ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 4, Pages 43–47 (June 11, 1998) S 1079-6762(98)00045-6

ON CHEREDNIK–MACDONALD–MEHTA IDENTITIES

PAVEL ETINGOF AND ALEXANDER KIRILLOV, JR.

(Communicated by David Kazhdan)

Abstract. In this note we give a proof of Cherednik’s generalization of Macdonald–Mehta identities for the root system A_{n−1}, using representation theory of quantum groups. These identities give an explicit formula for the integral of a product of Macdonald polynomials with respect to a “difference analogue of the Gaussian measure”. They were suggested by Cherednik, who also gave a proof based on representation theory of affine Hecke algberas; our proof gives a nice interpretation for these identities in terms of representations of quantum groups and seems to be simpler than that of Cherednik.

Introduction

In this note we give a proof of Cherednik’s generalization of Macdonald–Mehta identities for the root system A_{n−1}, using representation theory of quantum groups. These identities, suggested and proved in [Ch2], give an explicit formula for the integral of a product of Macdonald polynomials with respect to a “difference analogue of the Gaussian measure”. They can be written for any reduced root system, or, equivalently, for any semisimple complex Lie algebra 𝔤. Assuming for simplicity that 𝔤 is simple and simply laced, these identities are given by the following formula:

\begin{aligned}\frac{1}{|W |} ∫ δ_{k}\overline{δ_{k}}P_{λ}\overline{P_{μ}}γ dx & = q^{λ^{2}+(μ,μ+2kρ)}P_{μ}(q^{−2(λ+kρ)}) \\ & × q^{−2k(k−1)|R_{+}|} ∏_{α∈R_{+}} ∏_{i=0}^{k−1}(1 − q^{2(α,λ+kρ)+2i}) \end{aligned} (1)

where λ, μ are dominant integral weights, k is a positive integer, P_{λ} are Macdonald polynomials associated with the corresponding root system, with parameters q^{2}, t = q^{2k}(see [M1], [M2] or a review in [Kir2]), δ_{k} is the q-analogue of kth power of the Weyl denominator δ = δ_{1}:

δ_{k} = ∏_{α∈R^{+}} ∏_{i=0}^{k−1}(e^{α/2} − q^{−2i}e^{−α/2}), (2)

\begin{matrix}c \\ ◯ \end{matrix}1998 American Mathematical Society

Received by the editors April 14, 1998.

1991 Mathematics Subject Classification. Primary 05E35.

Key words and phrases. Macdonald polynomials.

and γ is the Gaussian, which we define by

γ = ∑_{λ∈P} e^{λ}q^{λ^{2}}, (3)

where P is the weight lattice. We consider γ as a formal series in q with coefficients from the group algebra of the weight lattice. In a more standard terminology γ is called the theta-function of the lattice P. All other notations, which are more or less standard, will be explained below.

These identities were formulated in the form we use in a paper of Cherednik [Ch2], who also proved them using double affine Hecke algebras (note: our notations are somewhat different from Cherednik’s ones). We refer the reader to [Ch2] for the discussion of the history of these identities and their role in difference harmonic analysis.

The proof uses the following identity for the Gaussian, due to Kostant [Kos]:

γ = ( ∏_{α∈R_{+}}(1 − q^{2(α,ρ)})) ∑_{ν∈P_{+}} q^{(ν,ν+2ρ)}(dim_{q} L_{ν})χ_{ν}. (4)

Here χ_{ν} is the character of the irreducible finite-dimensional module L_{ν} over 𝔤, and dim_{q} L_{ν} := χ_{ν}(q^{2ρ}) is the quantum dimension of L_{ν}.

Notations. We use the same notations as in [EK1], [EK2] with the following exceptions: we replace q by q^{−1} (note that this does not change Macdonald’s polynomials) and we use the notation φ_{λ} for “generalized characters” (see below), reserving the notation χ_{λ} for usual (Weyl) characters. In particular, we define \overline{e^{λ}} = e^{−λ}, q̄ = q, and for f ∈ ℂ_{q}[P], we define f(q^{λ}), λ ∈ P by e^{μ}(q^{λ}) = q^{(μ,λ)}. For brevity, we also write λ^{2} for (λ, λ). Finally, we denote by ∫ dx : ℂ_{q}[P] → ℂ_{q} the functional of taking the constant term: ∫ e^{λ} dx = δ_{λ,0}.

The proof

In this section, we give a proof of the Cherednik–Macdonald–Mehta identities (1) for 𝔤 = 𝔰𝔩_{n}. The proof is based on the realization of Macdonald’s polynomials as “vector-valued characters” for the quantum group U_{q}𝔰𝔩_{n}, which was given in [EK1]. For the sake of completeness, we briefly outline these results here, referring the reader to the original paper for a detailed exposition.

Let us fix k ∈ ℤ_{+} and denote by U the finite-dimensional representation of U_{q}𝔰𝔩_{n} with highest weight n(k − 1)ω_{1}, where ω_{1} is the first fundamental weight. We identify the zero weight subspace U[0], which is one-dimensional, with ℂ_{q}.

For λ ∈ P_{+}, we denote by Φ_{λ} the unique intertwiner

Φ_{λ} : L_{λ+(k−1)ρ} → L_{λ+(k−1)ρ} ⊗ U

and define the generalized character φ_{λ} ∈ ℂ_{q}[P] ⊗ U[0] ≃ ℂ_{q}[P] by φ_{λ}(q^{x}) = Tr_{L_{λ+(k−1)ρ}}(Φ_{λ}q^{x}).

We can now summarize the results of [EK1] as follows:

\begin{matrix}φ_{0} = ∏_{α∈R_{+}} ∏_{i=1}^{k−1}(e^{α/2} − q^{−2i}e^{−α/2}) = δ_{k}/δ, \\ φ_{λ}/φ_{0} = P_{λ}, \end{matrix} (5)

where P_{λ} is the Macdonald polynomial with parameters q^{2}, t = q^{2k}.

We can also rewrite Macdonald’s inner product in terms of the generalized characters as follows. Recall that Macdonald’s inner product on ℂ_{q}[P] is defined by

⟨f, g⟩_{k} = \frac{1}{|W |} ∫ δ_{k}\overline{δ_{k}}fḡ dx

(this differs by a certain power of q from the original definition of Macdonald). Obviously, one has

⟨P_{λ}, P_{μ}⟩_{k} = ⟨φ_{λ}, φ_{μ}⟩_{1}.

In order to rewrite this in terms of representation theory, let ω be the Cartan involution in U_{q}𝔰𝔩_{n} (see [EK1]). For a U_{q}𝔰𝔩_{n}-module V, we denote by V^{ω} the same vector space but with the action of U_{q}𝔰𝔩_{n} twisted by ω. Note that for finitedimensional V, we have V^{ω} ≃ V^{∗} (not canonically). Similarly, for an intertwiner Φ : L → L ⊗ U we denote by Φ^{ω} the corresponding intertwiner L^{ω} → U^{ω} ⊗ L^{ω}. Finally, for Φ_{1} : L_{1} → L_{1} ⊗ U, Φ_{2} : L_{2} → L_{2} ⊗ U, define Φ_{1} ⊙ Φ_{2}^{ω} ∈ End(L_{1} ⊗ L_{2}^{ω}) as the composition L_{1} ⊗ L_{2}^{ω} → L_{1} ⊗ U ⊗ U^{ω} ⊗ L_{2}^{ω} → L_{1} ⊗ L_{2}^{ω}, where the first arrow is given by Φ_{λ} ⊗ Φ_{μ}^{ω}, and the second by the invariant pairing U ⊗ U^{ω} → ℂ_{q} (which is unique up to a constant). Then it was shown in [EK1] that

(φ_{λ}\overline{φ_{μ}})(q^{x}) = Tr_{V} ((Φ_{λ} ⊙ Φ_{μ}^{ω})q^{Δ(x)}) = ∑_{ν∈P_{+}} χ_{ν}(q^{x})C_{λμ}^{ν}

where V = L_{λ+(k−1)ρ} ⊗ L_{μ+(k−1)ρ}^{ω} and C_{λμ}^{ν} is the trace of Φ_{λ} ⊙ Φ_{μ} acting in the multiplicity space Hom(L_{ν}, V). As a corollary, we get the following result:

\frac{1}{|W |} ∫ δδ̄φ_{λ}\overline{φ_{μ}}( ∑_{ν∈P^{+}} a_{ν}χ_{ν}) dx = ∑_{ν∈P^{+}} a_{ν^{∗}}C_{λμ}^{ν}, (6)

where ν^{∗} = −w_{0}(ν) is the highest weight of the module (L_{ν})^{∗} (here w_{0} is the longest element of the Weyl group).

Of course, the coefficients C_{λμ}^{ν} are very difficult to calculate. However, the formula above is still useful. For example, it immediately shows that ⟨φ_{λ}, φ_{μ}⟩_{1} = 0 unless λ = μ, which was the major part of the proof of the formula φ_{λ}/φ_{0} = P_{λ} in [EK1]. It turns out that this formula also allows us to prove the Cherednik– Macdonald–Mehta identities.

Theorem 1. Let φ_{λ} be the renormalized Macdonald polynomials for the root system A_{n−1} given by (5), and let γ be the Gaussian (3). Then

\begin{aligned}\frac{1}{|W |} ∫ δ\overline{δ}φ_{λ}\overline{φ_{μ}}γ dx & = q^{(λ+kρ)^{2}}q^{(μ+kρ)^{2}}φ_{μ}(q^{−2(λ+kρ)}) \\ & × ( ∏_{α∈R_{+}}(1 − q^{2(α,ρ)}))q^{−2ρ^{2}}∥P_{λ}∥^{2} dim_{q} L_{λ+(k−1)ρ}, \end{aligned} (7)

where ∥P_{λ}∥^{2} = ⟨P_{λ}, P_{λ}⟩_{k}.

Proof. From (6) and (4), we get

∫ δ\overline{δ}φ_{λ}\overline{φ_{μ}}γ dx = ( ∏_{α∈R_{+}}(1 − q^{2(α,ρ)})) ∑_{ν∈P^{+}} q^{(ν,ν+2ρ)}(dim_{q} L_{ν})C_{λμ}^{ν}. (8)

On the other hand, let C be the Casimir element for U_{q}𝔤 discussed above. Consider the intertwiner (Φ_{λ} ⊙ Φ_{μ}^{ω})Δ(C) : V → V, where, as before, V = L_{λ+(k−1)ρ} ⊗ L_{μ+(k−1)ρ}^{ω}. Then it follows from C|_{L_{λ}} = q^{(λ,λ+2ρ)} that

Tr_{V} ((Φ_{λ} ⊙ Φ_{μ}^{ω})Δ(C)Δ(q^{2ρ})) = ∑_{ν∈P_{+}} C_{λμ}^{ν}q^{(ν,ν+2ρ)} dim_{q} L_{ν},

which is exactly the sum on the right-hand side of (8). On the other hand, using Δ(C) = (C ⊗ C)(R^{21}R), we can write

Tr_{V} ((Φ_{λ} ⊙ Φ_{μ}^{ω})Δ(C)Δ(q^{2ρ}))
= q^{−2ρ^{2}}q^{(λ+kρ)^{2}}q^{(μ+kρ)^{2}} Tr_{V} ((Φ_{λ} ⊙ Φ_{μ}^{ω})(R^{21}R)Δ(q^{2ρ})).

This last trace can be calculated, which was done in [EK2, Corollary 4.2], and the answer is given by

Tr_{V} ((Φ_{λ} ⊙ Φ_{μ}^{ω})(R^{21}R)Δ(q^{2ρ})) = φ_{μ}(q^{−2(λ+kρ)})∥P_{λ}∥^{2} dim_{q} L_{λ+(k−1)ρ}.

Combining these results, we get the statement of the theorem.

The norms ∥P_{λ}∥^{2} appearing on the right-hand side of (7) are given by Macdonald’s inner product identities

∥P_{λ}∥^{2} = ∏_{α∈R_{+}} ∏_{i=1}^{k−1} \frac{1 − q^{−2(α,λ+kρ)−2i}}{1 − q^{−2(α,λ+kρ)+2i}} ,

which were conjectured in [M1], [M2] and proved for the root system A_{n−1} by Macdonald himself [M3]; see also [EK2] for the proof based on representation theory of U_{q}𝔰𝔩_{n}, and [Ch1] or a review in [Kir2] for a proof for arbitrary root systems. Using this formula and rewriting the statement of Theorem 1 in terms of Macdonald polynomials P_{λ} rather than φ_{λ}, we get the Cherednik–Macdonald–Mehta identities (1).

Remarks. 1. Note that the left-hand side of (7) is symmetric in λ, μ. Thus, the same is true for the right-hand side, which is exactly the statement of Macdonald’s symmetry identity (compare with the proof in [EK2]).

2. The proof of Cherednik–Macdonald–Mehta identities given above easily generalizes to the case when q is a root of unity (see [Kir1] for the discussion of the appropriate representation-theoretic setup). In this case, we need to replace the set P_{+} of all integral dominant weights by an appropriate (finite) Weyl alcove C (see [Kir1]), and the integral ∫ δδ̄f dx should be replaced by const ∑_{λ∈C} f(q^{2(λ+ρ)}) dim_{q} L_{λ}. Using the following obvious property of the Gaussian:

γ(q^{2(λ+ρ)}) = q^{−(λ,λ+2ρ)}γ(q^{2ρ})

(which in this case coincides with formula (1.7) in [Kir1]), it is easy to see that in this case the Cherednik–Macdonald–Mehta identities are equivalent to

S^{−1}T^{−1}S = T ST,

where the matrices S, T are defined in [Kir1, Theorem 5.4]. This identity is part of a more general result, namely, that these matrices S, T give a projective representation of the modular group SL_{2}(ℤ) on the space of generalized characters (see [Kir1, Theorem 1.10] and references therein).

References

[Ch1] I. Cherednik, Double affine Hecke algebras and Macdonald’s conjectures, Annals of Math. 141 (1995), 191–216. MR 96m:33010

[Ch2] , Difference Macdonald–Mehta conjecture, Internat. Math. Res. Notices 1997, 449– 467. CMP 97:12

[EK1] P. Etingof and A. Kirillov, Jr., Macdonald’s polynomials and representations of quantum groups, Math. Res. Let. 1 (1994), 279–296. MR 96m:17025

[EK2] , Representation-theoretic proof of the inner product and symmetry identities for Macdonald’s polynomials, Compos. Math. 102 (1996), 179–202. MR 97j:17013

[Kas] C. Kassel, Quantum groups, Springer-Verlag, New York, 1995. MR 96e:17041

[Kir1] A. Kirillov, Jr., On an inner product in modular tensor categories, J. Amer. Math. Soc. 9 (1996), 1135–1169. MR 97f:18007

[Kir2] , Lectures on affine Hecke algebras and Macdonald’s conjectures, Bull. Amer. Math. Soc. 34 (1997), 251–292. CMP 97:13

[Kos] B. Kostant, On Macdonald’s η-function formula, the Laplacian and generalized exponents, Advances in Math. 20 (1976), 179–212. MR 58:5484

[M1] I. G. Macdonald, A new class of symmetric functions, Publ. I.R.M.A. Strasbourg, 372/S-20, Actes 20 Séminaire Lotharingien (1988), 131–171.

[M2] ,Orthogonal polynomials associated with root systems, preprint (1988).

[M3] ,Symmetric functions and Hall polynomials, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1995. MR 96h:05207

Department of Mathematics, Harvard University, Cambridge, MA 02138

   E-mail address: etingof@math.harvard.edu

Department of Mathematics, MIT, Cambridge, MA 02139

   E-mail address: kirillov@math.mit.edu