Fibers of characters in Gelfand-Tsetlin categories

Abstract

Abstract. We solve the problem of extension of characters of commutative subalgebras in associative (noncommutative) algebras for a class of subrings (Galois orders) in skew group rings. These results can be viewed as a noncommutative analogue of liftings of prime ideals in the case of integral extensions of commutative rings. The proposed approach can be applied to the representation theory of many infinite dimensional algebras including universal enveloping algebras of reductive Lie algebras, Yangians and finite W -algebras. In particular, we develop a theory of Gelfand-Tsetlin modules for gl n . Besides classification results we characterize their categories in the generic case extending the classical results on gl 2 .

Eigenvalue coincidences and K-orbits, I

“…In the case we are considering, the argument outlined above is simpler and quite different than the existing arguments in the literature. Our approach may also provide interesting applications to geometric representation theory; in particular to the construction of certain generalized Harish-Chandra modules closely related to the Gelfand-Zeitlin modules studied by Drozd, Futorny, and Ovsienko [9,13,12]. It would also be interesting to relate our methods to results of Baruch on construction of invariant distributions on g [1].…”

“…By a linear algebra calculation from Proposition 5.9 of [8] (see Eqs. (5)(6)(7)(8)(9)(10)(11) and (5)(6)(7)(8)(9)(10)(11)(12) in [8]), elements of Ξ are matrices of the form ⎡…”

Gelfand–Tsetlin modules of quantum gln defined by admissible sets of relations

“…We define U q as a unital associative algebra generated by e i , f i (1 ≤ i ≤ n − 1) and q h (h ∈ P ) with the following relations: (6) e i e j = e j e i , f i f j = f j f i (|i − j| > 1). (7) Fix the standard Cartan subalgebra h and the standard triangular decomposition. The weights of U q will be written as n-tuples (λ 1 , ..., λ n ).…”

“…Further, infinite dimensional GelfandTsetlin modules for gl n were studied in [18], [25], [26], [3], [31], [7], [27], [28], [30], [8], [9], [10], [11], [12], [37], [32], [35], [36] among the others. These representations have close connections to different concepts in Mathematics and Physics (cf.…”

Bounds of Gelfand-Tsetlin multiplicities and tableaux realizations of Verma modules

“…Denote by GT the category of all Gelfand-Tsetlin gl(n, C)-modules, and for each equivalence class ζ ∈ C µ /(Z µ 0 #S µ ) denote by GT ζ the full subcategory of GT consisting of modules whose support is contained in ζ. We have a decomposition of GT into a direct sum of componentsGT ζ in the sense that Ext i GT (M, N) = 0 for all i ≥ 0 and for any M and N in different components (see [FO14, Corollary 3.4]).An upper bound for the Gelfand-Tsetlin multiplicities of any simple Gelfand-Tsetlin module was found in [FO14, Theorem 4.12(c)]. To write this bound, fix a seed v and consider the stabilizer S π(v) of v in S µ .…”