Lie algebra cohomology and generating functions (Q1770292)

Let \(\mathfrak g\) be a simple Lie algebra over \(\mathbb C\), \(\mathfrak b\) a Borel subalgebra of \(\mathfrak g\), \(\mathfrak h\) a Cartan subalgebra of \(\mathfrak b\), \(\mathfrak n=[\mathfrak b, \mathfrak b]\) the nilradical of \(\mathfrak b\), \(W\) the Weyl group, \(V\) an \(\mathfrak b\)-module. In the paper the cohomology \(H^*(\mathfrak n, V)\) is investigated. The cochain complex \(C^*(\mathfrak n, V)\) is a direct sum of weight subcomplexes \(C^*_{\nu }, ~\nu \in \mathfrak h^*\). The investigation is based on the study of the generating function \[ F_V(t,x)=\sum _{k, \nu }\dim C^k_{\nu }t^kx^{\nu }. \] Since \(\sum _k(-1)^k\dim H^k_{\nu }=\sum_k(-1)^k\dim C^k_{\nu },\) one has an inequality \[ \dim H^*\geq \sum _{\nu }\Bigl| \sum _k(-1)^k\dim C^k_{\nu }\Bigr| . \] The modules for which the equality is attained are called blue ones. It is proved that any irreducible \(\mathfrak g\)-module is blue. If \(\mathfrak g\) is of the type \(A_r\) then \(V=\mathfrak n\) is blue as well. It is shown that \(\dim H^*(\mathfrak n, V)\geq | W| \) for any blue \(\mathfrak b\)-module \(V\). Note that according to the Borel-Weil-Bott theorem \(\dim H^*(\mathfrak n, V)=| W| \) for any irreducible \(\mathfrak g\)-module \(V\). For \(\mathfrak g =A_r\) \(\dim H^*(\mathfrak n, \mathfrak n)\) and \(\dim H^k(\mathfrak n, \mathfrak n), ~k\leq 3\) are calculated by counting the coefficients of the function \(G_{\mathfrak n}(x)=F_{\mathfrak n}(-1,x)\); \(\dim H^k(\mathfrak n, \mathfrak n)\) as a function of \(k\) and \(r=rk g\) is investigated. At the end of the paper the results and open problems for \(\mathfrak g\neq A_r\) are discussed. Some of the results of the paper were announced earlier in [Russ. Math. Surv. 34, No. 1, 243--244 (1979); translation from Usp. Mat. Nauk 34, No. 1, 245--246 (1979; Zbl 0427.17010), Problems in group theory and homological algebra, 28--38 (1981; Zbl 0495.17008), Funct. Anal. Appl. 17, No. 3, 207--212 (1991); translation from Funkt. Anal. Prilozh. 17, No. 3, 55--60 (1983; Zbl 0533.17007), Ukr. Math. J. 42, No. 9, 1137--1141 (1990); translation from Ukr. Mat. Zh. 42, No. 9, 1278--1283 (1990; Zbl 0718.17018), Russ. Math. Surv. 48, No. 1, 193--194 (1993); translation from Usp. Mat. Nauk 48, No. 1 (289), 185--186 ) (1993; Zbl 0811.17023)].