Graded K-multiplicities

Siddhartha Sahi Rutgers University

October 10, 2005

Abstract Based on Vogan’s ideas, we define the graded K-multiplicity for an irreducible tempered (𝔤, K) module with real infinitesimal character. This gives rise to an interesting family of polynomials p_{μ} (t) in the Grothendieck ring of K.

1 Graded multiplicities

Let G be a real reductive group with maximal compact subgroup K. Let Ĝ_{rt} be the set of irreducible tempered unitary representations of G with real infinitesimal character, and let K̂ be the set of irreducible representations of K. Then there is a bijection from Ĝ_{rt} to K̂ which asigns to each real tempered representation its (unique) lowest K-type.

Let R be the Grothendieck ring of virtual representations of K, then K̂ forms a ℤ-basis for R. We will show that the above considerations allows us to construct a deformation of K̂. More precisely we construct a family of polynomials p_{μ}(t) in R[t] such that

p_{μ} (0) = μ.

For μ in K̂ let (π, V) be the Harish-Chandra module of the corresponding representation in Ĝ_{rt}. Then V has a natural K-stable filtration with

F_{0}(V) = μ and F_{k+1}(V) = F_{k}(V) + π(𝔤)[F_{k}(V)].

Let \overline{V} = ⊕_{k}V_{k} be the associated graded module with V_{k} = F_{k+1}(V)/F_{k}(V) and write

C_{μ}(t) = ∑ t^{k}V_{k}.

Then C_{μ}(t) is an element of the formal power series ring R[[t]].

Let 𝔤 = 𝔨 + 𝔭 be the Cartan decomposition with n = dim (𝔭), and write

D(t) = ∑_{k=0}^{n}(−t)^{k}Λ^{k}(𝔭) ∈ R[t].

We call D(t) the universal denominator. This terminology is motivated by the main result of the next section.

2 The polynomial p_{μ}

Our main result is the following:

Theorem 1 The product p_{μ}(t) := C_{μ}(t)D(t) is a polynomial in R[t] of degree at most n, with constant term p_{μ}(0) = μ.

For the proof of this we need to introduce a few concepts. A graded Kmodule X = ∑_{i≥0} X_{i} will be called admissible if each X_{i} is finite dimensional. The graded character of such a module, χ (t, X) = ∑ t^{i}χ (X_{i}) , belongs to R[[t]].

If X is admissible, so are the symmetric and exterior algebras S (X) and Λ (X). In the calculations below we will frequently use the following ‘Koszul’ lemma

Lemma 2 If X is an admissible graded K-module then following identity holds in R[[t]]

χ (−t, Λ (X)) χ (t, S (X)) = 1

Proof. (of Theorem 1) \overline{V} is a graded module for the symmetric algebra S(𝔤) which is the associated graded algebra of the enveloping algebra U(𝔤). Since π is irreducible, the subspace μ generates V as a U(𝔤)-module. Therefore μ = V_{0} generates \overline{V} as a S(𝔤)-module. Since μ is 𝔨-stable and S(𝔤) = S(𝔨)S(𝔭), it follows that V_{0} generates \overline{V} as a graded module for the algebra S(𝔭). By Hilbert’s syzygy theorem, we can construct a free resolution for \overline{V} of the form

0 → M_{n} → ... → M_{1} → M_{0} → \overline{V} → 0

where each M_{i} is a free S(𝔭)-module of the form E_{i} ⊗ S(𝔭), and each E_{i} is a finite dimensional graded K-module. Writing p_{μ}(t) = ∑_{k=0}^{n}(−1)^{k}E_{k} we have

C_{μ}(t) = p_{μ}(t)S(t)  (1)

where S(t) = ∑ t^{k}S^{k} (𝔭) is the graded character of the ring S(𝔭) as a Kmodule. In the case of the trivial module 1 such a resolution can be obtained using the Koszul complex

K_{n} → … → K_{1} → K_{0} → 1 → 0 where K_{i} = Λ^{i}(𝔭) ⊗ S (𝔭)

and Λ^{i}(𝔭) is homogeneous of degree i. This implies that

D(t)S(t) = 1  (2)

and part 1 follows by combining this with (1). For part 2 we combine the following (obvious) identites.

C_{μ}(0) = μ, D(0) = 1.

It would be interesting to obtain an explicit formula for the polynomials p_{μ} (t), especially in context of Vogan’s ideas for computing K-multiplicities (see notes by P. Trapa). In the following discussion we give such formulas in various special cases – our results depend on various classical facts from the literature on representation theory of semisimple Lie groups.

3 SL(2, ℝ)

We first consider the group G = SL(2, ℝ). We identify K = T = SO(2) with the unit circle {z ∈ ℂ : |z| = 1}. Then K̂ is isomorphic to the integers via the correspondence n (z) = z^{n}, and R can be identified with the ring of Laurent polynomials ℤ [z, z^{−1}]

For convenience we divide the real tempered representations into three kinds:

1. The (spherical) principal series representation V^{0} with minimal K-type 0

2. The discrete series and limits of discrete series representations V^{n} with minimal K-type n > 0.

3. The discrete series and limits of discrete series representations V^{−n} with minimal K-type −n < 0.

The K-type structures of the graded modules in these cases are as follows:

1. For n = 0, V_{0}^{0} = {0} , V_{k}^{0} = {−2k, 2k} for k > 0.

2. For n > 0, V_{k}^{n} = {n + 2k}.

3. For −n < 0, V_{k}^{−n} = {−n − 2k} .

Therefore the graded characters are given by

C_{0} (t) = 1 + ∑_{k≥1} t^{k} (z^{2k} + z^{−2k}) = \frac{1}{1 − tz^{2}} + \frac{1}{1 − tz^{−2}} − 1

C_{n} (t) = ∑_{k≥0} t^{k}z^{n+2k} = \frac{z^{n}}{1 − tz^{2}}

C_{−n} (t) = ∑_{k≥0} t^{k}z^{−n−2k} = \frac{z^{−n}}{1 − tz^{−2}}

For SL (2, ℝ) the K-module srtucture of 𝔭 is {−2, 2}. Therefore the universal denominator D (t) is given by

D (t) = 1 − t (z^{2} + z^{−2}) + t^{2} = (1 − tz^{2}) (1 − tz^{−2})

A simple calculation now shows that the polynomials p_{μ} (t) are given as follows:

p_{0} (t) = (1 − tz^{−2}) + (1 − tz^{2}) − (1 − t (z^{2} + z^{−2}) + t^{2}) = 1 − t^{2}
p_{n} (t) = z^{n} (1 − tz^{−2}) = z^{n} − tz^{n−2}
p_{−n} (t) = z^{−n} (1 − tz^{2}) = z^{−n} − tz^{−n+2}

4 The spherical principal series

We now consider the case of the trivial K-type for general G. Under the bijection of K̂ with Ĝ_{rt} this corresponds to the spherical principal series representation with infinitesimal character 0. For consistency with the previous section, we write V^{0} for this representation, C_{0} (t) for its graded character, and p_{0} (t) for the associated polynomial. We will show that p_{0} (t) can be computed using ideas of Kostant and Rallis.

It is known from [ B. Kostant and S. Rallis, Amer. J. Math. 93 (1971), 753-809] that as a graded K-module \overline{V^{0}} is isomorphic to the subspace H of ‘harmonic’ polynomials in S (𝔭). Also we have a decomposition

S (𝔭) = H ⊗ I

where I is the ring of invariants I = S (𝔭)^{K}. Finally I is itself a polynomial ring generated by ‘fundamental’K-invariants in degees in d_{1}, …, d_{n}.

Theorem 3 We have p_{0}(t) = [∏(1 − t^{d_{i}})]1.

Proof. Taking graded characters of the decomposition above, we have

S(t) = C_{0}(t)I (t)

and multiplying by D (t) we get

1 = D (t) C_{0}(t)I (t) = p_{0} (t) I (t)

Since we have

I (t) = ∏_{i} (1 + t^{d_{i}} + t^{2d_{i}} + ⋯) 1 = ∏_{i} (1 − t^{d_{i}})^{−1} 1

the result follows