Lie algebra homology and the Macdonald-Kac formulas
I. G. Macdonald's noted identities which are a highly compact generalization of famous ``theta-function Identity of Jacobi'' were originally obtained by means of ``affine root systems'' [Invent. Math. 15, 91--143 (1972; Zbl 0244.17005)]. These formulas have subsequently been interpreted by V. G. Kac and R. V. Moody as precise analogues of Weyl's denominator formula for the ``Euclidean Lie algebras''. Kac also sketches a new proof of the much wider class of identities -- the analogues of both Weyl's character and denominator formulas for a family of Lie algebras considerably more general than Euclidean Lie algebras which are introduced also by Kac and Moody by symmetrizable Cartan matrices. The main purpose of the present paper is to generalize B. Kostant's fundamental result on the homology (or cohomology) of nilradicals of parabolic subalgebras in certain modules from finite dimensional complex semisimple Lie algebras to the Kac-Moody Lie algebras. The authors obtain the results including the Macdonald identities, as immediate consequences of the Euler-Poincaré principle just as B. Kostant derived Weyl's character and denominator formulas from his homology theorem [\textit{B.~Kostant}, Ann. Math. (2) 74, 329--387 (1961; Zbl 0134.03501)]. Let \(\mathfrak g=\coprod\limits_{(n_1,\dots,n_\ell)\in\mathbb Z^\ell}\mathfrak g(n_1,\dots,n_\ell)\) be the Lie algebra defined by an \(\ell\times\ell\)-Cartan matrix \(A=(A_{ij})\). Let \(D_i\) \((1\leq i\leq\ell)\) be the \(i\)-th degree derivation of \(\mathfrak g\) that is the derivation which acts on \(\mathfrak g(n_1,\dots,n_\ell)\) as scalar multiplication by \(n_i\). Then \(D_1,\dots,D_\ell\) span an \(\ell\)-dimensional space \(\mathfrak v_0\) of commuting derivations of \(\mathfrak g\). Let \(\mathfrak v\) be a subspace of \(\mathfrak v_0\). Since \(\mathfrak v\) may be regarded as an abelian Lie algebra acting on the \(\mathfrak v\)-module \(\mathfrak g\) by derivations, we may form the semi-direct product Lie algebra \(\mathfrak g^e=\mathfrak v\times\mathfrak g\) with respect to this action. Then \(\mathfrak h^e=\mathfrak v+\mathfrak h\), where \(\mathfrak h=\mathfrak g(0,\dots,0)\) is an abelian Lie algebra of \(\mathfrak g^e\) which acts via scalar multiplication on each space \(\mathfrak g(n_1,\dots,n_\ell)\) Define \(\alpha_1,\dots,\alpha_\ell\in(\mathfrak h^e)*\) by conditions \([h,e_i]=\alpha_i(h)e_i\) for all \(h\in\mathfrak h^e\) and all \(i = 1,2,\dots,\ell\). Note that \(\alpha_j(h_i)=A_{ij}\) for all \(i,j=1,\dots,\ell\) and \(\mathfrak g=\coprod\limits_{\varphi\in(\mathfrak h^e)^*}\mathfrak g^\varphi\) where \(\mathfrak g^\varphi =\{x\in\mathfrak g;[h,x]=\varphi(h)x\}\). Assume that \(\mathfrak v\) is chosen so that \(\alpha_1,\dots,\alpha_\ell\) are linearly independent. Every subset \(S\) of \(\{1,\dots,\ell\}\) defines in the obvious way a square submatrix \(B\) of the Cartan matrix \(A\). Assume that \(S\) is of finite type, namely, \(B\) is a classical Cartan matrix of finite type. Let \(\mathfrak g_S=\coprod\limits_{\varphi\in\Delta^s}\mathfrak g^\varphi\) where \(\Delta^S=\Delta\cap\coprod\limits_{i\in S}\mathbb Z\alpha_i\) be the subalgebra of \(\mathfrak g(A)=\mathfrak g\) which is isomorphic to the finite dimensional semisimple Lie algebra \(\mathfrak g(B),\mathfrak h_s\) be the span of \(\{h_i\}_{i\in s}\) \(\mathfrak r=\mathfrak g_s+\mathfrak h\) and \(\mathfrak p=\mathfrak r\oplus\mathfrak u\) (resp. \(\mathfrak p^e=\mathfrak r^e\oplus\mathfrak u\)) where \(\mathfrak u=\coprod\limits_{\varphi\in\Delta_+,\notin\Delta^S}\mathfrak g^\varphi\) is called the \(F\)-parabolic subalgebra of \(\mathfrak g\) (resp. of \(\mathfrak g^e\)). The authors simplify I. N. Bernstein, I. M. Gelfand and S. I. Gelfand's results an the complex semisimple Lie algebras by using Casimir element in place of the Harish-Chandra theorem and refine and extend their method so as to obtain Kostantis homology theorem in full generality and carry out the argument for the present Lie algebras. Let \((\mathfrak h^e)_Z^*\) be the set of integral linear forms, i.e., the set of all \(\lambda\in(\mathfrak h^e)^*\) such that \(\lambda(h_i)\in\mathbb Z\) for all \(i=1,\dots,\ell\). Now, let \(\mathfrak E\) be the category of all \(\mathfrak h^e\)-modules \(X\) such that \(X\) has a direct sum decomposition \(\coprod\limits_{\lambda\in(\mathfrak h^e)_{\mathbb Z}^*}X_\lambda\) and \(\dim X_\lambda<\infty\). Define the formal character \(\chi(X)=\sum\limits_{\lambda\in(\mathfrak h^e)_{\mathbb Z}^*}(\dim X_\lambda)e(\lambda)\) where \(e(\lambda)\) be an element of the abelian group of all formal integral combinations of elements of \((\mathfrak h^e)_{\mathbb Z}^*\). Let \(C_*\) be an admissible chain complex in \(\mathfrak E\), that is a chain complex such that \(\coprod\limits_jC_j\in\mathfrak E\),then \(\coprod\limits_jH_j\in\mathfrak E\) where \(H_j\) is the \(j\)-th homology group and \(\sum\limits_{j\in \mathbb Z_+}(-1)^j\chi(C_j)=\sum\limits_{j\in \mathbb Z_+}((-1)^j\chi(H_j)\). The authors show the following identities: \[ \prod\limits_{\varphi\in\Delta^+}(1-e(-\varphi))^{\dim\mathfrak g^\varphi}=\sum\limits_{w\in W}(\det w)e(w\rho-\rho)\tag{1} \] \[ \chi(X)=\sum\limits_{w\in W}(\det w)e(w(\mu+\rho))/\sum\limits_{w\in W}(\det w)e(w\rho).\tag{2} \] Macdonald's identity comprises the special case of (1) in which \(\mathfrak g\) is a Euclidean Lie algebra. When \(\mathfrak g^e=\mathfrak g\) is a finite dimensional split semi-simple Lie algebra, (2) is just Weyl's character formula and (1) is Weyl's denominator formula and Macdonald's specialized identity for Laurent series Lie algebra corresponding to (2) is Jacobi's identity \(\prod\limits_{n\geq 1}(1-x^n)^3 = \sum(-1)^n(2n+1)x^{n(n+1)/2}\) (\(x\) an indeterminate).
scientific article; zbMATH DE number 3558014
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