Hessenberg Varieties and Generalized Eulerian Numbers for Semisimple Lie Groups: The Classical Cases.

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Certain subvarieties of flag manifolds arise from the study of Hessenberg and banded forms for matrices. For a matrix A gl(n,C) (or sl(n,C)) and a nonnegative integer p, the p^th Hessenberg variety of A is the subvariety of the (complete) flag manifold consisting of those flags (S_1, . . . ,S_{n-1} ) satisfying the condition AS_i {IS A SUBSET OF, SYMBOL} S_{i+p}, {UPSIDE DOWN A}i. The definition of these varieties extends to an arbitrary connected complex semisimple Lie group G with Lie algebra g using the root-space decomposition. We investigate the topology of these varieties for the classical linear Lie algebras. If A is a regular element, then for p > 1, the pth Hessenberg variety is smooth and connected. The odd Betti numbers vanish, while the even Betti numbers represent (apparently new) generalizations of the classical Eulerian numbers which are determined by the height function on the root system of the Lie algebra. In particular, for g = sl(n, C) they yield a family of symmetric unimodal sequences which link the classical Eulerian numbers (p = 1) to the classical Mahonian numbers (p = n-1 ), while if g = sp(n, C) and p = 1, they are f- Eulerian numbers in the sense of R.P. Stanley.