Certain topics on the representation theory of the Lie algebra \(gl(\lambda)\)
The paper surveys recent results in the representation theory of the Lie algebra \(g=gl(\lambda)\) and some applications to combinatorial identities. The Lie algebra \(g\), where \(\lambda\) is a complex parameter, is a continuous version of the Lie algebra \(gl_{\infty}\) and arises from the homology ring of the Lie algebra of differential operators on the line. It is \({\mathbb Z}\)-graded and possesses an analogue of the Cartan decomposition. It is known that the problem of irreducibility of \(g\)-modules can be reduced to the corresponding problem for \(\widehat{gl}_{\infty}\)-modules. In Chapter 1 the author describes the structure of \(\widehat{gl}_{\infty}\)-modules induced from the largest parabolic subalgebra. In particular, he describes the Jantzen filtration and computes the determinant of the Shapovalov form. As a corollary, one obtains a character formula for the first consecutive module of the Jantzen filtration, which can be applied to get the classical and ``higher'' Euler identities. The subject of Chapter 2 is the representation theory of \(g\). The author describes embeddings into \(\widehat{gl}_{\infty}\) and deduces a criterion for reducibility of those representations of \(g\) that are induced from the special parabolic subalgebra. This is applied to obtain certain identities called ``local''. Finally, in Chapter 3 the author considers the Lie algebra of functions on a hyperboloid, which can be flat deformed in \(g\). Then the author applies all previous results to obtain a new combinatorial identity, which is called ``global''.
scientific article; zbMATH DE number 1269458
scientific article