Generalized Harish-Chandra morphism on Reflection Equation algebras
Dimitry Gurevich, Pavel Saponov
TL;DR
This work develops a quantum analogue of the Harish-Chandra morphism for Reflection Equation algebras associated with skew-invertible Hecke symmetries by exploiting the Cayley-Hamilton identity for the generating matrix $L$ to define central eigenvalues $\mu_i$. It provides explicit parametrizations of central elements, such as power sums $p_k(L)$ and Schur polynomials, in terms of $\mu_i$ (and their modified counterparts $\hat{\mu}_i$), with precise weighting factors $d_i$ and $\hat{d}_i$, within the representation theory of modules $V_\lambda$ where the $\mu_i$ act as scalars. The authors introduce quantum weight systems by mapping Hecke algebra elements into the centers of RE algebras and applying the quantum Harish-Chandra morphism to obtain (shifted) symmetric polynomials in the eigenvalues, providing concrete examples that illuminate the distinction between modified and non-modified RE algebras. The construction is extended to general-type Hecke symmetries with bi-rank $(m|n)$, yielding two eigen-value families $\mu_i$ and $\nu_j$ and super-symmetric expressions for central elements, thereby unifying and extending prior results (e.g., quantum Gelfand invariants) in a universal framework not tied to RTT realizations.
Abstract
We consider the so-called generalized Harish-Chandra morphism, taking the center of the enveloping algebra U(gl(N)) to the commutative algebra generated by eigenvalues of the generating matrix of this algebra, and generalize this construction to Reflection Equation algebras. To this end we introduce the eigenvalues of the generating matrix of the Reflection Equation algebra (modified or not), corresponding to a skew-invertible Hecke symmetry and define the generalized Harish-Chandra morphism in a similar way. We use this map in order to introduce quantum analogs of the so-called weight systems.