M ar 2 00 8 On q-deformed gl l + 1-Whittaker function I
M ar 2 00 8 On q-deformed gl l + 1-Whittaker function I
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Mar 2 00 8 关于 q 变形 gl l 1-Whittaker 函数 I
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发表时间：
2008
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通讯作者：
S. Oblezin
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作者：
A. Gerasimov;D. Lebedev;S. Oblezin
We propose new explicit form of q-deformed Whittaker functions solving q-deformed gll+1-Toda chains. In the limit q → 1 constructed solutions reduce to classical class one gll+1Whittaker functions in the form proposed by Givental. An important property of the proposed expression for the q-deformed gll+1-Whittaker function is that it can be represented as a character of C∗ × GL(l + 1). This provides a q-version of the Shintani-Casselman-Shalika formula for p-adic Whittaker function. The Shintani-Casselman-Shalika formula is recovered in the limit q → 0 when the q-deformed Whittaker function is reduced to a character of a finite-dimensional representation of gll+1 expressed through Gelfand-Zetlin bases. Introduction Whittaker functions corresponding to semisimple finite-dimensional Lie algebras arise in various parts of modern mathematics. In particular, these functions appear in representation theory as matrix elements of infinite-dimensional representations, in the theory of quantum integrable systems as common eigenfunction of Toda chain quantum Hamiltonians, in string theory as generating functions of correlators in Type A topological string theory on flag manifolds and in number theory in a description of local Archimedean L-factors corresponding to automorphic representations. Although much studied, Whittaker functions seems have some deep properties that are not yet fully revealed. In this paper we study the q-deformed gll+1-Whittaker functions. The q-deformed Whittaker function can be identified with a common eigenfunction of a set of commuting q-deformed Toda chain Hamiltonians. This q-deformed Toda chain (also known as the relativistic Toda chain [Ru]) was discussed in terms of representation theory of quantum groups in [Se1], [Et], [Se2] and an integral representation for the q-deformed gll+1-Whittaker function was constructed in [KLS]. Recently the q-deformed Toda chain attracts special interest due to its connection with quantum K-theory of flag manifolds [GiL]. In this paper we pursue another direction. Our principal motivation to study q-deformed Whittaker functions is that in this, more general setting, some important hidden properties of classical Whittaker functions become visible. The main result of the paper is given by Theorem 2.1 where a new expression for the qdeformed gll+1-Whittaker function ( for q < 1) is introduced. As a simple corollary of Theorem 2.1, the q-deformed gll+1-Whittaker function can be represented as a character of C ∗ ×GL(l+ 1). In the limit q → 1 this leads to a similar representation of classical gll+1-Whittaker function. This representation is not easy to perceive looking directly at the classical Whittaker functions. The importance of this representation of (q-deformed) gll+1-Whittaker function becomes obvious if we notice that in the limit q → 0 the constructed q-deformed Whittaker function reduces to p-adic Whittaker function. In this limit the representation as a character reduces to well-known Shintani-Casselman-Shalika representation of p-adic GLl+1-Whittaker function as a character of a 1 finite-dimensional representation of GLl+1 [Sh],[CS]. Thus the representation of (q-deformed) gll+1Whittaker function as a character can be considered as a q-version of Shintani-Casselman-Shalika representation. Indeed, the constructed q-deformed Whittaker function is equal to zero outside a dominant weight cone of gll+1 similarly to the Shintani-Casselman-Shalika p-adic Whittaker function. We expect that the representation of the classical Whittaker function as a character should provide important insights into the arithmetic geometry at an infinite place of Spec(Z). Let us also remark that taking into account results [CS] one should expect that in the case of an arbitrary semisimple Lie algebra g, q-deformed g-Whittaker function should be given by a character of C × LG(C) where Lie(LG) = Lg is a Langlands dual Lie algebra. It is worth mentioning that the q → 1 limit of the explicit expression of the q-deformed Whittaker function proposed in this paper reduces to the integral representations for classical Whittaker functions introduced by Givental [Gi],[GKLO]. We consider this as a sign of an “arithmetic nature” of this integral representation. On the other hand the explicit solution has an obvious relation with Gelfand-Zetlin parametrization of finite-dimensional representations of gll+1 (and precisely reproduces Gelfand-Zetlin form of characters of finite-dimensional representations in the limit q → 0). This duality of Gelfand-Zetlin and Givental representations was already noticed in [GLO]. Let us comment on our approach to derivation of explicit expressions for q-deformed Whittaker functions. It is known [Et] that defining difference equations for Macdonald polynomials are transformed into q-deformed Toda chain eigenfunction equations in a certain limit. This is a simple generalization of the Inozemtsev limit [I] transforming Calogero-Sutherland integrable model into standard Toda chain. The other ingredient we use is a recursive construction of Macdonald polynomials (analogous to the recursive construction for (q-deformed) Toda chain eigenfunctions [KL1], [KLS]). We combine these results to obtain recursive expression for q-deformed gll+1-Whittaker functions satisfying q-deformed gll+1-Toda chain eigenfunction equations. The explicit form of the q-deformed Whittaker function implies various interesting interpretations. This includes connections with representation theory (via characters of Demazure modules), geometry of quiver varieties, quantum cohomology of flag manifolds and will be discussed elsewhere [GLO2]. Finally note that eigenfunctions of q-deformed Toda chain were discussed previously (e.g. [KLS],[GKL1],[BF] and [FFJMM]). The relation of these constructions with the one proposed in this paper is an interesting question which deserves further considerations. The paper is organized as follows. In Section 1 we recall a systems of mutually commuting difference Macdonald-Ruijsenaars operators and recursive construction of their common eigenfunctions. In Section 2 we derive recursive expression for solutions of q-deformed gll+1-Toda chain. In Section 3 various limiting cases elucidating the construction of the q-deformed gll+1-Whittaker functions are discussed. In Section 4 details of the proof of the Theorem 2.1 are given. Acknowledgments: The research of AG was partly supported by SFI Research Frontier Programme and Marie Curie RTN Forces Universe from EU. The research of SO is partially supported by RF President Grant MK-134.2007.1. 1 Macdonald-Ruijsenaars difference operators In this section we recall relevant facts from the theory of Macdonald polynomials ( see e.g. [Mac], [Kir], [AOS]). 2 Consider symmetric polynomials in variables (x1, . . . , xl+1) over the field Q(q, t) of rational functions in q, t. Given a partition Λ = (0 ≤ Λ1 ≤ Λ2 ≤ . . . ≤ Λl+1), denote by the same symbol Λ the Young diagram containing l+ 1 rows with Λk boxes in the k-th row; and the upper row having the maximal length Λl+1. Let mΛ and πΛ be polynomial basises of the space of symmetric polynomials indexed by partitions Λ: mΛ = ∑ σ∈Sl+1 x1 σ(1)x Λ2 σ(2) · . . . · x Λl+1 σ(l+1), πΛ = πΛ1πΛ2 · . . . · πΛl+1, πn = l+1 ∑ k=1 xk , where Sl+1 is the permutation group. Define a scalar product 〈 , 〉q,t on the space of symmetric functions over Q(q, t) as follows 〈πΛ, πΛ′〉q,t = δΛ,Λ′ · zΛ(q, t), where zΛ(q, t) = ∏
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