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**Bassey UN ^{1*} and Oyadare OO^{2}**

^{1}Department of Mathematics, University of Ibadan, Ibadan, Nigeria

^{2}Department 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.

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

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 -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.

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

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, is a maximal compact
subgroup of G: We say K and θ are associated whenever . In this
case, set and Then t is the Lie
algebra of K and we have the decompositions 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 consider the subspace
g_{λ} of g defined as λ is called a
root of the pair (g , a) whenever λ ≠ 0 and g λ ≠ {0}. We therefore have
the root-space decomposition, 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 then

Put a lexicographic ordering on a* and denote the subset of Δ
consisting of positive roots of (g, a) as Δ^{+}. Define 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* 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 * *so that every
x∈G may be decomposed as x = K(x)a(x)n(x). Since we find that a(x) = exp H(x) where is the composition of the
maps . 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 rank of G; denoted as m; is defined as the dimension of a.
Since it is therefore not unexpected that the analytic map should have a relationship with the -rank of G: We refer
to as the H-function of G.

For any G; with -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

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

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 -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 where is a finite dimensional *irreducible holomorphic* representation of , simply connected group such that , 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 Let us write the subgroup of G as Then, from the above relation, it may be shown that for every The c-function in this case is then given as which, by an ingenious substitution becomes the product

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

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

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

Where In particular, each 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 -rank (G)=m=dim (a).
For each j∈{1,...,m}choose a semisimple subalgebra g_{j} of g with a
Cartan decomposition such that and 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 α_{j} and 2α_{j} Thus,
denoting by the set of restricted roots of the pair (g_{j}, a_{j}),
then with a corresponding positive system We denote by μ_{j} the linear functional on a_{j} which
equals one half the largest positive restricted root of Δ_{j}. We decompose
a into a direct sum of one-dimensional m subspaces that
is, with .

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

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

We may then choose

as a_{1} and a_{2}, respectively, each of which is one-dimensional. In
the case of G =

a maximal abelian subspace is

Thus and 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 To this end let

**Corollary**

We have

This corollary generalizes an equivalent expression for 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 we establish the following proposition.

**Proposition**

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

**Proof**

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

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

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

and

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, also vanish on K.

The H-function is known to be completely defined on where g_{α} 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 t_{m,j} on the direct sum of
eigenspaces corresponding to the positive restricted roots in .Hence
a procedure for deriving an explicit expression for each of t_{m,j} is to be
accomplished on where This, among other things, will be achieved in 3 below.

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 It is, however, customary to use the understanding of the function in order to study the c function. Note that We consider first the example of

**Example**

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

With It is known [6] that the H-function, relative to a minimal parabolic subgroup S=MAN; is given by the relation where is a finitedimensional irreducible holomorphic representations of , a simply connected group such that with highest weight 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 Where

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

and the c-function on is given as

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

The above situation may be generalised to the c-function on To this end we take to be a lower triangular matrix, with 1’s on the diagonal. For each l with a generalisation of the above computations is obtained by forming the sum of the squares of minors of size l-by-l obtained from the first l columns of The result is raised to a power depending on l, and the analogue of the c-function above is the integral over 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 do not extend to other real semisimple Lie groups with finite center. For this reason the earlier expression given as 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 such that and with a_nite-dimensional irreducible holomorphic representation, . 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

**Theorem**

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

Where α_{j} is chosen appropriately and is a quadratic form.

**Proof**

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

Hence we restrict our computations to

If we recall the definition of μ_{j} above, then

each of which is not a root of the pair Hence μ_{j} is a
short root of and we have the root-space decomposition where is the centraliser of a_{j} in By construction each is stable
under the restriction of the Cartan involution of g and is therefore
simple.

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

It therefore follows that is the for the real
rank one semisimple Lie group (with μ_{j} given in terms of α_{j} as above). Hence

as required.

**Corollary**

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

**Computation of : the case of **

We start by restricting the members of to a_{1} and a_{2} to have

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

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

For This is as computed earlier in Example 3.1.

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

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

from which we now define

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

**Proposition**

The map for is an analytic -valued cocycle.

**Proof**

By Proposition 2.3.

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

**Proposition**

is an irreducible unitary representation of on for and irreducible σ_{j} It reduces to the left-regular
representation on

**Proof**

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

If then from the same cocycle properties of t_{m,j} ,we have
that Thus

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

**Corollary**

Let σ_{j} be a finite-dimensional irreducible unitary representation of and The representations exhausts the unitary
principal series of .

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

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

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,

(3.1)

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

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 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, is of class-1 if, and
only if, σ is the trivial representation on M. Let us therefore denote and set the matrix coefficient of π_{λ} defined by the function 1, as φ_{λ} given as

Where and (.,.) is an inner product on L^{2}(K) . The Function φ_{λ} is spherical and, has the integral representation as given above.

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

**Theorem**

Every spherical function, , on G is of the form

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

**Proof**

We first note that

which is substituted into gives

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

As

The product formula above explains that spherical functions, on any real rank m group G, is the product of its resolutions, 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.

- Warner G (1972) Harmonic analysis on semisimple Lie groups II. GrundlehrendermathematischenWissenschaften 189.
- Knapp AW (1986) Representation theory of semisimple groups, an overview based onexamples. Princeton University Press, Princeton.
- Bassey UN, Oyadare OO (2013) A Theorem on the Iwasawa Projection and Applications to Some Representations of Reductive Groups. Universal J Math Math Sci 3: 157-164.
- Varadarajan VS (1999) An introduction to harmonic analysis on semisimple Lie groups. Cambridge University Press .
- Bassey UN, Oyadare OO (2009) Remarks on the principal series of representations:the case of SL(2,). JP J Geometry Topology 9: 249-262.
- Knapp AW (2002) Lie groups beyond an introduction. Progress in Mathematics 140.
- Bhanu MTS (1960) The asymptotic behaviour of zonal spherical functions on the Siegel upper-half plane. DocladyAkadNauk SSSR 135: 1025-1029.
- Gangolli R, Varadarajan VS (1988) Harmonic analysis of spherical functions on real reductive groups. Ergebnisse der Mathematik und ihrer Grenzgebiete 101.

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