Representation theory of very non-standard quantum $so(2N-1)$

Abstract

We classify the finite dimensional representations of the quantum symmetric pair coideal subalgebra $B_{\mathbf c}$ of type $DII$ corresponding to the symmetric pair $(so(2N),so(2N-1))$. For $B_{\mathbf c}$ defined over an arbitrary field $k$ and $q\in k$ not a root of unity we establish a one-to-one correspondence between finite dimensional, simple $B_{\mathbf c}$-modules and dominant integral weights for $so(2N-1)$. We use specialisation to show that the category of finite dimensional $B_{\mathbf c}$-modules is semisimple if $\mathrm{char}(k)=0$ and $q$ is transcendental over ${\mathbb Q}$. In this case the characters of simple $B_{\mathbf c}$-modules are given by Weyl's character formula. This means in particular that the quantum symmetric pair of type $DII$ can be used to obtain Gelfand-Tsetlin bases for irreducible representations of the Drinfeld-Jimbo quantum group $U_q(so(2N))$.