Category O over a deformation of the symplectic oscillator algebra

Abstract

We discuss the representation theory of $H_f$, which is a deformation of the symplectic oscillator algebra $sp(2n) \ltimes h_n$, where $h_n$ is the ((2n+1)-dimensional) Heisenberg algebra. We first look at a more general setup, involving an algebra with a triangular decomposition. Assuming the PBW theorem, and one other hypothesis, we show that the BGG category $\mathcal{O}$ is abelian, finite length, and self-dual. We decompose $\mathcal{O}$ as a direct sum of blocks $\calo(\la)$, and show that each block is a highest weight category. In the second part, we focus on the case $H_f$ for $n=1$, where we prove all these assumptions, as well as the PBW theorem.Comment: 42 pages, LaTeX, 11pt; Typos removed, references added, presentation improved, minor corrections and additions, Section 16 modified, and Standing Assumption added in Section 17; Final form, to appear in the Journal of Pure and Applied Algebr

Quantized symplectic oscillator algebras of rank one

“…The representation theory of the symplectic oscillator algebras of rank 1 was studied by Khare in [12]. In our present paper, we show that the main results of [12] can naturally be q-deformed.…”

“…In our present paper, we show that the main results of [12] can naturally be q-deformed. One of our main results is that, in the q-deformed setting, there exist PBW deformations whose finite-dimensional representations are completely reducible.…”

“…(We remark that this constant α rm is different from the constant that was also denoted by α rm in [12].) Let = ±1 henceforth.…”

“…This method can also be applied to H f , and provides a simpler proof than in [12]. We may define a Z 0 -filtration on A by assigning deg E = deg F = 1, deg K = deg K −1 = 0, and deg X = deg Y to be some sufficiently big number so that, by Theorem 2.1, the associated graded algebra gr A is a skew-Laurent extension of a quantum affine space, cf.…”

Center and Representations of Infinitesimal Hecke Algebras of 𝔰𝔩2

“…Then his result ( [Kh,Theorem 11]) says that V (r) is finite-dimensional if and only if there exists a nonnegative integer s ≤ r such that α r,r−s+2 = 0. Let us explain why this can not happen as long as z = 0 and r is large enough.…”