Kazhdan-Lusztig-type multiplicity formula for symmetrizable generalized Kac-Moody algebras
This is an announcement of the author's following result. Let \({\mathfrak g}(A)\) be a generalized Kac-Moody algebra with a symmetrizable GGCM \(A=(a_{ij})_{i,j\in I}\) satisfying the condition that \(a_{ii}\neq 0\) \((i\in I)\). Let \(\Lambda\in P_+\) and \((w,\beta),(z,\beta)\in W\times A(\Lambda)\), where \(A(\Lambda)\) is a certain subset of an imaginary root lattice. The Verma module \(V((w,\beta)\circ\Lambda)\) and its irreducible quotient \(L((w,\beta)\circ\Lambda)\) are defined and the multiplicity \([V((w,\beta)\circ\Lambda):L((z,\beta)\circ\Lambda)]\) makes sense. The main result says that this multiplicity equals the value at \(q=1\) of the Kazhdan-Lusztig polynomial \(P_{w,z}(q)\).
scientific article; zbMATH DE number 701458