Representations of shifted Yangians and finite 𝑊-algebras

Abstract

We give a presentation for the finite W -algebra associated to a nilpotent matrix in the general linear Lie algebra over C. In the special case that the nilpotent matrix consists of n Jordan blocks each of the same size l, the presentation is that of the Yangian of level l associated to gl n , as was first observed by Ragoucy and Sorba. In the general case, we are lead to introduce some generalizations of the Yangian which we call the shifted Yangians.

Graded decomposition numbers for cyclotomic Hecke algebras

“…For this we repeat an easy argument from the proof of [13,Theorem 4.4], as follows. Proceed by induction on ht(α), the statement being trivial for ht(α) = 0.…”

The Gelfand–Kirillov conjecture and Gelfand–Tsetlin modules for finite W-algebras

“…The growing interest in finite W -algebras theory is due, on the one hand, to their geometric realizations as quantizations of Slodowy slices (Premet [34] and Gan and Ginzburg [19]) and, on the other hand, to their close connections with Yangian theory, which was initially proposed by Ragoucy and Sorba [35], and was developed in full by Brundan and Kleshchev [7]. The latter of these pieces may well be regarded as a substantial step forward in understanding the structure of finite W -algebras associated with gl m .…”

“…, p n ); see Section 2 for the precise definition of and the relationship of W (π) to the shifted Yangian. One of the surprising consequences of the results of [7] is that the isomorphism class of W (π) depends only on the sequence of row lengths (p 1 , . .…”

Shifted Yangians and finiteW-algebras

“…In a subsequent article [BK2], we will use the presentation for W ( ) obtained here to study its highest weight representation theory. In particular, we will explain the sense in which rational representations of W ( ) categorify the polynomial representation of GL ∞ parametrized by the partition .…”