Last update: 20 February 2014
Let be the character group of the discrete group is the product of circles, and may be identified with the torus where as in (3.3.13) is the lattice of linear forms on such that for all Let be Haar measure on the compact group normalized so that the total mass of is
We shall assume in this section that Call this the standard case. The exceptional case, where for some will be examined separately in (5.2). Here we shall prove
Assume that for all Then the Plancherel measure (1.5.1) on the space of positive definite spherical functions on relative to is concentrated on the set and is given there by where is the order of the Weyl group
For we define We have to show that By linearity it is enough to take to be characteristic functions and we may assume that with respect to the total ordering denned in (3.3.9). Clearly we have
Now consider To compute this we shall express as an infinite ascending series, and as a descending polynomial, with at most one term in common. Term by term integration over the torus will then give the required result.
Since we have for each hence the product being over all the roots, positive and negative. This shows that is hence for all
Now where and is any element of Hence if is nonsingular we have from (4.1.2) and (5.1.4) or equivalently Now if then and if then (because Hence the right-hand side of (5.1.5) is denned for all and therefore (5.1.5) is valid for all
Moreover, under the assumption (5.1.1) we have for all and therefore these series being absolutely and uniformly convergent on the compact group Multiplying them all together we shall get from (5.1.5) say, with coefficients independent of and the summation being over all of the form with integer exponents It follows that with respect to the ordering (3.3.9); also it is easy to see that so that (replacing by we have for all the series on the right being uniformly convergent on
On the other hand, from (3.3.4') and (3.3.8'),
Furthermore, we have by orthogonality of characters on Hence, multiplying the infinite series (5.1.6') by the polynomial (5.1.7) and integrating term by term over we shall have, using (5.1.8), Hence and
In this section we shall consider the case excluded from the considerations of (5.1), namely where for some First of all, this implies that for some so that the root system is not reduced. Since it is irreducible, it must be of type and can therefore be described as follows. There exists an orthonormal basis of such that, if are the coordinate functions relative to this basis, the root system consists of and consists of and is of type We choose the set of simple roots as follows: The Weyl group is the group of all signed permutations of and has order
Let so that
Next consider the translation group Put Then and hence The are a basis of
If put Then is the product of the factors for and the factors for
Finally, let Then we have to compute the integral where and Proceeding as in (5.1), we have Since the Haar measure on is invariant under it follows that the integral (5.2.1) is equal to On the other hand, from (3.3.8'), so that (5.2.1) now takes the form In this expression, is a sum of terms Since for all in order to compute (5.2.2) we have to compute integrals of the form where and Since we may express (5.2.3) as a multiple contour integral, each contour being the unit circle in the complex plane. If we have and therefore Consequently, if the integral (5.2.3) takes the form Since the first of the exponents which is non-zero is positive.
We shall establish a reduction formula for For this purpose we have to introduce more notation. Let be a sequence of integers satisfying and let be the set of all such that Also define a function inductively as follows: is a function of the complex variables but we regard it as a function on It is easily checked from the product which defines that has as a factor in the numerator, so that the right-hand side of (5.2.4) is well defined.
Let where the measure on is given by is a translate of a subtorus of and this measure is the translation of the normalized Haar measure on that subtorus.)
Finally, let (resp. denote the product obtained from (resp. by deleting all factors involving and let be the subgroup of generated by Define and as above for subsequences of and put for the measure being here
The final form of the Plancherel measure will depend on how many of the numbers are less than So we define for as follows: and for
Since it follows that
Having set all this up, the key to the calculation of the integral in (5.2.3) is the following lemma:
Let be a subsequence of and put Then
If then also (because and there is nothing to prove. So assume that Consider the integral and integrate it round the unit circle with respect to the variable To do this we must pick out the factors in the product which involve An elementary calculation shows that they are, after some cancellations, Hence, since for the integrand considered as a function of has poles at the points and also at the origin, if
The points all lie outside the unit circle (for and and are both The point lies inside if and only if (If then the factor in the denominator of (5.2.7) cancels into the factor in the numerator, so there is never a pole on the unit circle.) The lemma now follows by computing the residues at the poles in the usual way.
where the sum is over all subsequences of and
By induction on The induction can start with when the result is easily verified.
If we have from (5.2.6)
If again from (5.2.6) we have the sum on the right being over all subsequences of The result now follows from the inductive hypothesis.
Now let be the mapping of onto the set of all z.s.f. on relative to The fibres of are the orbits of the action of on Clearly depends only on the number of elements in and not on the particular sequence. Let
From the definition of the z.s.f. belonging to are the for which of the numbers take the values (or their inverses), and the remaining have absolute value Since is invariant under the action of it follows that depends only on the number of elements in so we may define
The spherical functions belonging to are positive definite if
We leave the proof until later.
We have already defined a measure on each giving it total mass Now define a measure on as follows: for any continuous function on with compact support.
The Plancherel measure on the space of positive definite spherical functions is concentrated on the sets such that where is defined in (5.2.5). On is given by where is the order of the normalizer in of any of the sets such that
For define where is as above. Then we have to show that As before we take with and Let be the sequence Then exactly as in (5.1) Now it is easily verified that if and then unless maps to respectively with Hence Consequently and by (5.2.8) the inner sum is or according as or Hence, finally, since
We have still to prove (5.2.9). Consider first the set this consists of a single spherical function, say where is given by
Consider where and One sees immediately that some factor in the numerator vanishes, so that if Also a simple verification shows that and hence by (4.5.8). The lemma now follows from (4.1.2).
If then is square-integrable.
We have where by (3.2.15). Hence we have to show that the series converges if for
Since we have with Let be a sequence of integers such that and consider the such that will be the same for all these because they all have the same stabilizer in the Weyl group Hence the series (5.2.13) splits up into a sum of series, one for each sequence satisfying (5.2.14), and each of these series is a product of geometric series whose common ratios are products of the Hence all these series converge and therefore so does the series (5.2.13).
Remark. It is not difficult to show, using (5.2.11), that is a z.s.f. on relative to the Iwahori subgroup By (1.2.6) the z.s.f. on relative to correspond bijectively to the homomorphisms Let (resp. be the set of simple roots for (resp. (so that where Then is generated as by the characteristic functions of and the homomorphism is given by
In general, the z.s.f. on relative to correspond naturally to the linear characters of the Weyl group So they are always finite in number, equal to the order of made abelian.
We shall now sketch a proof of (5.2.9). Consider a zonal spherical function where We may assume that Let (resp. be the subspace of spanned by the first basis vectors (resp. by The restrictions to of the roots (resp. affine roots which vanish identically on form a subsystem of (resp. a subsystem of The Weyl group and affine Weyl group of can be regarded as subgroups of and respectively. The translation subgroup of is generated by Likewise, starting with we define etc.
Let be the subgroup of generated by the with and by and define similarly. Then is a subgroup of
Let Then is a square-integrable spherical function on by (5.2.12), and hence by (1.4.8) is positive definite. Again, since is a character of it follows from (3.3.1) (applied to that is positive definite.
Now let be the (parabolic) subgroup of generated by and and let be the modulus function on is the semi-direct product of and the subgroup generated by the root subgroups corresponding to negative roots which do not belong to either or (i.e. with and The spherical function on extends to a spherical function on say By (1.4.7), the spherical function induced by on is positive definite, and by transitivity of induction (1.3.3) it is equal to This completes the proof of (5.2.9) and hence of (5.2.10).
Let be a local field, that is to say a non-discrete locally compact field. Then is either or or a field, and the additive group of is self-dual. Associated canonically with there is a meromorphic function of a complex variable sometimes called the gamma-function of We recall its definition briefly (see Tate's thesis [Tat1967], where it is denoted by
If is any well-behaved function on let be its Fourier transform with respect to the additive group structure. Since is self-dual, is a function on (The Haar measure is normalized so that If define where is the normalized absolute value on (i.e. where is additive Haar measure) and is a Haar measure on the multiplicative group Then has a functional equation with independent of This equation defines for and is then extended by analytic continuation to a meromorphic function in the whole complex plane.
We can compute from (5.3.1) by choosing the function intelligently.
take then and one finds that so that where is Euler's beta-function.
take the ordinary absolute value on in fact then again and we have so that
Taking to be the characteristic function of the ring of integers of one finds that where is the discriminant of and is the number of elements in the residue field. Hence in this case we have
Now let be the universal Chevalley group as in (2.5), where is any local field. If or let be any maximal compact subgroup of (they are all conjugate). If is let be the maximal compact subgroup of defined as hitherto in terms of the affine root structure on given in (2.5).
If or the z.s.f. on relative to are parametrized by the mappings and the Plancherel measure for the positive definite spherical functions is supported on the space of pure imaginary which is naturally a real vector space of dimension If is a Euclidean measure on this space, then (Harish-Chandra [Har1966]) the Plancherel measure is of the form where is a constant (depending on the choices of and of Haar measure on and and Hence, from (5.3.2), we have in both cases, apart from a constant factor in the case
On the other hand, if is then as we have seen in (5.1) the support of the Plancherel measure is the character group of To bring out the analogy with the real and complex cases we shall replace the multiplicative parametrization of the spherical functions, which we have used hitherto, by an additive parametrization. If we define by the rule where as before is the number of elements in the residue field of Then from (5.1.2) the Plancherel measure is of the form where is Euclidean measure on the space of pure imaginary is a constant, and (because in the present situation all the are equal to Hence, from we have apart from a constant factor depending only on
So, to sum up:
If is any local field and is a universal Chevalley group, then the Plancherel measure on the space of positive definite spherical functions on relative to the maximal compact subgroup is of the form where is a constant.
Also for non-split groups there is a strong resemblance between the Plancherel measure in the real case and in the case.
Let be a simply-connected simple algebraic group denned over a local field and let (We can exclude the case from here on, because in that case is necessarily split.)
Let be an Iwasawa decomposition, and the Lie algebras of and respectively, the set of 'restricted roots' of relative to and for each let be the multiplicity of Then the z.s.f. on relative to are parametrized by the elements of and the Plancheral measure is supported on the subspace consisting of the pure imaginary Write for the z.s.f. corresponding to Then [Har1966] the Plancherel measure is given by where is a constant and (see [GKa1962]) is a product of beta-functions, namely Let which is the local zeta-function for the function In this notation we have where is independent of
If the number of elements in the residue field of is then each index is a power of We shall therefore write and call the formal multiplicity of the root Next, to bring our notation into line with that used above in the case we shall replace the parameter for the spherical functions by defined by for all is not uniquely determined by but it is unique modulo the lattice where as in (2.5) is the lattice of all such that for all In particular, if then is pure imaginary. Writing in place of we have from (4.1) Now let which is the local zeta-function for the characteristic function of the ring of integers of Then in this notation we have
For real or the Plancherel measure is where is a constant and the functions being defined by (5.3.5) and (5.3.7).
Remark. There is one important difference between the real and cases: in the case the formal multiplicity can be negative if is also a root. This is precisely the 'exceptional case' dealt with in (5.2).
This is a typed version of the book Spherical Functions on a Group of Type by I. G. Macdonald, Magdalen College, University of Oxford. This book is copyright the University of Madras, Madras 5, India and was first published November 1971.
Published by the Ramanujan Institute, University of Madras, and printed at the Baptist Mission Press, 41A Acharyya Jagadish Bose Road, Calcutta 17, India.