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ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Helgason-Schiman Formula for Semisimple Lie Groups of Arbitrary Rank

Bassey UN1* and Oyadare OO2

1Department of Mathematics, University of Ibadan, Ibadan, Nigeria

2Department of Mathematics, Obafemi Awolowo University, Ile-Ife, Nigeria

*Corresponding Author:
Bassey UN
Department of Mathematics
University of Ibadan, Ibadan, Nigeria
E-mail: [email protected]

Received date: July 21, 2014; Accepted date: November 29, 2014; Published date: December 05, 2014

Citation: Bassey UN, Oyadare OO (2015) Helgason-Schiman Formula for Semisimple Lie Groups of Arbitrary Rank. J Generalized Lie Theory Appl 9:216. doi:10.4172/1736-4337.1000216

Copyright: © 2015 Bassey UN, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

This paper extends the Helgason-Schiffman formula for the H-function on a semisimple Lie group of real rank one to cover a semisimple Lie group G of arbitrary real rank. A set of analytic Equation -valued cocycles are deduced for certain real rank one subgroups of G. This allows a formula for the c-function on G to be worked out as an integral of a product of their resolutions on the summands in a direct-sum decomposition of the maximal abelian subspace of the Lie algebra g of G. Results about the principal series of representations of the real rank one subgroups are also obtained, among other things.

Keywords

Helgason-Schiffman formula; Spherical functions; H− function; Semi simple Lie group

Introduction

Let G be a semisimple Lie group with finite center and Lie algebra, g. Define a Cartan involution on G as an involutive automorphismθ of G whose set of fixed points, Equation is a maximal compact subgroup of G: We say K and θ are associated whenever Equation. In this case, set Equation and Then t is the LieEquation algebra of K and we have the decompositions Equation and G=K exp P commonly called the Cartan decompositions of g and G; respectively, associated to θ . Now choose a maximal abelian subspace, a, of p and let a* be its dual vector space. For any Equation consider the subspace gλ of g defined as Equation λ is called a root of the pair (g , a) whenever λ ≠ 0 and g λ ≠ {0}. We therefore have the root-space decomposition, Equation of g; where m is the centralizer of a in g and Δ = Δ (g,a) denotes the set of all roots of (g, a).m is θ -stable and, hence, reductive in g. If we set Equation then Equation

Put a lexicographic ordering on a* and denote the subset of Δ consisting of positive roots of (g, a) as Δ+. Define Equation and N=exp n. Then n is nilpotent subalgebra of g, N is the closed analytic subgroup of G defined by n, and exp (n → N) is an analytic diffeomorphism. We now have the Iwasawa decompositions Equation and G=KAN of g and G, respectively, with the abelian subgroup, A, defined as A = exp a: This decomposition of G gives rise to the projection maps Equation so that every x∈G may be decomposed as x = K(x)a(x)n(x). Since Equation we find that a(x) = exp H(x) where Equationis the composition of the maps Equation. The maps K, a, n, and H are analytic maps on G and are known to contribute to many discussions of the harmonic analysis of G. The Equation rank of G; denoted as m; is defined as the dimension of a. Since Equation it is therefore not unexpected that the analytic map Equation should have a relationship with the Equation-rank of G: We refer to Equation as the H-function of G.

For any G; with Equation-rank one and Lie algebra g, there is an explicit expression for the H-function which was independently established by Helgason and Schiffman [1]. Indeed the expression is completely defined on θ (N) and we have it as

Equation

Where λ* is half of the only positive real root of (g, a), Equation and B is the Killing form on g, This may also be written as Equation.

An analogous expression has been sought for other examples of G; starting in 1960 with the work of Bhanu-Murthy, whose study entails a group-by-group consideration, while the case of an arbitrary G is not known. A common feature of the computation of the H-function for higher-than-one Equation-rank groups, which is used to compute the H-function on a group-by-group basis, is its relationship with the finite-dimensional representations of G. The above mentioned relationship is as follows: the H-function of G relative to a minimal parabolic subgroup satisfies the relation Equation where Equation is a finite dimensional irreducible holomorphic representation of Equation, simply connected group such that Equation, with highest weight λ and u is any unit vector in the sum of the weight spaces for weights that restricts to λ on a [2].

We give the computation in the case of Equation Let us write the subgroup Equation of G as Equation Then, from the above relation, it may be shown that Equation for every Equation The c-function in this case is then given as Equation which, by an ingenious substitution becomes the product

Equation

of three one-dimensional integrals. This is the Gindikin-Karpelevic formula for Equation, which may be expressed in terms of gamma function. However, our interest here is to find the generalization of the expression for Equation , that would work for every semisimple group G [3], In order to generalize the methods in the last paragraph to every semisimple Lie group G we seek the earlier mentioned relationship of H in terms of Equation-rank (G): In this paper, we give an expression, in 2 for H which makes the harmonic analysis on G Equation-rank dependent. Indeed this expression leads to a generalization of the Equation-rank one Helgason-Schiffman formula [1] to arbitrary rank as contained in 3. This general formula reduces to the H-function for Equation without using the method of the highest weight theorem for finite dimensional representations of G.

The Decomposition of the H-function

We start with Theorem 2.1 below which plays a fundamental role in what follows.

Theorem Let G be of Equation-rank m. Then we have

Equation

Where Equation In particular, each Equation a logarithm function and is analytic on G.

Proof:

The proof is essentially the same as in ([3], Theorem 2.1) and so is omitted

Before going on, we give the following notations which are required for what follows below. We know that the Equation-rank (G)=m=dim (a). For each j∈{1,...,m}choose a semisimple subalgebra gj of g with a Cartan decomposition Equation such that Equation and Equation Fix a maximal abelian proper subspace aj of pj (assume throughout that aj is one-dimensional). Fix also a compatible order on non-zero restricted roots; here there are at most two roots which are positive with respect to this order, which we denote by αj and 2αj Thus, denoting by Equation the set of restricted roots of the pair (gj, aj), then Equation with a corresponding positive system Equation We denote by μj the linear functional on aj which equals one half the largest positive restricted root of Δj. We decompose a into a direct sum of one-dimensional m subspaces Equation that is, Equation with Equation.

We employ the groups Equation and Equation to illustrate examples of the decomposition in the Theorem 2.1 above.

For the real rank 2 group Equation a maximal abelian subspace, a, of p is

Equation

We may then choose

Equation

as a1 and a2, respectively, each of which is one-dimensional. In the case of G =

Equation a maximal abelian subspace is Equation

Thus Equation and Equation may be chosen as a1 and a2; respectively.

It is clear that the case m=1 reduces to the situation of Helgason- Schiffmann. Next we discuss some of the properties of each of the maps Equation To this end let Equation

Corollary

We have Equation

This corollary generalizes an equivalent expression for Equation established in [4] to any semisimple Lie group with finite center and of any real rank. One of the major applications of the H-function, and now of Theorem 2.1, is its contribution to the compact picture of the induced representations on semisimple Lie groups. This contribution relies on the cocycle nature of H. In anticipation of a similar use to be made of the maps Equation we establish the following proposition.

Proposition

Let there be given Equation the map Equation induces an analytic Equation-valued cocycle on G.

Proof

Since Equation, the subgroup K may be regarded as a transitive homogeneous space for G acting from the left. We denote this action as Equation In this context the function Equation induces an A-valued map Equation given simply as Equation and which satisfies

Equation

Equation

Equation

Now going over, from the map Equation to a (via the H-function) and then to Equation(via each of tm,j), we may define the map Equation and denote it by Equation

Using Theorem 2:1 above, properties (i), (ii) and (iii) of a (x : k) become

Equation

Equationand

Equation

The real rank 1 case of the last proposition is contained in Proposition 3.1 of [5]. It is known that the H-function vanishes on the maximal compact subgroup K. The implication is that each of the coefficient maps, Equation also vanish on K.

The H-function is known to be completely defined on Equation where Equation gα and θ is the Cartan involution of G associated to K. The decomposition of a in Theorem 2.1 means we consider the complete understanding of each of tm,j on the direct sum of eigenspaces corresponding to the positive restricted roots in Equation .Hence a procedure for deriving an explicit expression for each of tm,j is to be accomplished on Equation where Equation This, among other things, will be achieved in 3 below.

The c-function and zonal Spherical Functions

We now study the contributions of the decomposition of the H-function in Theorem 2.1 to some aspects of harmonic analysis on G. These include the structure of spherical and c-functions and representations on G. Here we consider the c-function which appears as the coefficient-function of the eigenspace expansion of spherical functions.

Let ρ be the half-sum of the positive roots of the pair (g, a) with multiplicity. The c-function is given by the integral Equation It is, however, customary to use the understanding of the function Equation in order to study the c function. Note that Equation We consider first the example of Equation

Example

Equation :Take m=2 for a start and introduce real parameters for members of Equation to have

Equation

With Equation It is known [6] that the H-function, relative to a minimal parabolic subgroup S=MAN; is given by the relation Equation where Equation is a finitedimensional irreducible holomorphic representations of Equation, a simply connected group such that Equation with highest weight Equation being any unit vector in the sum of the weight spaces for weights that restrict to λ on a

The roots of the pair (g, a) are Equation Where

Equation

The corresponding positive system of restricted roots is Equation on the requirements that Equation [1]. It may be shown that Equation and Equation Now since Equation and Equation then Equation [6] Thus if we write complex numbers to describe the behaviour of λ on a, then

Equation

and the c-function on Equation is given as

Equation

Equation We then have an expression for the c-function on Equation as the integral of complex indices of two polynomials.

The above situation may be generalised to the c-function on Equation To this end we take Equation to be a lower triangular matrix, Equation with 1’s on the diagonal. For each l with Equation a generalisation of the above computations is obtained by forming the sum of the squares of Equation minors of size l-by-l obtained from the first l columns of Equation The result is raised to a power depending on l, and the analogue of the c-function above is the integral over Equation of the product of m expressions raised to their respective powers.

It is however known that the above construction techniques given for the c-function of Equation do not extend to other real semisimple Lie groups with finite center. For this reason the earlier expression given as Equation is always resorted to when ever the c-function of specific groups are needed, with the attendant restriction that there exists a simply connected group Equation such that Equation and with a_nite-dimensional irreducible holomorphic representation, Equation. We give here an approach for the computation of the above j-function (hence the c-function) for any real rank m connected semisimple Lie group with finite center, which will establish the exact contribution of m as earlier seen in the case of Equation

Theorem

Let Equation and Equation where every Equation is of the form Equation Introduce parameters that describe members of each Equation such that Equation Then, for every Equation

Equation

Where αj is chosen appropriately and Equation is a quadratic form.

Proof

If Equation then a choice may be made to have Equation Hence if Equation then Equation while if Equation, then Equation where is as defined under Theorem 2.1. Therefore

Equation

Hence we restrict our computations to Equation

If we recall the definition of μj above, then

Equation

each of which is not a root of the pair Equation Hence μj is a short root of Equation and we have the root-space decomposition Equation where Equation is the centraliser of aj in Equation By construction Equation each Equation is stable under the restriction of the Cartan involution of g and is therefore simple.

Denote by Equation the analytic subgroup of G corresponding to Equation , while the K and A for Equation may be taken to be the connected groups Equation and Equation with Equation as the corresponding M group. Thus the symmetric space Equation has rank one, where each Equation is a real rank one semisimple Lie group with finite center. Hence we may define a quadratic form, Equation, as Equation where Equation is such that Equation and Equation is the restriction of the Killing form to Equation

It therefore follows that Equation is the Equation for the real rank one semisimple Lie group Equation (with μj given in terms of αj as above). Hence

Equation as required.

Corollary

Let Equation Then the function Equation on Equation are polynomials in the Lie algebra coordinates on Equation

Computation of Equation : the case of Equation

We start by restricting the members of Equation to a1 and a2 to have Equation

If we now require, in addition to the earlier requirements of Example 3.1, that Equation and Equation , we may define Equation and Equation as Equation and Equation respectively. These are respectively the restrictions Equation and Equation with Equation and Equation.

If we then define Equation and Equation (since m=0), then Equation and Equation with Equation. The restriction of members of Δ+ to aj shows that Equation and we may conclude that each Equation is isomorphic with a real rank one (semi-) simple Lie algebra with Equation so that

Equation

For Equation This is as computed earlier in Example 3.1.

Another approach to the construction of Equation is as follows. Let Equation be the centraliser of aj in g. It may be shown that Equation is stable under the restriction of the Cartan involution and that the analytic subgroup, Equation of G corresponding to Equation , is the centraliser of aj in G. We set Equation and Equation

Let us now choose α to be a short root of the pair (g,a), i.e., Equation such that Equation. We may choose αj by restrictions as in Computation 3:4 and compute the algebra

Equation

from which we now define Equation

We are now in a position to employ Proposition 2:3 to construct the compact picture of the induced representation on Equation. Fix Equation. Let Equation, Equation and define Equation by the requirement Equation . Equation is a quasi-character of Aj and is unitary iff Equation We therefore have the following.

Proposition

The map Equation for Equation is an analytic Equation-valued cocycle.

Proof

By Proposition 2.3.

Setting Equation we define Equation as Equation Equation with Equation where Equation, σj a finite-dimensional unitary representation on Equation Details of the construction of Equation may be found in [5].

Proposition

Equation is an irreducible unitary representation of Equation on Equation for Equation and irreducible σj It reduces to the left-regular representation on Equation

Proof

The cocycle relations proved in Proposition 2.3 for tm,j give Equation and Equation while the continuity of the map Equation of Equation into Equation the irreducibility and unitarity of Equation are established exactly as in the case of the principal series on G.

If Equation then from the same cocycle properties of tm,j ,we have that Equation Thus Equation

It is known that each of the real rank one semisimple Lie groups, Equation admits the induced representations, Equation which may be restricted to Equation to get all the principal series of representations of Equation. In this light a consequence of the above Proposition is the following.

Corollary

Let σj be a finite-dimensional irreducible unitary representation of Equation and Equation The representations Equationexhausts the unitary principal series of Equation.

We are now encouraged to define the spherical functions Equation corresponding to the class 1 members of Equation With respect to the spherical function, Equation of G, we refer to Equationas the resolution of the spherical function Equation.

The Plancherd measure μ is supported on the set of real-valued λ and is of the form

Equation

Where dλ is the Lebesgue measure on the dual of the real vector space a and the function c is given explicitly as a product of betafunctions by the following formula,

Equation (3.1)

where the product is over the positive roots relative to some ordering, ma is the multiplicity of the root a, and Equation is the dual root corresponding to a, that is,

Equation

The explicit calculation (3.1) of c(λ ) is due to Bhanu - Murthy [7] for the split groups and to Gindikin and Karpelevic in the general case [1].

We define a representation π on a (locally convex) space V to be of class-1 whenever the subspace Equation of all K-invariant vectors in V, is of dimension 1. It is known [8] that class-1 representations are associated with spherical functions on G (which are the matrix coefficients of these representations), and that, for irreducible σ, the (unitary) principal series, Equation is of class-1 if, and only if, σ is the trivial representation on M. Let us therefore denote Equation and set the matrix coefficient of πλ defined by the function 1, as φλ given as

Equation

Where Equation and (.,.) is an inner product on L2(K) . The Function φλ is spherical and, has the integral representation Equation as given above.

The result of Theorem 3.2 leads to the following product formula for the spherical functions, φλ, in a direction different from the Gindinkin-Karpelevic product formula for spherical functions.

Theorem

Every spherical function, Equation, on G is of the form

Equation

where each Equation is the resolution of Equation on each summand in the direct sum Equation

Proof

We first note that

Equation

which is substituted into Equation gives Equation

The expression Equation is the resolution of Equation on each aj and is denoted

As Equation

The product formula above explains that spherical functions, Equation on any real rank m group G, is the product of its resolutions, Equation on each of the 1-dimensional subspaces, aj of a. It implies that spherical functions on real rank m groups can be studied through its resolutions, on some 1-dimensional subspace.

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