Description
Consider the rank 1 DAHA of type $\check C_1 C_1$ with generators $T_1$, $T_0$, $Z$, $Z^{-1}$ and depending on the Askey--Wilson parameters $a,b,c,d,q$. Its basic representation is a faithful representation on the space of Laurent polynomials in $z$. We extend the generators $Z,Z^{-1}$ to all rational functions in $Z$. Then the basic representation extends to a faithful representation on the space of rational functions in $z$. Then the mapping sending a rational function $f(z)$ to $f(z^{-1})$ can be lifted to the extended DAHA, and similarly for the mapping sending $f(z)$ to $f(qz)$.
A DAHA automorphism is a parameter transformation $(a,b,c,d)\to(a'.b',c',d')$ together with an algebra isomorphism, depending on $a,b,c,d$, from the (extended) DAHA $H_{a,b,c,d}$ to $H_{a',b',c',d'}$. For each parameter transformation we define explicit isomorphisms $H_{a,b,c,d}\to H_{a',b',c',d'}$ which we call canonical. This can be done in particular for the $a \leftrightarrow c$ flip discussed in Mazzocco's lecture. More general isomorphisms $H_{a,b,c,d}\to H_{a',b',c',d'}$ factorize as a canonical isomorphism followed by a DAHA automorphism $H_{a',b',c',d'}\to H_{a',b',c',d'}$, which turns out to be a conjugation in many examples.
Many special DAHA automorphisms can be easily read off from the relations for the DAHA generators, but they are usually not canonical. We discuss some examples, and also their factorization. Some examples imply transformation formulas of non-symmetric Askey--Wilson polynomials (or functions), which functions occur as eigenfunctions of $Y=T_1 T_0$ in the basic representation.