A Note on Exponents vs Root Heights for Complex Simple Lie Algebras

Abstract

We give an elementary combinatorial proof of a special case of a result due to Bazlov and Ion concerning the Fourier coefficients of the Cherednik kernel. This can be used to give yet another proof of the classical fact that for a complex simple Lie algebra g, the partition formed by the exponents of g is dual to that formed by the numbers of positive roots at each height. Let g be a finite dimensional, complex simple Lie algebra of rank n with associated root system ∆, simple roots αi (i = 1 · · ·n) and set of positive roots ∆ . Let Q be the root lattice of g and Q denote the set comprising Z linear combinations of the αi. For each α ∈ Q, let e denote the corresponding formal exponential; these satisfy the usual rules: e = 1 and e = e e. We define A := Q[t] [[e1 , · · · , en ]]. Thus a typical element of A is a power series of the form ∑ β∈Q+ cβ(t)e −β where each cβ(t) ∈ Q[t]. Consider the element ξ ∈ A defined by: ξ := ∏ α∈∆+ 1 − e 1 − te−α = ∏ α∈∆+ (1 + (t − 1)e + t(t − 1)e + t(t − 1)e + · · · ) (1) Given β = ∑n i=1 biαi ∈ Q , define its height to be ht β := n ∑ i=1 bi The main objective of this short note is to give an elementary combinatorial proof of the following proposition: the electronic journal of combinatorics 13 (2006), #N22 1 Proposition 1 For β ∈ ∆, the coefficient of e in ξ is ( t − t ) This proposition is the q = 0 case of a more general (q, t) theorem obtained by Bazlov [1] and Ion [2]. They consider K̃(q, t) = ∏ α∈∆+ ∏ i≥0 (1 − qe)(1 − qe) (1 − tqie−α)(1 − tqi+1eα) If [K̃(q, t)] denotes the constant term (coefficient of e) of K̃(q, t), one defines C̃(q, t) := K̃(q, t)/[K̃(q, t)] (upto a minor difference in convention, this is called the Cherednik kernel in [2] ). Bazlov and Ion compute the coefficient of e in C(q, t) for β a positive root of g. Their approaches use techniques from Cherednik’s theory of Macdonald polynomials. When q = 0, C̃(0, t) reduces to ξ introduced above. Though proposition 1 is only a special case, it has a very interesting consequence. Ion showed [2] that it can be used to give a quick and elegant proof of the classical fact that for a finite dimensional simple Lie algebra g, the partition formed by listing its exponents in descending order is dual to the partition formed by the numbers of positive roots at each height (see below). This fact, first observed empirically by Shapiro and Steinberg was later proved by Kostant [3] using his theory of principal three dimensional subalgebras of g and by Macdonald [4] via his factorization of the Poincaré series of the Weyl group of g. The motivation for our approach to proposition 1 is to thereby obtain a proof of this classical fact via elementary means (bypassing Macdonald-Cherednik theory). For completeness sake, we first quickly recall [2] how one can use proposition 1 to deduce the classical fact concerning exponents and heights of roots. Let P denote the weight lattice of g and W its Weyl group. For our definition of exponents, we use the Kostka-Foulkes polynomial Kα̃,0(t) where α̃ is the highest long root of g. It is well known that this is given by

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