On generalized Peterson variety
Date:
Abstract: The Peterson variety is a subvariety of the full flag variety and arises as a special case of regular nilpotent Hessenberg varieties. In type A , it can be defined as the set of flags \(\left(F_1, F_2, \ldots, F_n\right)\) such that \(X F_i \subseteq F_{i+1}\), where \(X\) is a fixed regular nilpotent operator. First studied by D. Peterson and B. Kostant, later by E. Insko, A. Yong, and also others, the Peterson variety has deep connections to Lie theory, algebraic geometry, and geometric representation theory. In this talk, we will review the known results about its geometric structure.
Meanwhile, much less is known about the generalized case where \(X\) is allowed to be an arbitrary (possibly non-regular) nilpotent operator. In this talk, we will explore the geometry of these generalized Peterson varieties, based on the speaker’s ongoing research. Surprisingly, these varieties exhibit rich structure and turn out to be closely connected to the representation theory of the special linear Lie algebra \(\mathfrak{s} \mathfrak{l}_r\).