Quantum Conjugacy Classes of Simple Matrix Groups

Abstract

Let G be a simple complex classical group and g its Lie algebra. Let U (g) be the Drinfeld-Jimbo quantization of the universal enveloping algebra U (g). We construct an explicit U (g)-equivariant quantization of conjugacy classes of G with Levi subgroups as the stabilizers.

Non-Levi Closed Conjugacy Classes of SP q (2n)

“…[1]. Its eigenvalues are pairwise distinct in the classical limit, apart from lim q→1 µ ℓ+1 = lim q→1 µ −1 ℓ+1 q −4n+2(m−1) = −1.…”

“…It generalizes for the orthogonal groups in the obvious way, with the only stipulation for the D-series: the traces of matrix powers are not enough to fix a class, and one has to add one more condition on the invariants of G, see e.g. [1].…”

“…Here we need the information on its eigenvalues, which are found in [1]. Specifically onM 1 andM 2 , the operator Q is a scalar multiplier by, respectively, q 2(λ,ε 1 ) and q 2(λ,ε m+1 )−2m .…”

“…We should stress that the methods of [1] are inapplicable, as they are, for the non-Levi classes, whose quantization is still an open problem. In our recent paper [2] we have shown how to approach it on the simplest example of SP (4)/SP (2) × SP (2).…”

Quantum Flag Manifolds as Quotients of Degenerate Quantized Universal Enveloping Algebras

“…In [6], see also [14], flag manifolds were considered with a Poisson structure obtained from the associated dynamical r-matrix together with a character on the Lie algebra of the stabilizer. In case this data satisfied a certain regularity condition, it was shown (in the formal deformation setting) that the quantization of this flag manifold could be constructed as a quotient by the kernel of a representation on a suitable generalized Verma module.…”