DIMENSIONS OF SPACES OF

GENERALIZED SPHERICAL FUNCTIONS

Jing-Song Huang, Toshio Oshima and Nolan Wallach

1. Introduction Let G be a connected semi-simple Lie group with finite center and K be a maximal compact subgroup of G. We denote by 𝔤 the complexified Lie algebra of G which we look upon as left invariant vector fields on G. Let Z(𝔤) denote the center of the enveloping algebra of 𝔤, which we look upon as right and left invariant differential operators on G. If (τ, V) is a finite dimensional representation of K × K with the first factor acting on the left and the second acting on the right (we will write τ(k_{1}, k_{2})v = τ_{1}(k)vτ_{2}(k)) then we denote by C^{∞}(G, τ) the space of all C^{∞} functions from G to V such that f(k_{1}gk_{2}) = τ_{1}(k_{1})f(g)τ_{2}(k_{2}). If D is a subalgebra of Z(𝔤) and χ is a homomorphism from D to ℂ then we consider the space

E_{χ}(G, τ, D) = {f ∈ C^{∞}(G, τ) | Df = χ(D)f, D ∈ D}.

If D is understood we will use the notation E_{χ}(G, τ). If D = Z(𝔤) then this space is usually called the space of τ-spherical functions with infinitesimal character χ. Let (P, A) be a minimal K-standard parabolic pair for G. Then P = MAN with N the unipotent radical of P and M the centralizer of A in K.

Before we get to the more technical aspects of the results of this paper, we will first describe our main results in the case when G is split over ℝ and D = Z(𝔤). We first note that (τ, V) splits into a direct sum ⊕_{j}(τ^{j}, V^{j}) as a representation of K × K, with each (τ^{j}, V^{j}) irreducible. Clearly, E_{χ}(G, τ) = ⊕_{j}E_{χ}(G, τ_{j}). Thus we may assume that (τ, V) is irreducible. One of our main results in this case is

dim E_{χ}(G, τ) = dim{v ∈ V |τ_{1}(m)v = vτ_{2}(m)}.

The point here is that the dimension is independent of χ. If χ is “generic” then this result is just the (relatively well known) assertion that every element of E_{χ}(G, τ) is an Eisenstein integral for P. For general χ our technique implies that one must include derivatives at zeros of Eisenstein integrals.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 22E47.

The first author is partially supported by NSF Grant DMS 9306138 and RGC Competitive Earmarked Research Grant HKUST 588/94P

The third author is partially supported by NSF Grant

We will now describe the general context of the paper. Let 𝔥 be a Cartan subalgebra of 𝔤 containing 𝔞 (the complexified Lie algebra of A). Let W (𝔤, 𝔥) denote the (complex) Weyl group of 𝔤 on 𝔥 and let W (A) denote the (little) Weyl group of G on A (and of course 𝔞). Let γ denote the Harish-Chandra isomorphism of Z(𝔤) onto S(𝔥)^{W (𝔤,𝔥)}. We look upon S(𝔥) as the polynomials on 𝔥^{∗}. Using the Killing form, B, of 𝔤, we identify 𝔞^{∗} with the subspace of 𝔥^{∗} of all λ such that λ(h) = 0 if B(h, 𝔞) = (0). We first assume that S(𝔥)^{W (𝔤,𝔥)} |_{𝔞^{∗}} = S(𝔞)^{W (A)}. This is clearly the case for G split (𝔞 = 𝔥). Helgason has shown that this condition is satisfied for all semisimple G unless G contains simple factors of type E_{6(−26)}, E_{6(−14)}, E_{7(−25)} and E_{8(−24)}. If G does not contain any simple factors of these four exceptional types, then S(𝔞)^{W (A)} is a polynomial ring in generators u_{1}, ..., u_{l} (l = dim 𝔞). We can choose D_{1}, ..., D_{l} ∈ Z(𝔤) such that γ(D_{i})|_{𝔞} = u_{i}. If G does contain a simple factor of one of the four exceptional groups, we prove in Section 2 that we can choose D_{1}, ..., D_{l} ∈ Z(𝔤) such that γ(D_{i})|_{𝔞} = u_{i} and S(𝔞)^{W (A)} is a free module for ℂ[u_{1}, ⋯, u_{l}]. In all cases, let D denote the (polynomial algebra) generated by D_{1}, ..., D_{l}. One of our main result is the same dimension formula for E_{χ}(G, τ). The characters χ of D are simply evaluations of a polynomial of l-variables at some points. The other main result of this paper is proof of the existence of a global basis f_{χ,τ}^{1} (g), ⋯, f_{χ,τ}^{k} (g) for E_{χ}(G, τ), which is holomorphic in χ.

If D is as above, denote by E_{χ}(G) the space of all left and right K−finite smooth functions on G such that Df = χ(D)f for D ∈ D. Then E_{χ}(G) is naturally an admissible (𝔤 × 𝔤, K × K)-module. As an application of our main results we calculate the distribution character of E_{χ}(G) for all split groups, G, over ℝ.

The paper is organized as follows. In Section 2 we determine for all cases the differential algebra D as a subalgebra of Z(𝔤). In Section 3 we give the definition of the generalized spherical functions. In Section 4 we prove a formula for the dimension of the generalized spherical functions with fixed K-types on both sides. In the proof we use the a well-known theorem of Kostant and Rallis [KS]. In Section 5 we show that there exists a global basis for E_{χ}(G, τ). The main results are used in Section 6 to prove a character formula for E_{χ}(G) for a split group G.

Part of this work was done while the third named author was enjoying the hospitality of the University of Tokyo (Fall of 1990). The first named author would like to thank David Vogan for helpful discussions at the early stage of this work. Frederic Bien has pointed out to us that he was aware of similar results in the larger context of semi-simple symmetric pairs (as has the second named author).

2. Restrictions of invariants. Let G be a connected simple Lie group over ℝ with finite center and let K be a maximal compact subgroup of G. Let 𝔤 denote the complexified Lie algebras of G. Denote by θ the Cartan involution of 𝔤 corresponding to K. Fix 𝔞_{o} a maximal abelian subalgebra of Lie(G) contained in {X ∈ 𝔤 | θX = −X}. Let 𝔞 be the complexification of 𝔞_{o} in 𝔤. Set

W (𝔞) = {s ∈ GL(𝔞) | there exits k ∈ K, Ad(k)|_{𝔞} = s}

the “little Weyl group”. Let 𝔥 be a Cartan subalgebra of 𝔤 containing 𝔞. Let W denote the Weyl group of (𝔤, 𝔥). We denote by O(𝔥), O(𝔞) respectively the complex valued polynomial functions on 𝔥 and 𝔞. Let O(𝔥)^{W} (resp. O(𝔞)^{W (𝔞)}) denote the W - (resp. W (𝔞)-) invariants in O(𝔥) (resp. O(𝔞)). Then it is a theorem of Chevalley that O(𝔥)^{W} (resp. O(𝔞)^{W (𝔞)}) is a polynomial ring in l = dim 𝔥 (resp. r = dim 𝔞) homogeneous generators u_{1}, ..., u_{l} (resp. v_{1}, ..., v_{r}). Let Res_{𝔥/𝔞} = Res denote the restriction homomorphism of O(𝔥) onto O(𝔞). Then it is well known that Res(O(𝔥)^{W}) ⊂ O(𝔞)^{W (𝔞)}.

In [H] Helgason showed that if 𝔤 is not of type E then Res(O(𝔥)^{W}) = O(𝔞)^{W (𝔞)} and for the E series Res is surjective except for the following 4 cases: in E_{6} when dim 𝔞 = 2 (two cases), in E_{7} when dim 𝔞 = 3 and in E_{8} when dim 𝔞 = 4. We will call these examples the exceptional examples. Our main result in this section is

Proposition 2.1 For the exceptional examples of type E_{6} there exist homogeneous elements w_{1}, w_{2} of O(𝔥)^{W} such that Res(w_{1}) and Res(w_{2}) are algebraically independent and if J = ℂ[Res(w_{1}), Res(w_{2})] then O(𝔞)^{W (𝔞)} is a free J-module (under multiplication) on two generators. For the exceptional example of type E_{7} then there exist homogeneous elements w_{1}, w_{2}, w_{3} ∈ O(𝔥)^{W} such that Res{w_{1}, w_{2}, w_{3}} is an algebraically independent subset (of 3 elements) of O(𝔞)^{W (𝔞)} and if J is the (polynomial) algebra generated by Res{w_{1}, w_{2}, w_{3}} then O(𝔞)^{W (𝔞)} is a free J-module on two generators. For the exceptional example of type E_{8} there exist homogeneous elements w_{1}, w_{2}, w_{3}, w_{4} in O(𝔥)^{W} such that Res{w_{1}, w_{2}, w_{3}, w_{4}} is an algebraically independent set (of 4 elements) which generates a (polynomial) subalgebra, J, of O(𝔞)^{W} such that O(𝔞)^{W} is free as a J-module on three generators.

The proof of this result is based on (complicated) calculations in the E_{7} and E_{8} cases. We will just give the high points for those two cases. The key to the proof is the following (simple) result.

Lemma 2.2 Let R = ℂ[x_{1}, ..., x_{p}] be a polynomial algebra with deg x_{i} = d_{i} ≥ 1. Let y ∈ ℂ[x_{1}, ..., x_{p}] be homogeneous of degree md_{1}. Let S be the subalgebra of R generated by y, x_{2}, ..., x_{p}. Assume that there exist a_{j}∈ S, j = 0, ..., m − 1 such that

x_{1}^{m} + ∑_{i=0}^{m−1} a_{i}x_{1}^{i} = 0.

Then y, x_{2}, ..., x_{p} are algebraically independent and R is a free S-module on the m generators 1, x_{1}, ..., x_{1}^{m−1}.

Proof: It is clear that

R = ∑_{i=0}^{m−1} Sx_{1}^{i}.

Let R_{j} denote the subspace of elements of R homogeneous of degree j here we use the weighted notion of homogeneity. Set S_{j} = R_{j} ∩ S. Then the Hilbert series of R is

∑_{j≥0} q^{j} dim R_{j} = \frac{1}{∏_{i=1}^{p}(1 − q^{d_{i}})}.

Now

dim R_{j} ≤ ∑_{k=0}^{m−1} dim S_{j−kd_{1}}.

Clearly, dim S_{j} is less than or equal to the coefficient, h_{j}, of q^{j} in

\frac{1}{(1 − q^{md_{1}}) ∏_{i=2}^{p}(1 − q^{d_{i}})}.

The above inequality implies that dim R_{j} is less than or equal to the coefficient of q^{j} in

\frac{1 + t^{d_{1}} + ... + t^{(m−1)d_{1}}}{(1 − q^{md_{1}}) ∏_{i=2}^{p}(1 − q^{d_{i}})} = \frac{1}{∏_{i=1}^{p}(1 − q^{d_{i}})}.

Thus all inequalities must be equalities and the lemma follows.

Q.E.D.

The rest of the proof of the proposition involves the following idea. Let Φ = Φ(𝔤, 𝔥) be the root system of 𝔤 with respect to 𝔥. Fix P a system of positive roots. Let

z_{2k} = ∑_{α∈P} α^{2k}.

Then clearly z_{2k} ∈ O(𝔥)^{W}. Let Σ(𝔤, 𝔞) be the restricted root system of 𝔤 with respect to 𝔞. Choose, Q a system of positive (restricted) roots. If α ∈ Q let

m_{α} = |{β ∈ Φ | β_{|𝔞} = α}|.

Then

Res(z_{2k}) = ∑_{α∈Q} m_{α}α^{2k}.

We now look at each case:

E_{6}: There are two real forms in question in the first case the restricted root system is of type BC_{2}. O(𝔞)^{W (𝔞)} = ℂ[v_{1}, v_{2}] with deg v_{1} = 2, deg v_{2} = 4. We now make things a bit more explicit. Relative to a basis of 𝔞 we can take the elements of Q to be 2x_{1}, 2x_{2}, x_{1}, x_{2}, x_{1} − x_{2}, x_{1} + x_{2}. With multiplicities given by 1,1,8,8,6,6. Thus

Res(z_{2k}) = (2^{2k} + 8)(x_{1}^{2k} + x_{2}^{2k}) + 6((x_{1} − x_{2})^{2k} + (x_{1} + x_{2})^{2k}).

We take v_{i} = x_{1}^{2i} +x_{2}^{2i}, i = 1, 2. Then Resz_{2} = 24v_{1}. Now φ =Res(z_{8}) = av_{1}^{4} +bv_{1}^{2}v_{2} +cv_{2}^{2}. If c = 0 then v_{1} must divide φ. Substituting in x_{2} = ±ix_{1} in φ we have a strictly positive multiple of x_{1}^{8}. Thus c ≠ 0. Now Lemma 2.2 implies the proposition in this case.

In the second case for E_{6} the restricted root system is of type A_{2}. We realize it, as usual, in the space where x_{1} + x_{2} + x_{3} = 0 and Q has elements x_{1} − x_{2},x_{1} − x_{3}, x_{2} − x_{3} all with multiplicity 8. We take v_{1} = x_{1}^{2} + x_{2}^{2} + x_{3}^{2}, v_{2} = x_{1}^{3} + x_{2}^{3} + x_{3}^{3}. As before, Res(z_{2}) is a positive multiple of v_{1}. This time we must analyze

φ = \frac{1}{8}Res(z_{6}) = (x_{1} − x_{2})^{6} + (x_{1} − x_{3})^{6} + (x_{2} − x_{3})^{6}.

Now φ = av_{1}^{3} + bv_{2}^{2}. Thus if b = 0 then φ is a multiple of v_{1}^{3}. We leave it to the reader to check that this is false. So Lemma 2.2 implies the result in this case.

E_{7}. This time the restricted root system is of type C_{3}. We realize Q as 2x_{1}, 2x_{2}, 2x_{3}, and the elements of the form x_{i} ± x_{j}, 1 ≤ i < j ≤ 3. The elements 2x_{i} have multiplicity 1 the others each have multiplicity 8. Thus

φ_{k} = Res(z_{2k}) = 2^{2k}(x_{1}^{2k} + x_{2}^{2k} + x_{3}^{2k}) + 8 ∑_{i<j}((x_{i} − x_{j})^{2k} + (x_{i} + x_{j})^{2k}).

We take v_{i} = ∑ x_{j}^{2i}, i = 1, 2, 3. As usual, φ_{1} is a positive multiple of v_{1}. φ_{3} = av_{1}^{3} + bv_{1}v_{4} + cv_{3}. We leave it to the reader to check that c ≠ 0. Thus we can replace v_{3} with φ_{3}. The last step in this case is to consider

φ_{4} = av_{1}^{4} + bv_{1}v_{3} + cv_{1}^{2}v_{2} + dv_{2}^{2}.

We leave it to the reader to show that d ≠ 0 (Mathematica or Maple would help). Thus Lemma 2.2 prevails in this case.

E_{8}. This time the restricted root system is of type F_{4}. We take Q (as in [B]) to be x_{i}, i = 1, ..., 4, \frac{1}{2}(x_{1} ± x_{2} ± x_{3} ± x_{4}), x_{i} ± x_{j}, 1 ≤ i < j ≤ 4. The short roots each have multiplicity 8 the long ones have multiplicity 1. So in this case we have

φ_{k} = Res(z_{2k}) = 8(x_{1}^{2k} + ... + x_{4}^{2k})+

2^{3−2k} ∑(x_{1} ± x_{2} ± x_{3} ± x_{4})^{2k} + ∑_{i<j}((x_{i} − x_{j})^{2k} + (x_{i} + x_{j})^{2k}).

Following [M] we set f_{k} = ∑_{i<j}((x_{i} − x_{j})^{2k} + (x_{i} + x_{j})^{2k}) and we may take v_{1} = f_{1}, v_{2} = f_{3}, v_{3} = f_{4}, v_{4} = f_{6}. As before φ_{1} is a positive multiple of v_{1}. Also, one can show that

φ_{4} = av_{1}^{4} + bv_{1}v_{2} + cv_{3}

with c ≠ 0 and

φ_{6} = av_{1}^{6} + bv_{1}^{3}v_{2} + cv_{1}^{2}v_{3} + dv_{2}^{2} + ev_{4}

with e ≠ 0 (not to be confused with the base of natural logarithms). Thus ℂ[φ_{1}, v_{2}, φ_{4}, φ_{6}] = O(𝔞)^{W (A)}. Finally one considers φ_{9} which is a polynomial in v_{1}, ..., v_{4} which we write the form

a_{1}v_{1}^{9} + a_{2}v_{1}^{5}v_{3} + a_{3}v_{1}v_{3}^{2} + a_{4}v_{1}^{3}v_{4}+
(b_{1}v_{1}^{6} + b_{2}v_{4} + b_{3}v_{1}^{2}v_{3})v_{2} + cv_{1}^{3}v_{2}^{2} + dv_{2}^{3}.

One can show that d ≠ 0. Thus Lemma 2.2 implies the result in this case.

Let 𝔭 = {X ∈ 𝔤 | θX = −X}. Let O(𝔭) (resp. O(𝔤)) denote the algebra of complex valued polynomials on 𝔭 (resp. 𝔤). Let O(𝔭)^{K} (resp. O(𝔤)^{G}) denote the algebra of K- (resp. G-)invariant polynomials on 𝔭 (res. 𝔤). Let Res_{𝔤/𝔭} (resp. Res_{𝔤/𝔥}) denote the restriction homomorphism f ↦ f_{|𝔭}(resp. f_{|𝔥}) from O(𝔤) onto O(𝔭) (resp. O(𝔥)). Also we use the notation Res_{𝔭/𝔞} for the restriction homomorphism from 𝔭 to 𝔞. Then Chevalley’s restriction theorem says that

Res_{𝔤/𝔥} : O(𝔤)^{G} ⟶ O(𝔥)^{W}

and

Res_{𝔭/𝔞} : O(𝔭)^{K} ⟶ O(𝔞)^{W (𝔞)}

are algebra isomorphisms. We combine this with the above proposition and Helgason’s restriction theorem and have

Theorem 2.3 There are homogeneous elements U_{1}, ..., U_{l} ∈ O(𝔤)^{G} such that if W_{i} = Res_{𝔤/𝔭}U_{i} then the polynomials W_{1}, ..., W_{l} are algebraically independent and O(𝔭)^{K} is free as a ℂ[W_{1}, ..., W_{l}] module on a finite number, d, of generators.

Note: If G does not correspond to one of the 4 exceptional examples then the U_{i} can be chosen so that d = 1. The above proposition says that we can make our choices so that d = 2 for the exceptional examples of type E_{6} and E_{7} and d = 3 for the exceptional example of type E_{8}.

Recall that 𝔤 = 𝔨 + 𝔭 is the (complexified) Cartan decomposition. We denote by S(𝔭) the symmetric algebra of polynomial functions over 𝔭^{∗}. Let S(𝔭)^{K} be the K-invariants of S(𝔭). For any X ∈ 𝔭 let ∂(X) be the differential operator defined by

(∂(X)f)(Y) = \frac{d}{dt}|_{t=0}f(Y + tX), f ∈ C^{∞}(𝔭).

We can extend the definition of ∂(X) to any X ∈ S(𝔭). Denote by S_{+}(𝔭)^{K} the set of K-invariant polynomials with zero constant terms. A polynomial h ∈ O(𝔭) is called harmonic if ∂(J)h = 0 for all J ∈ S_{+}(𝔭)^{K}.

Let H denote the space of all K−harmonic polynomials on 𝔭. Then Kostant and Rallis [K-R] have shown that the map

H ⊗ O(𝔭)^{K} ⟶ O(𝔭)
h ⊗ u ↦ hu

is a linear bijection. This theorem combining with Theorem 2.3 implies (𝔨 is the complexified Lie algebra of K and ker Res_{𝔤/𝔭}O(𝔨) = 0)

Theorem 2.4 (Notation as in Theorem 2.3.) Let E be a subspace of O(𝔭)^{K} of dimension d generating O(𝔭)^{K} as a D = ℂ[U_{1}, ..., U_{l}]-module under U_{i} acts by multiplication by W_{i}. Then the map

H ⊗ E ⊗ D ⊗ O(𝔨) ⟶ O(𝔤)

h ⊗ e ⊗ d ⊗ k ↦ hedk

is a linear bijection.

Let U^{j}(𝔤) ⊂ U^{j+1}(𝔤) be the standard filtration of the universal enveloping algebra, U(𝔤), of 𝔤. Let U_{i} be as in Lemma 2.3. Let symm denote the usual symmetrization mapping from S(𝔤) onto U(𝔤). Let Z(𝔤) denote the center of U(𝔤). Let B be the Killing form of 𝔤. Using B we can identify S(𝔤) with O(𝔤), S(𝔨) with O(𝔨), S(𝔭) with O(𝔭), etc. Let D_{i} = symm(U_{i}) and let D denote the subalgebra of Z(𝔤) generated by D_{1}, ..., D_{l}. If we compare the grade with the filtration and argue as in [W, 11.2.2], we have

Theorem 2.5 The map

H ⊗ E ⊗ D ⊗ U(𝔨) ⟶ U(𝔤)
h ⊗ e ⊗ D ⊗ k ↦ symm(h)symm(e)Dk

is a linear isomorphism.

3. Generalized spherical functions We retain the notations in Section 2. The spherical functions, which were first defined and studied by Harish-Chandra [HC], can be characterized by being K-bi-invariant and being eigenfunctions of each operator in D(G/K). Here D(G/K) is the algebra of G-invariant differential operators on G/K, which is isomorphic to S(𝔞)^{W (A)} as an algebra over ℂ. The center Z(𝔤) of the universal enveloping algebra can be identified with the algebra of G-bi-invariant differential operators on G. Let γ : Z(𝔤) → S(𝔥)^{W (𝔤,𝔥)} be the Harish-Chandra isomorphism.

Let D = ℂ[D_{1}, ⋯, D_{l}] be the algebra of differential operators defined before Theorem 2.5. The projection p : 𝔥 = 𝔞 + 𝔱 → 𝔞 induces a map of S(𝔥)^{W (𝔤,𝔥)} to S(𝔞)^{W (A)}, which we also denoted by p. Let l be the real rank of G. Then S(𝔞)^{W (A)} is isomorphic to a polynomial algebra in l generators. Note that if G does not contain any of four exceptional groups as a simple factors, then D is isomorphic to S(𝔞)^{W (A)}. In particular, when G is split, D is Z(𝔤). If G does contain some of the four exceptional groups as a simple factor, then our main results in the previous section implies that D is a subalgebra of Z(𝔤) such that S(𝔞)^{W (A)} is a simple module E for p ∘ γ(D) ≅ D. In all cases, D is a polynomial algebra generated by l homogeneous elements, which are algebraically independent. For any ν ∈ 𝔞^{∗}, we will write χ_{ν} for the corresponding character of D.

Let τ = (τ_{1}, τ_{2}) be a pair of two unitary representations of K with representation spaces V_{τ_{1}} and V_{τ_{2}}. For simplicity we assume that both τ_{1} and τ_{2} are irreducible. Set V = Hom(V_{τ_{2}}, V_{τ_{1}}). Then (τ, V) can be made into a double representation of K × K, in the sense τ(k_{1}, k_{2})v = τ_{1}(k_{1})vτ_{2}(k_{2}).

Definition 3.1 Let τ be a double representation of K × K as above. We define a generalized spherical function f of type τ to be a V -valued joint eigenfunction of D such that f(k_{1}gk_{2}) = τ_{1}(k_{1})f(g)τ_{2}(k_{2}). For simplicity, a generalized function of type τ will be also called a τ-spherical function.

The space of all τ-spherical functions with eigenvalue χ will be denoted by E_{χ}(G, τ). The spherical functions defined by Harish-Chandra are obviously a generalized spherical functions of the trivial type τ = (1, 1).

Now let G = KAN be the Iwasawa decomposition of G. Let M = Z_{K}(𝔞) be the centralizer of 𝔞 in K. If ϕ ∈ C^{∞}(M, V) is such that ϕ(k_{1}mk_{2}) = τ_{1}(k_{1})ϕ(m)τ_{2}(k_{2}) for k_{1}, k_{2}, m ∈ M, then we set

ϕ(kman) = τ_{1}(k)ϕ(m)

with k ∈ K, m ∈ M, a ∈ A, n ∈ N. If ν ∈ 𝔞^{∗}, we put

E(P, ϕ, ν)(g) = ∫_{K} a(gk)^{ν−ρ}ϕ(gk)τ_{2}(k)^{−1}dk.  (3.2)

Following Harish-Chandra, we call such a function an Eisenstein integral. Note that all Eisenstein integrals are generalized spherical functions. When G is split and ν is generic, our main results actually implies that Eisenstein integrals exhaust all generalized spherical functions. In general case, when G is not necessarily split, we will see in the next section that a generalized function is a matrix coefficient of some admissible (𝔤, K)-module.

4. Dimensions of spaces of τ-spherical functions In this section we show that the dimension of the space of τ-spherical functions is an integer d(τ), which only depends on the bi-K-type τ. We denote by E_{χ}(G, τ) the space of all τ-spherical functions with eigenvalue χ. We prove that the dim E_{ν}(G, τ) is equal to

d(τ) = dimV^{M} · dim E = dim Hom_{M}(V_{τ_{1}}|_{M}, V_{τ_{2}}|_{M}) · dim E.  (4.1)

Here E is the space in Theorem 2.5. Note that d(τ) is an integer which is independent of the eigenvalues χ. In particular, we know there does not exist any nonzero generalized spherical functions of type τ if Hom_{M}(V_{τ_{1}}|_{M}, V_{τ_{2}}|_{M}) = 0. Our approach is first to prove a theorem which gives an upper bound of the dimension of the space of τ-spherical functions. Then we show this upper bound is a lower bound as well.

Theorem 4.1 The dimension of the space of generalized spherical functions of type τ = (τ_{1}, τ_{2}) is less than or equal to d(τ).

Proof: Recall that V = Hom(V_{τ_{2}}, V_{τ_{1}}). For any f ∈ E_{χ}(G, τ), we define T_{f} : U(𝔤) → V by T_{f}(X) = X · f(1) with X ∈ U(𝔤). Then

T_{f}(symm(h)symm(e)Dk) = χ(D)f(symm(h)symm(e))τ_{2}(k),

with h ∈ H, e ∈ E, D ∈ D and k ∈ U(𝔨). Here τ_{2} is actually the differential of the K representation τ_{2}. By Theorem 2.5 T_{f} is completely determined by its evaluation on symm(H) ⊗ E. Now we set A_{f}(h ⊗ e) = T_{f}(symm(h)symm(e)). If A_{f} = 0, then f is a constant zero function. It is clear that A_{f}(Ad(k)h ⊗ e) = τ_{1}(k)A_{f}(h ⊗ e)τ_{2}(k)^{−1}. Hence A_{f} is contained in Hom_{K}(H ⊗ E, V). In other words, A_{f} is an injection from E_{χ}(G, τ) into the K-finite dual of Hom_{K}(H ⊗ E, V) ≅ V^{M} ⊗ E^{∗}. This proves the theorem.

Q.E.D.

Now we show that the dimension of E_{χ}(G, τ) is actually equal to d(τ). We use a theorem of Casselman on the existence of globalization of admissible (𝔤, K)-modules.

Proposition 4.2 (cf. [W, 4.2.4]) Let W be an admissible (𝔤, K)-module. Then there exists a Hilbert space representation (π, H) of G, such that the Harish-Chandra module of H is isomorphic to W.

Theorem 4.3 The dimension dim E_{χ}(G, τ) of the space of τ-spherical functions is equal to d(τ) = dimV^{M} · dim E, which is independent of eigenvalues. In particular, there does not exist any nonzero generalized spherical functions of type τ if V^{M} = 0.

Proof: Let W = U(𝔤) ⊗_{U(𝔨)D} V_{τ_{2},χ}. Here V_{τ_{2},χ} is the finite dimensional representation space of V_{τ_{2}} so that D acts via the character χ. Then W is an admissible (𝔤, K)-module. Let (π, H) be a globalization of W and H_{∞} be the space of smooth vectors of H. Now let T be an arbitrary K-intertwining map T : U(𝔤) ⊗_{U(𝔨)D} V_{τ_{2},χ} → V_{τ_{1}}. The map T extends continuously to T : H_{∞} → V_{τ_{1}}. We define (f_{T} (g))(v) = T (π(g)(1 ⊗ v)), for g ∈ G and v ∈ V_{τ_{2}}. Then we have f_{T} (g) ∈ Hom(V_{τ_{2}}, V_{τ_{1}}) and

f_{T} (k_{1}gk_{2})(v) = T (π(k_{1}gk_{2})(v))
= τ_{1}(k_{1})T (π(g)(1 ⊗ τ_{2}(k_{2})v))
= τ_{1}(k_{1})f_{T} (g)τ_{2}(k_{2})v.

That is f_{T} (k_{1}gk_{2}) = τ_{1}(k_{1})f_{T} (g)τ_{2}(k_{2}). So f_{T} is an τ-spherical function. If f_{T} = 0, then T (π(g)(1 ⊗ v)) = 0 for all g ∈ G and v ∈ V_{τ_{2}}. Then T = 0. Hence by Theorem 2.5 we have

dimE_{χ}(G, τ) ≥ dim Hom_{K}(U(𝔤) ⊗_{U(𝔨)D} ⊗V_{τ_{2},χ}, V_{τ_{1}})
= dim Hom_{K}(H ⊗ E ⊗ V_{τ_{2}}, V_{τ_{1}})
= dim Hom_{K}(H ⊗ E, V_{τ_{1}} ⊗ V_{τ_{2}}^{∗})
= dim Hom_{K}(H, V_{τ_{1}} ⊗ V_{τ_{2}}^{∗}) · dimE
= dim Hom_{M}(ℂ, V_{τ_{1}} ⊗ V_{τ_{2}}^{∗}) · dimE
= dimV^{M} · dimE.

Q.E.D.

For any Harish-Chandra module V let V^{∗} be its contragredient. The admissible (𝔤, K)- module V (resp. V^{∗}) may be realized as the K-finite vectors of some admissible representation (π, V_{π}) (resp. (π^{∗}, V_{π^{∗}})). We define the matrix coefficient map

C : V × V^{∗} → C^{∞}(G)

by C_{v,v^{∗}}(g) = ⟨π(g)v, v^{∗}⟩. The definition of the matrix coefficient map C depends on the globalization of V and V^{∗}. But the matrix coefficients C_{v,v^{∗}}(g) of V are invariants of Harish-Chandra modules. As a corollary in the proof of Theorem 4.3, we have

Corollary 4.4 The generalized functions are matrix coefficients of some admissible (𝔤, K)- module.

5. Analytic dependence on parameters.

If ν ∈ 𝔞^{∗} then we define, as usual, χ_{ν}(D) = Res(γ(D))(ν) for D ∈ D. The purpose of this section is to prove

Theorem 5.1 Assume that the Casimir operator, C, of 𝔤 is in D. Let (τ, V) be a double representation of K. Then there exist functions f_{i,ν} ∈ E_{χ_{ν}} (G, τ) i = 1, ..., d = dim V^{M} dim E, ν ∈ 𝔞^{∗} satisfying the following two conditions:

(1) {f_{1,ν}, ..., f_{d,ν}} is a basis of E_{χ_{ν}} (G, τ) (2) The map ν, g ↦ f_{i,ν}(g) is real analytic from 𝔞^{∗} × G to V and holomorphic in ν.

Note. D can always be chosen so that C ∈ D (see section 2). Also, Theorem 5.1 is true without the assumption that C ∈ D but the proof is (even) more complicated.

The proof of this result will take up the rest of this section.

We maintain the notation of the previous sections. Let 𝔭 be the complexified Lie algebra of P = MAN and let 𝔪, 𝔞, 𝔫 be respectively the Lie algebras of M, A, N. As is well known (cf. [H], [W, Vol. II]) U(𝔞) is a free U(𝔞)^{W (𝔤,𝔞)}-module on w = |W (𝔤, 𝔞)| homogeneous generators, a_{1} = 1, a_{2}, ..., a_{w}. Set A = Span_{ℂ}{a_{1}, ..., a_{w}}. We also identify O(𝔞) with U(𝔞) = S(𝔞) using the Killing form. Thus we look upon E as a subspace of U(𝔞). We also assume (as we may) that the Casimir operator of 𝔤 is in D.

Lemma 5.2 The map U(𝔫)⊗E⊗A⊗D⊗U(𝔨) ⟶ U(𝔤) given by n⊗e⊗a⊗D⊗k ↦ neaDk is a linear bijection.

The proof of this lemma is essentially the same as that of Lemma 11.2.1 in [W].

It is enough to prove Theorem 5.1 in the case when (τ, V) is irreducible. We may thus assume that there are irreducible finite dimensional representations (τ_{i}, V_{i}), i = 1, 2 such that V = Hom_{ℂ}(V_{2}, V_{1}) and τ(k_{1})vτ(k_{2}) = τ_{1}(k_{1})vτ_{2}(k_{2}).

If ν ∈ 𝔞^{∗} then we make V_{2} into a DU(𝔨)-module by

Dk · v = χ_{ν}(D)τ_{2}(k)v, v ∈ V_{2}, k ∈ U(𝔨), D ∈ D.

We denote this DU(𝔨)-module by (V_{2})_{ν}. We set

Y_{τ_{2},ν} = U(𝔤) ⊗_{DU(𝔨)} (V_{2})_{ν}.

If τ_{2} is understood we will write Y^{ν} for Y_{τ_{2},ν}. We note that Y^{ν} is a free U(𝔫)-module on dim A · dim E-generators. Thus in particular, it is admissible and finitely generated (cf. [W, 4.2.6]).

If Z is an 𝔫-module then we set 𝔫Z = ∑_{X∈𝔫} XZ and 𝔫^{k+1}Z = ∑_{X∈𝔫} X𝔫^{k}Z. Lemma 5.2 implies that each of the modules Y^{ν}/𝔫^{k}Y^{ν} is a finite dimensional (𝔭, M)-module. Since AN is simply connected, this (𝔭, M)-module structure integrates to a representation (σ_{ν}, W_{k,ν}) of P with W_{ν} = Y^{ν}/𝔫^{k}Y^{ν}. We note that as a space W_{k,ν} = U(𝔫)/𝔫^{k}U(𝔫)⊗A ⊗ E⊗V_{2} and that the action of M and 𝔫 on W_{k,ν} is independent of ν. We will identify W_{k,ν} with this fixed space which we denote by W_{k}.

Lemma 5.3 Let H ∈ 𝔞 then ν ↦ σ_{k,ν}(H) is a polynomial map of 𝔞^{∗} → End_{ℂ}(W_{k}).

This lemma can be proved using the ideas in [W, 11.3.5].

Let (σ, W) be a finite dimensional (smooth) representation of P. We set I_{P,σ}^{∞} equal to the space of all C^{∞} f : G → W such that f(pg) = σ(p)f(g), g ∈ G, p ∈ P. We set π_{σ}(x)f(g) = f(gx). Then (π_{σ}, I_{P,σ}^{∞}) is an admissible, smooth, Fréchet representation of G. We set I_{P,σ} equal to the (𝔤, K)-module of K-finite vectors in I_{P,σ}^{∞}.

Let π_{k} : Y^{ν} → W_{k} be the canonical projection to W_{k,ν} followed by identification with W_{k}. If v ∈ Y^{k} define T_{k,ν}(v)(k) = π_{k}(kv). Then as in [W, 4.2.3], T_{k,ν} induces a (𝔤, K)- homomorphism of Y^{ν} into I_{P,σ_{k,ν}} which is injective for large k.

Lemma 5.4 If ν_{o} ∈ 𝔞^{∗} then there exist an open neighborhood U of ν_{o} in 𝔞^{∗} and a positive integer k = k(ν_{o}) such that T_{k,ν} is injective for all ν ∈ U.

Proof: Let C be the Casimir operator of 𝔤. Then C acts on Y^{ν} by χ_{ν}(C) = ⟨ν, ν⟩ − ⟨ρ, ρ⟩. We note that we have the projections Y^{ν}/𝔫^{k+1}Y^{ν} ↦ Y^{ν}/𝔫^{k}Y^{ν} with kernels Z_{k,ν} = 𝔫^{k}Y^{ν}/𝔫^{k+1}Y^{ν}. We note that 𝔫Z_{k,ν} = 0. If H ∈ 𝔞 then H on Z_{k,ν} has generalized eigenvalues of the form λ(H) + ξ(H) with ξ a weight of 𝔞 on Z_{k−1,ν}. Let μ_{k,ν} be the representation of P on Z_{k,ν}. It is easily seen from the above observations that if ξ is an eigenvalue of C on I_{P,μ_{k,ν}} then Reξ ≥ C_{1}k^{2} − C_{2}(ν) with C_{1} > 0 and C_{2}(ν) continuous in ν (see the proof of [W, 4.1.4] for the basic ideas). Thus if U is chosen so that |C_{2}(ν)| is bounded by a constant C_{3} and if k_{o} is chosen so that C_{1}k^{2} − C_{3} > Re⟨ν, ν⟩ − ⟨ρ, ρ⟩ for k ≥ k_{o} then C cannot have eigenvalue ⟨ν, ν⟩ − ⟨ρ, ρ⟩ on any I_{P,μ_{k,ν}}. We assert that we can chose k(ν_{o}) = k_{o}. Indeed, if T_{k,ν} is not injective but T_{k+1,ν} is injective then we have the commutative diagram

\begin{aligned}& T_{k+1,ν} \\ Y^{ν} & ⟶ & I_{P,σ_{k+1,ν}} \\ ↓ & T_{k,ν} & ↓ \\ Y^{ν} & ⟶ & I_{P,σ_{k,ν}} \end{aligned} .

Thus ker T_{k,ν} must inject into the kernel of the right vertical arrow in the diagram which is isomorphic with I_{P,μ_{k,ν}}. Hence, ker T_{k,ν} injects into I_{P,μ_{k,ν}}. This is impossible if k ≥ k_{o} and ν ∈ U. The lemma now follows.

Fix ν_{o}, k and U as in the preceding lemma. We note that as a representation of K, I_{P,σ_{k,ν}}^{∞} ≅ (Ind_{M}^{K})^{∞}(σ_{k}) (C^{∞} induction from M to K) where (σ_{k}, W_{k}) is the M-module (U(𝔫)/𝔫^{k}U(𝔫)) ⊗ A ⊗ E ⊗ V_{2}. We write I^{∞}(σ_{k}) for (Ind_{M}^{K})^{∞}(σ_{k}) and we look upon I_{P,σ_{k,ν}}^{∞} as I^{∞}(σ_{k}) with action π_{ν} depending on ν. We note that Lemma 5.2 implies that

1. The map ν, g, f ↦ π_{ν}(g)f from 𝔞^{∗} × G × I^{∞}(σ_{k}) to I^{∞}(σ_{k}) is real analytic and holomorphic in ν.

We assume that S_{1}, ..., S_{d} ∈ Hom_{K}(Y^{ν_{o}}, V_{1}) are such that (in the notation of the proof of Theorem 4.3) f_{S_{1}}, ..., f_{S_{d}} form a basis of E_{χ_{ν}} (G, τ).

We note that with our identifications the map v ↦ T_{k,ν}(1 ⊗ v) of V_{2} into I^{∞}(σ_{k}) is independent of ν ∈ U. We will denote T_{k,ν}(1 ⊗ v) by T (v). Set T_{ν} = T_{k,ν}. As in Theorem 2.4, we look upon Y^{ν} as Y = H ⊗ E ⊗ V_{2} with the (𝔤, K)-module structure depending on ν. We write μ_{ν}(g) for the action of g ∈ U(𝔤) on Y. 1. above now implies that T_{ν}(h ⊗ e ⊗ v) = μ_{ν}(he)T (v). Thus 1. implies that if z ∈ Y then the map ν ↦ T_{ν}(z) is holomorphic from 𝔞^{∗} to I^{∞}(σ_{k}).

Fix S̃_{1}, ..., S̃_{d} ∈ Hom_{K}(I^{∞}(σ_{k}), V_{1}) such that S̃_{i} ∘ T_{ν_{o}} = S_{i}. If ν ∈ U then set f_{i,ν}(g)v = S̃_{i}(π_{ν}(g)T (1 ⊗ v)) for v ∈ V_{2}. f_{i,ν}(g) is holomorphic in ν and real analytic in G. Then {f_{1,ν_{o}}, ..., f_{d,ν_{o}}} is a basis of E_{χ_{νo}} (G, τ) (see the proof of Theorem 4.2). Thus if a_{i} ∈ ℂ, i = 1, ..., d and if for all λ ∈ (Hom_{ℂ}(V_{2}, V_{1}))^{∗}, g ∈ G we have

∑_{i} a_{i}λ(f_{i,ν_{o}}(g)) = 0

then a_{i} = 0 for i = 1, .., d. This implies that we can choose λ_{i} ∈ (Hom_{ℂ}(V_{2}, V_{1}))^{∗}, g_{i} ∈ G, i = 1, ..., d such that

det[λ_{i}(f_{j,ν_{o}}(g_{i}))] ≠ 0.

Thus there is an open subset U(ν_{o}) of U such that ν_{o} ∈ U(ν_{o}) and det[λ_{i}(f_{j,ν}(g_{i}))] ≠ 0 for all ν ∈ U(ν_{o}). Thus {f_{1,ν}, ..., f_{d,ν}} is a basis of E_{χ_{ν}} (G, τ) for ν ∈ U(ν_{o}).

We are now almost done. Indeed, we note that we have proven that ν ↦ E_{χ_{ν}} (G, τ) is a holomorphic vector bundle over 𝔞^{∗} and so the proof can be completed using general results in complex analysis. However, we will finish the proof of Theorem 5.1 by more elementary means.

Let U_{1}, ..., U_{d} be a basis of Hom_{K}(H ⊗ E ⊗ V_{2}, V_{1}). If ν_{o} ∈ 𝔞^{∗} then we write f_{i,ν}^{ν_{o}} for the elements constructed above for ν ∈ U(ν_{o}). Then

T_{f_{i,ν}^{νo}} = ∑_{j} b_{ji}(ν)U_{j}

with ν ↦ [b_{ij}(ν)] a holomorphic map of U(ν_{o}) to GL(d, ℂ). Set [c_{ij}(ν)] = [b_{ij}(ν)]^{−1}. Set g_{k,ν}^{ν_{o}} = ∑ c_{ik}(ν)f_{i,ν}^{ν_{o}}, ν ∈ U(ν_{o}). Then T_{g_{i,ν}^{νo}} = U_{i}. This implies that if ν ∈ U(μ) ∩ U(μ′)

then g_{i,ν}^{μ} = g_{i,ν}^{μ′}. Thus we can define f_{i,ν} = g_{i,ν}^{ν} for i = 1, ..., d and ν ∈ 𝔞^{∗}. {f_{1,ν}, ..., f_{d,ν}} is the desired family. This completes the proof of Theorem 5.1.

6. Split case: a character formula We assume throughout this section that the semisimple G is split. Let 𝔞 be the split Cartan subalgebra of 𝔤. For each λ ∈ 𝔞^{∗}, we define a character of Z(𝔤)

χ_{λ} : Z(𝔤) → ℂ

to be the composition of Harish-Chandra homomorphism with evaluation at λ. Let W be the Weyl group. Note that the character χ_{λ} depends only on the W -orbit of λ. Now we define

E_{λ}(G) = {f ∈ C^{∞}(G)|(z − χ_{λ}(z))f = 0, for all z ∈ Z(𝔤)}.

Let P = MAN be Langlands decomposition of the minimal parabolic subgroup. Denote by Ind_{P}^{G}(δ ⊗ λ) the induced representation from the minimal parabolic subgroup.

Definition 6.1 A functional λ ∈ 𝔞^{∗} is called generic, if Ind_{P}^{G}(δ ⊗ λ) is irreducible for any δ ∈ M̂.

Theorem 6.2 For λ ∈ 𝔞^{∗} generic, E_{λ}(G) is a space of functions generated by matrix coefficients of all principal series with infinitesimal character χ_{λ}and as (𝔤 × 𝔤, K × K)- modules

E_{λ}(G) ≅ ⨁_{δ∈M̂}Ind_{P}^{G}(δ ⊗ λ) ⊗ Ind_{P}^{G}(δ^{∗} ⊗ −λ).

Proof: The assumption implies that all Ind_{P}^{G}(δ ⊗ λ) ⊗ Ind_{P}^{G}(δ^{∗} ⊗ −λ) occur in E_{λ}(G) as irreducible (𝔤 × 𝔤, K × K)-module. Then the dimension formula in Theorem 4.3 implies that they exhaust all generalized spherical functions.

Q.E.D.

Remark 6.3 If λ is not generic, the set of matrix coefficients of principal series is smaller. In other words, E_{λ}(G) can be much larger than the matrix coefficients of all principal series with infinitesimal character χ_{λ}. For example, let G = SL(2, ℝ) and λ = ρ the half sum of all positive roots. We can construct a Harish-Chandra module V with infinitesimal character χ_{λ} and with composition series

0 = V_{0} ⊂ V_{1} ⊂ V_{2} ⊂ V_{3} = V

so that V_{3}/V_{2} ≅ D^{+}, V_{2}/V_{1} ≅ ℂ and V_{1}/V_{0} ≅ D^{−}, where ℂ is the 1-dimensional representation and D^{+} and D^{−} are two discrete series. The matrix coefficients of V is contained in E_{λ}(G) but is not contained in the space generated by the matrix coefficients of all principal series.

Let (π, H) be an admissible representation of G. If f is a compactly supported smooth function on G, then π(f) is of trace class. We set Θ_{π}(f) = trπ(f). Thus Θ_{π} defines a continuous linear functional on C_{o}^{∞}(G). We call Θ_{π} the distribution character of π, which is also denoted by θ(π). Let E_{γ} is the orthogonal projection of H to the K-finite isotropic subspace H(γ) for γ ∈ K̂. Then

trE_{γ}π(g)E_{γ} = ∑_{j}⟨π(f)v_{j}, v_{j}⟩  (6.1)

where v_{j}’s are a basis for H(γ). Set ϕ_{p}i^{γ}(g) = trE_{γ}π(g)E_{γ}. Then ϕ_{π}^{γ}(g) is a real analytic functional on G and

Θ_{π}(f) = ∑_{γ∈K̂} ∫_{G} f(g)ϕ_{π}^{γ}(g)dg.  (6.2)

Theorem 6.4 The distribution character of (𝔤 × 𝔤, K × K)-modules E_{λ}(G) is equal to the sum of distribution characters of principal series tensoring its contragredient. That is

θ(E_{λ}(G)) = ∑_{δ∈M̂} θ(Ind_{P}^{G}(δ ⊗ λ) ⊗ Ind_{P}^{G}(δ^{∗} ⊗ −λ)),

where P = MAN is a minimal parabolic subgroup of G.

Proof: The dimension formula in Theorem 4.3 implies that E_{λ}(G) is an admissible (𝔤 × 𝔤, K ×K)-module. For any irreducible K ×K-type (τ, V), we know in Section 5 there exist a global basis for E_{λ}(G), which depend holomorphically on λ. Then the theorem follows from (6.1), (6.2) and Theorem 6.2.

Q.E.D.

References

[B] N. Bourbaki, Groupes et algèbres de Lie. Hermann, Paris, 1968.

[HC] Harish-Chandra, Spherical functions on a semisimple Lie group, I, II, Amer. J. Math. 80 (1985), 241-310, 553-613.

[H] S. Helgason Some results on invariant differential operators on symmetric spaces, Amer. J. Math. 114 (1992), 789-811.

[KR] B. Kostant and S. Rallis, Orbits and Lie algebra representations associated to symmetric spaces, Amer. J. Math. 93 (1971), 753-809.

[W] N. Wallach, Real Reductive Groups, Volume I, II. Academic Press 1988, 1992.

Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

E-mail address: mahuang@uxmail.ust.hk

Department of Mathematics, University of California, San Diego, La Jolla, CA 92093 E-mail address: nwallach@math.ucsd.edu

Department of Mathematical Sciences, University of Tokyo, Tokyo 113, Japan E-mail address: t-oshima@tansei.cc.u-tokyo.ac.jp