A geometric realization of quantum groups of type \(\mathbf D\)
There are several models to realize quantum groups, such as the Ringel-Hall algebra approach and the geometry approach, etc. \textit{A. A. Beilinson, G. Lusztig} and \textit{R. MacPherson} [Duke Math. J. 61, No. 2, 655--677 (1990; Zbl 0713.17012)] gave a geometric realization of the quantum deformation of \(\mathfrak{gl}_n\). First, the \(q\)-Schur algebra \(K_d\), a quotient of the quantum \(\mathfrak{gl}_m\), is constructed by using the geometry of pairs of \(m\)-step flags on a \(d\)-dimensional vector space over a finite field. Then, the behavior of the structure constants of \(K_d\) is studied when \(d\) increases by a multiple of \(m\). Finally, the quantum \(\mathfrak{gl}_m\) is obtained via taking limit. The generalization of this approach to quantum affine \(\mathfrak{gl}_m\) was done by \textit{V. Ginzburg} and \textit{E. Vasserot} [Int. Math. Res. Not. 1993, No. 3, 67--85 (1993; Zbl 0785.17014)], and independently by \textit{G. Lusztig} [Asian J. Math. 3, No. 1, 147--177 (1999; Zbl 0983.17011)]. The paper under review considers a variety consisting of certain ``ramified'' \((2m-2)\)-step flags in a fixed \(d\)-dimensional vector space over a finite field. It is shown that the space of \(\text{GL}(d)\)-invariant functions on the double \(X\times X\), equipped with a natural convolution product, receives an algebra homomorphism from the quantum group of type \(D_m\). Therefore, the author gives a geometric realization of certain quotients of the quantum group of type \(D_m\), in analogy with the geometric construction of \(K_d\).
scientific article; zbMATH DE number 5712554