Description
The Riemann--Liouville fractional integral interpolates iterated integration to non-integer orders and turns a hypergeometric solution of rank $p$ into one of rank $p+1$. Katz recast this into the so-called \emph{middle convolution}, an operation on local systems on the punctured Riemann sphere that preserves the rigidity index. Dettweiler and Reiter subsequently made the operation explicit at the matrix level, as an action either on the residues of a Fuchsian system (additive middle convolution) or on the monodromy representation (multiplicative middle convolution). In this talk I will first recall the Riemann--Liouville transform and both the additive and the multiplicative middle convolutions of Dettweiler--Reiter, and then describe a noncommutative analogue of the multiplicative case --- a \emph{quantum middle convolution} --- built by enlarging the standard Hecke triple of the rank-one DAHA of type $\check{C}_1 C_1$ to a cyclic Noumi--Stokman-type quadruple and performing an explicit two-step quotient by the resulting invariant subspaces. Applied to the basic representation, this construction yields an explicit intertwiner between the basic representations with parameters $(a,b,c,d)$ and $(c,b,a,d)$. On the spherical subalgebra this operation reduces to the identity, recovering the $W(D_4)$ action on the Askey--Wilson / Zhedanov algebra; on the full DAHA basic representation it provides a new and explicit incarnation of the parameter symmetry $a \leftrightarrow c$.