforms

ABSTRACT

This work studies the canonical representations (Berezin representations) for para-Hermitian symmetric spaces of rank one. These spaces are exhausted up to the covering by spaces with,. For Hermitian symmetric spaces, canonical representations were introduced by Berezin and Vershik-Gelfand-Graev. They are unitary with respect to some invariant non-local inner product (the Berezin form). We consider canonical representations in a wider sense: we give up the condition of unitarity and let these representations act on spaces of distributions. For our spaces, the canonical representations turn out to be tensor products of representations of maximal degenerate series and contragredient representations. We decompose the canonical representations into irreducible constituents and decompose boundary representations.

1. Introduction

Canonical representations on Hermitian symmetric spaces were introduced by Vershik-Gelfand-Graev [1] (for the Lobachevsky plane) and Berezin [2]. They are unitary with respect to some invariant non-local inner product (the Berezin form). Molchanov’s idea is that it is natural to consider canonical representations in a wider sense: to give up the condition of unitarity and let these representations act on sufficiently extensive spaces, in particular, on distributions. Moreover, the notion of canonical representation (in this wide sense) can be extended to other classes of semisimple symmetric spaces, in particular, to para-Hermitian symmetric spaces, see [3]. Moreover, sometimes it is natural to consider several spaces together, possibly with different, embedded as open -orbits into a compact manifold, where acts, so that is the closure of these orbits.

Canonical representations can be constructed as follows. Let be a group containing (an overgroup), a series of representations of induced by characters of some parabolic subgroup associated with and acting on functions on. The canonical representations of are restrictions of to.

In this talk we carry out this program for para-Hermitian symmetric spaces of rank one. These spaces are exhausted up to the covering by spaces with,. For these spaces, an overgroup is the direct product and canonical representations turn out to be tensor products of representations of maximal degenerate series and contragredient representations. These tensor products are studied in [4], see also [5]. So we lean essentially on these papers [4,5]. We decompose canonical representations into irreducible constituents and decompose boundary representations. Notice that in our case the inverse of the Berezin transform can be easily written: precisely it is the Berezin transform.

Canonical and boundary representations for in the case (then is the hyperboloid of one sheet in) were studied in [6]. For the two-sheeted hyperboloid in, it was done in [7].

In this paper we present only the main results. The detailed theory of canonical and boundary representations, for example, on a sphere with an action of the generalized Lorentz group, can be seen in [8].

Let us introduce some notation and agreements.

By we denote. The sign denotes the congruence modulo 2.

For a character of the group we shall use the following notation

where, ,.

For a manifold, let denote the Schwartz space of compactly supported infinitely differentiable -valued functions on, with a usual topology, and the space of distributions on—of anti-linear continuous functionals on.

2. The Space and the Manifold

We consider the symmetric space where, ,.

The group acts on the space by

Let us write matrices in in block form according to the partition of. Let us take the matrix

The subgroup is just the stabilizer of this point, this subgroup consists of block diagonal matrices:

Thus, our space is the -orbit of, it consists of matrices of rank one and trace one.

Equip with the standard inner product, let. Let be the sphere. Let be the Euclidean measure on. The group acts on by.

Let be a cone in consisting of matrices of rank one. Therefore, the space is the section of by the hyperplane.

Introduce a norm in by

where the prime denotes matrix transposition.

Let be the section of by.

Define a map by

It is a two-fold covering. The measure defines a measure on by

The action of the group on gives the following action of on:

In particular, the subgroup, a maximal compact subgroup, acts on by translations:

Let us consider on the function

(1)

The action on has three orbits: namely, two open orbits (of dimension): and and one orbit of dimension: . The orbit is a Stiefel manifold, it is the boundary of. Denote. Each of orbits can be identified with the space. The map is constructed by means of generating lines of the cone.

3. Maximal Degenerate Series Representations

Recall [4] maximal degenerate series representations, , , of the group. Let be the subspace of consisting of functions of parity:. The representations act on by

4. Representations of Associated with

Recall [5] a series of representations of the group associated with the space.

Denote by the space of functions in of parity:

The representation acts on by

(2)

Let denote the following sesqui-linear form

(3)

Define an operator on by

It intertwines and. The operator is a meromorphic function of. Let us normalize this operator (multiplying it by a function of) such that the normalized operator is an entire non-vanishing function of.

There are three series of unitarizable irreducible representations. The continuous series consists of with, , the inner product is (3). The complementary series consists of with

, the inner product is

with a factor. The discrete series consists of the representations where,

, , which are factor representations of

on the quotient spaces. The representations with the same and different

are equivalent. It is convenient to take where for odd and for even. The inner product is induced by the form.

5. Canonical Representations

We define canonical representations, , , of the group as tensor products:

They can be realized on: let denote the subspace of consisting of functions of parity:, then the representation acts on by a formula similar to (2):

The inner product

(4)

is invariant with respect to the pair, i.e.

(5)

Consider an operator on defined by

It turns out that the composition is equal to the identity operator up to a factor. We can take such that

namely,

With the form (4) the operator interacts as follows:

(6)

This operator intertwines the representations and, i.e.

Let us call it the Berezin transform.

Let be the space of distributions on of parity. We extend and to by (5) and (6) respectively and retain their names and the notation.

Let us introduce the following Hermitian form on:

Let us call this form the Berezin form.

6. Boundary Representations

The canonical representation gives rise to two representations and associated with the boundary of the manifolds (boundary representations). The first one acts on distributions concentrated at, the second one acts on jets orthogonal to.

We can introduce “polar coordinates” on corresponding to the foliation of into -orbits. The - orbits are level surfaces of the function, see (1). For the -orbits are diffeomorphic to. In these coordinates the measure on is

where is the measure on.

Let be a function in. Consider it as a function of polar coordinates. Consider its Taylor series in powers of. Here are functions in. Denote by, , , the space of distributions in, having the form

where, is the Dirac delta function on the real line, its derivatives. Let .

Denote by Taylor coefficients of the function

. The distribution acts on a function as follows:

(7)

Denote by the restriction of to. This representation is written as a upper triangular matrix with the diagonal,.

Distributions in can be extended in a natural way to a space wider than. Namely, let

be the space of functions of class on and of parity and having the Taylor decomposition of order:

where. Then (7) keeps for with.

Let denote the column of Taylor coefficients. The representation acts on these columns:

It is written as a lower triangular matrix with the diagonal,.

The boundary representations and are in a duality.

7. Poisson and Fourier Transforms

Let us write operators and intertwining representations and. We call them Poisson and Fourier transforms associated with canonical representations.

The Poisson transform is a map given by

It intertwines with. Here we consider

as the restriction to of the representation acting on distributions in.

For a -finite function and the Poisson transform has the following decomposition in powers of:

where has polar coordinates. Here and

are certain operators acting on. The factors and give poles of the Poisson transform in depending on:

(8)

where and,. If a pole belongs only to one of series (8), then the pole is simple, and if a pole belongs to both series (8), then and the pole is of the second or first order.

Let the pole, , be simple. The residue of at this pole is an operator

. Denote the image of this operator by.

The Fourier transform is a map given by

It intertwines with.

The Fourier and Poisson transforms are conjugate to each other:

Poles in of the Fourier transform are situated at points

(9)

where and,. If a pole belongs only to one of the series (9), then the pole is simple, and if a pole belongs to both series (9), then and the pole is of the second or first order.

Let the pole, , be simple.

The residue of at this pole is a “boundary” operator,. The operator is defined in terms of Taylor coefficients: it is a linear combination of functions. Therefore, we may consider the following operator acting on columns

of functions: this operator to any column assigns the column

of functions in the same space—by the same formulas without. This operator is given by a lower triangular matrix.

8. Decomposition of Boundary Representations

The meromorphic structure of the Poisson and Fourier transforms is a basis for decompositions of boundary representations and.

Let the pole of the Poisson transform is simple, in particular, it happens when. Then the boundary representation is diagonalizable which means that decomposes into the direct sum of, and the restriction of

to is equivalent to (by means of).

If a pole is of the second order, then the decomposition of contains a finite number of Jordan blocks, this number depends on.

Let the pole of the Fourier transform is simple, in particular, when. Then the matrix is diagonalizable which means that

is a diagonal matrix. Its diagonal is,.

If a pole is of the second order, then the decomposition of contains a finite number of Jordan blocks, this number depends on.

9. Decomposition of Canonical Representations

Let us write decomposition of canonical representations. We restrict ourselves to a generic case: lies in strips

Case (A):.

**Theorem 1** Let. Then the canonical representation decomposes—as the quasiregular representation [5]—into irreducible unitary representations of continuous and discrete series with multiplicity one. Namely, let us assign to a function the family of its Fourier components, , , , and

,. This correspondence is equivariant. There is an inversion formula:

(10)

and a “Plancherel formula” for the Berezin form:

(11)

Here and stand for the Plancherel measure for, see [5], the factor is given by following formula:

Case (B):.

Here we continue decomposition (10) analytically in from to,. Some poles in of the integrand intersect the integrating line—the line . They are poles and of the Poisson transform with. They give additional summands to the right hand side. So after the continuation we obtain:

(12)

where the integral and the series mean the same as in (10) and

are some numbers.

Similarly, the continuation of (11) gives

(13)

where the integral and the series mean the same as in (11) and are some numbers.

The operators, , can be extended from

to the space and therefore to the sum

Then these operators turn out to be projection operators onto. Moreover, there are some “orthogonality relations” for them. Decomposition (13) can also be extended to the space. This decomposition is a “Pythagorean theorem” for decomposition (12).

**Theorem 2** Let,. Then the space has to be completed to. On this space the representation splits into the sum of two terms: the first one decomposes as does in Case (A), the second one decomposes into the sum of irreducible representations,. Namely, let us assign to any the family

where, ,. This correspondence is -equivariant. There is an inverse formula, see (12), and a “Plancherel formula”, see (13).

Case (C):.

Now we continue decomposition (10) analytically in from to. Here poles and, , , of the integrand (they are poles of the Fourier transform) give additional terms. We obtain

(14)

where the integral and the series mean the same as in (10) and

some numbers. The operators can be extended to the space,. Denote by the image of. It turns out that the operators are projection operators onto and for them there are some “orthogonality relations”.

Now we continue decomposition (11) from to. Poles of the integrand which intersect the integrating line and give additional terms (they are poles of both Fourier transforms) turn out fortunately to be of the first order, since at these points the function as a function of has zero of the first order. After the continuation we obtain:

(15)

where the integral and the series mean the same as in (11), some numbers. It is a “Pythagorean theorem” for decomposition (14).

**Theorem 3** Let,. Then the representation considered on the space splits into the sum of two terms. The first one acts on the subspace of functions such that their Taylor coefficients are equal to 0 for and decomposes as does in Case (A), the second one decomposes into the direct sum of irreducible representations, acting on the sum of the spaces. There is an inversion formula, see (14), and a “Plancherel formula” for the Berezin form, see (15).

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