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A Recursive Formula for Characters of Simple Lie Algebras View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 137, 126-144 (1991) A Recursive Formula for Characters of Simple Lie Algebras STEVEN N. KASS* Centre de recherches mathkmatiques, UniversitP de MontrPal C. P. 6128, &cc. “A”, MontrPal, Qutbec, Canada H3C 357 Communicated by Robert Steinberg Received July 20, 1988; accepted July 18, 1989 In 1964, Antoine and Speiser published succinct and elegant formulae for the characters of the irreducible highest weight modules for the Lie algebras A, and B,. (The result for A, may have been known as early as 1957.) These can be derived from a recursive formula for characters valid for all simple complex Kac-Moody Lie algebras for which the Weyl-Kac character formula holds. 0 1991 Academic Press. Inc. 1. INTRODUCTION To avoid the encumbrance of technicalities, we present the results first for simple finite-dimensional Lie algebras over C. We use the following notation, generally following Humphreys [H]. Let L be a finite-dimensional complex simple Lie algebra of rank 1. We denote by H a Cartan subalgebra of L, by ai (1 Q i < I) a choice of simple roots, by Q the integral lattice they generate, and by R, R+, and R- the root system, positive roots, and negative roots with respect to H and the ai, respectively. Let A be the weight lattice for L, and li (1~ i,< I) the fundamental weights corresponding to the choice of simple roots. The set of dominant weights will be denoted A +. For two weights p, A, we write p<I when A-pee+, the positive root lattice. We define the height of an element of Q+ to be the sum of its coefficients when expressed in the basis of simple roots. The lattices Q and A lie in H*, the dual vector space of H. We denote the non&simple positive roots by yi (1s i < r = $(dim L - 31)). This is our first non-standard definition. Let YV be the Weyl group of L. The Weyl group acts on the real sub- space of H spanned by the cli and is generated by reflections ri (1~ i < 1), * Present address: Department of Mathematics and Computer Science, Drew University, Madison, NJ 07940. 126 0021-8693/91 $3.00 Copyright 0 1991 by Academic Press, Inc. All rights of reproduction in any form reserved. CHARACTERSOFSIMPLELIE ALGEBRAS 127 in hyperplanes orthogonal to the ai. Let sgn(w) be the sign (or determi- nant) of the element w of YY. Let V(A) be the irreducible L-module of highest weight A. This module is finite-dimensional if and only if A E A +. Let Z?(A) be the set of weights of V(A), giving the weight space decomposition V(A) = OfiEncAJV(A),, where the dimension of V(A),, written mi(p), is the multiplicity of ,n in V(A). PROPOSITION 1.1. The dominant weights in l7(A), A+ n n(k), are the set ofweights {PEA+ Ip<l}. ProoJ [H, Sections 13.4, 21.31. 1 The function mA-:A + B taking p to m,(p) is called the formal character of V(A), and will usually be written x1. If we write ep for the characteristic function of {p} on A, we have xA = CA. P ml(p) e”. Let Z be the set of all complex-valued functions on A whose support lies in a finite union of sets of the form {CLE A ( p < v }. The characters of highest weight modules lie in Z, the representation ring of L, in which it will be convenient to work. With the product * given by (g * h)(l) = C,f(p) g(n - p), 6 becomes a commutative, associative complex algebra and an integral domain. This product gives ea * e” = ep+ ’ . We will generally omit the symbol *. To elements g of B with finite support we extend the action of YV from .4 by the rule (wg)(p) =g(w -lp). This is consistent with. the identification of p with e”, since w(e”) = ewcI.The Weyl group respects addition, scalar multi- plication and the product in E. The multiplicative identity in E, e”, is denoted 1. The Weyl group leaves invariant the character xl for each AEA+ [H]. For PEQ+, define the elementfP8:= (1+e-8+e-28+ ...)EZ. Note that f@ is not an element of 8. Then we have PROPOSITION 1.2. In 2, and for /?E Q +, (1 -e-B)-‘=f-B and (1 -eB)-’ = -e-Bf-P 2. KNOWN RESULTS The most important expression for x1 is due to Weyl. Let 6=1,+ ... + A,, and define N(q), for q E H*, by N(q) = CwaW sgn(w) e”“I. We will refer to N(6) often, and will denote it by A. THEOREM 2.1. (Weyl’s character formula). If A E A +, then xn = N(1+ 6)/A. Proof: [H, Section 241. m 481/137/l-9 128 STEVEN N.KASS THEOREM 2.2 (Weyl’s denominator formula). d = e6 n, ER+ (I- e ~ “). Proof: [H, Section 241. 1 Another formula for the character, due to Kostant, requires the follow- ing definition. DEFINITION 2.3. For PE/~, define p(n) to be the number of distinct ways in which -,u can be written as a sum of positive roots, allowing repetition. THEOREM 2.4. (Kostant). IfA E A+, then or equivalently, xi =p * ec’ * N(I + 6). Proof: [H, Section 24.21. 1 In [AS], Antoine and Speiser developed a geometric interpretation of Theorem 2.4, equivalent to Kostant’s weight multiplicity formula [H, Sec- tion 24.21. Applied to characters of the simple algebras A2 and B,, explicit formulae were obtained. Define F(J) E E to be the characteristic function of n(J). From now on we will sometimes refer to the weight 1 =Ci= l,...,rmijli by the I-tuple (ml, m2, .. ml). THEOREM 2.5 [AS]. (A) Let L = A, and let m and n be non-negative integers. Then ifman, ~~,,,~)=F(m,n)+F(m- l,n- l)+ ... +F(m-n,O). If m < 4 x(~,~) =F(m,n)+F(m-l,n-l)+ . +F(O,m-n). (B) Let L = B, and let m and n be non-negative integers. Then if n is odd. xc,,,, = F(m, n) + F(m, n - 2) + . + F(m, 1) +F(m-l,n)+F(m-l,n-2)+ ... +F(m-1,l) + . +F(O,n)+F(O,n-2)+ ... +F(O, 1). If n is even, CHARACTERS OF SIMPLE LIE ALGEBRAS 129 x(~,~) = F(m, n) + F(m, n - 2) + . + F(m, 0) +F(m-l,n)+F(m-l,n-2)+ ... +F(m-1,2) +F(m-2,n)+F(m-2,n-2)+ ... +F(m-2,O) + . + F(0, n) + F(0, n - 2) + . + F(O,j), where j is 0 or 2 depending on whether m is even or odd. Proof: [B]. These theorems will also follow from our Corollary 4.5. 1 A geometric interpretation of Theorem 2.5B for the weight (2,2) is given in Fig. 1. Y2,21 . F(2,2)+FIl,2)+F~O,2)+F~2,O)+F(o.oI FIGURE I Braden used the following lemmas to prove Theorem 2.5: LEMMA 2.6A. Let L = A, and let ,I = (m, n) E A+. (a) Zfm>O andn>O, then AF(m,n)=N(m+l,n+l)-N(m,n). (b) Zf mn = 0, then AF(m, n) = N(m + 1, n + 1). LEMMA 2.6B. Let L = B, and let 1= (m, n) E A+. (a) Ifm+n<l, then AF(m,n)=N(m+l,n+l). (b) Zfm=O, n>l, then AF(m,n)=N(l,n+l)-N(l,n-1). (c) Zfm>l,n=O, thenAF(m,n)=N(m+l,l)-N(m-1,l). 130 STEVEN N. KASS (d) Zfm>,l, n=l, then dF(m,n)=N(m+1,2)-N(m,2). (e) rfma 1, n> 1, or ifm> 1, n> 1, then dF(m,n)=N(m+1,n+1)-N(m,n+1)-N(m+1,n-1)+N(m,n-1). In the next section we generalize these lemmas and derive the main theorem of the paper. 3. THE MAIN THEOREM The casesinto which Lemmas 2.6 were divided are unnecessary.To show this, we need the following lemma, which follows immediately from the definition of N(A). LEMMA 3.1. For AEH* andw~W, N(A)=sgn(w)N(wil). COROLLARY3.2. Zf ,I E H* and a1 = 1 for some CJE W, o # l,, then N(1) = 0. Proof: In this case, I must be fixed by some element w E w of sign - 1 (see [H, Section 241). Using this w in the lemma gives N(A) = -N(1). 1 With this observation, we can simplify Lemmas 2.6 to LEMMA 3.3A. Let L= A, and let ;1= (m, n)~ A+. Then AF(m, n) = N(m + 1, n + 1) - N(m, n). LEMMA 3.3B. Let L= B, and let A= (m, n)EA+. Then AF(m, n)= N(m+1,n+1)-N(m,n+1)-N(m+1,n-1)+N(m,n-l). Table I in the Appendix gives the relevant information about the Weyl groups of A, and B, needed to confirm this assertion. Lemmas 3.3 will follow from a single lemma which holds for all L. For this we need two more definitions. Recall that yi (1 < i< r) are the non- simple positive roots of L, and define pi (0 < i < n = 2’) to be the sums of distinct yi.
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