Helgason-Schiman Formula for Semisimple Lie Groups of Arbitrary Rank

Copyright: © 2014 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. 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


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, { : ( ) }, G x G x x θ θ = ∈ = is a maximal compact subgroup of G: We say K and θ are associated whenever In this case, set t { g : } X X X θ =∈ =and p={X g: X= -X} θ ∈ Then t is the Lie algebra of K and we have the decompositions g = t p ⊕ 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 * a λ ∈ consider the subspace g λ of g defined as g { g : [ , ] = ( ) }. X H X H X H a λ λ = ∈ ∀ ∈ λ is called a root of the pair (g , a) whenever λ ≠ 0 and g λ ≠ {0}. We therefore have the root-space decomposition, g =m , g λ λ ∈∆ ⊕ ⊕ of g; where m is the centralizer of a in g and ∆ = ∆ (g,a) denotes the set of all roots of (g, a).m is θ -stable and, hence, reductive in g. If we set 1 m =m t, ∩ then 1 m=m t. ⊕ Put a lexicographic ordering on a* and denote the subset of ∆ consisting of positive roots of (g, a) as ∆ + . Define n = g λ λ ∈∆ + ∑ 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 g t a n = ⊕ ⊕ 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 : , : , : , k G K a G A n G N → → → so that every x G ∈ may be decomposed as x = K(x)a(x)n(x). Since ( ) exp a x A a ∈ = we find that a(x) = exp H(x) where : H G a → is the composition of the maps G A a → → .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. 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 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 λ Φ is a finite dimensional irreducible holomorphic representation of G  , simply connected group such that , G G ⊆  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 formula for (3, ), SL  which may be expressed in terms of gamma function. However, our interest here is to find the generalization of the expression for

( ( ))
H n e ρ − , 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 : m =  -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 SL (3,  ), 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  -rank m. Then we have x t x  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 choose a semisimple subalgebra g j of g with a Cartan decomposition j j 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 j α and 2 j α Thus, denoting by the set of restricted roots of the pair (g j , a j ), then We denote by j µ the linear functional on a j which equals one half the largest positive restricted root of j ∆ .We decompose a into a direct sum of one-dimensional m subspaces ,1 , We employ the groups (3, ) SL  and (2, ) Sp  to illustrate examples of the decomposition in the Theorem 2.1 above.
For the real rank 2 group (3, ) SL  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 = Sp  a maximal abelian subspace is 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 This corollary generalizes an equivalent expression for (m 1, ), SL +  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    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  It is known [6] that the H-function, relative to a minimal parabolic subgroup S=MAN; is given by the relation It is however known that the above construction techniques given for the c-function of   µ is a short root of ( , ).  Another approach to the construction of ( ) j g µ is as follows. Let  We therefore have the following.
where the product is over the positive roots relative to some ordering, a m is the multiplicity of the root a, and a a ν ∈ is the dual root corresponding to a, that is, a a a a ν λ λ = 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 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.