Principal Galois orders and Gelfand-Zeitlin modules

Abstract

We show that the ring of invariants in a skew monoid ring contains a so called standard Galois order. Any Galois ring contained in the standard Galois order is automatically itself a Galois order and we call such rings principal Galois orders. We give two applications. First, we obtain a simple sufficient criterion for a Galois ring to be a Galois order and hence for its Gelfand-Zeitlin subalgebra to be maximal commutative. Second, generalizing a recent result by Early-Mazorchuk-Vishnyakova, we construct canonical simple Gelfand-Zeitlin modules over any principal Galois order.As an example, we introduce the notion of a rational Galois order, attached an arbitrary finite reflection group and a set of rational difference operators, and show that they are principal Galois orders. Building on results by Futorny-Molev-Ovsienko, we show that parabolic subalgebras of finite W-algebras are rational Galois orders. Similarly we show that Mazorchuk's orthogonal Gelfand-Zeitlin algebras of type A, and their parabolic subalgebras, are rational Galois orders. Consequently we produce canonical simple Gelfand-Zeitlin modules for these algebras and prove that their Gelfand-Zeitlin subalgebras are maximal commutative.Lastly, we show that quantum OGZ algebras, previously defined by the author, and their parabolic subalgebras, are principal Galois orders. This in particular proves the long-standing Mazorchuk-Turowska conjecture that, if q is not a root of unity, the Gelfand-Zeitlin subalgebra of Uq(gl n ) is maximal commutative and that its Gelfand-Zeitlin fibers are non-empty and (by Futorny-Ovsienko theory) finite.Notation. The multiplicative identity element in a monoid or ring S is denoted 1 S . All rings are assumed to have an identity. Frac A denotes the field of fractions of an Ore domain A. For a group G acting by automorphisms on a ring A we let A G = {a ∈ A | ∀g ∈ G : g(a) = a} denote the subring of G-invariants.

A study of Galois and flag orders

“…In [FMO10] they showed that ι(W (π)) is a Galois order. In [Har20] it was further shown to be a principal Galois order. Here is the realization.…”

“…They form a collection of algebras that contains many important examples including: generalized Weyl algebras defined independently by Bavula [Bav92] and Rosenberg [Ros95] in the early nineties, U (gl n ), shifted Yangians and finite W -algebras [FMO10], Coulomb branches [Web19], and U q (gl n ) [FH14]. Their structures and representations have been studied in [Fut+18], [FS18a], [Har20], and [MV18].…”

“…We need the following data: (Λ, G, M ), where Λ is an integrally closed domain, G is a finite subgroup of Aut(Λ), and M is a submonoid of Aut(Λ). Additionally, this data adheres to the following assumptions from [Har20]:…”

“…Definition 1.2.1 ( [Har20]). For any element X ∈ Frac(Λ)#M we define a Z-bilinear map from (Frac(Λ)#M ) × Frac(Λ) → Frac(Λ), called the evaluation of X at f for an element f ∈ Frac(Λ), by:…”

“…In [Har20] it was shown that any (co-)principal Galois Γ-order is a Galois order in the sense of Definition 2.2.6.…”