Outlook

Dualizing the coalgebra map $A^{{\rm s}}_{R,q}(n,r-2)\stackrel{\cdot d_q}{\rightarrow } A^{{\rm s}}_{R,q}(n,r)$ from the sequence (9), one obtains an epimorphism of algebras from $S^s_{q}(n,r)$ to $S^s_{q}(n,r-2)$. On a basis element $C^{\underline{\lambda}}_{{\bf i}, {\bf j}}$ it is given by subtracting $1$ from $l$ in $\underline{\lambda}=(\lambda, l)$ and keeping ${\bf i}, {\bf j}$ fixed. Its kernel is the linear span of those basis elements which occur in the case $l=0$. This forces a recursive structure on the representation theory of these algebras in a similar way as is known for the Birman-Murakami-Wenzl algebras (see [BW]). In addition these epimorphisms can be used to define an inverse limit of the symplectic $q$-Schur algebras in a similar way as has been worked out for the type $A$ $q$-Schur algebra in [GR, section 6.4]. It seems to be plausible that accordingly the quantized universal enveloping algebra embeds into this inverse limit.

Concerning analogues to the orthogonal case, note that Lemma 11.7 will not work here. Maybe, a way out is to consider coefficient functions of the symmetric algebra, i.e. the elements

$$\sum_{w \in {\cal S}_{\lambda}} y^{-l(w)}\beta (w)\wr x_{{\bf i}... ... \sum_{w \in {\cal S}_{\lambda}} y^{-l(w)}x_{{\bf i} {\bf j}}\wr \beta (w)$$

instead of bideterminants, which are coefficient functions of the exterior algebra.