## Confluences of the Painleve equations, Cherednik algebras and q-Askey scheme    [PDF]

Marta Mazzocco
In this paper we show that the Cherednik algebra of type \$\check{C_1}C_1\$ appears naturally as quantisation of the monodromy group associated to the sixth Painlev\'e equation. As a consequence we obtain an embedding of the Cherednik algebra of type \$\check{C_1}C_1\$ into \$SL(2,\mathbb T_q)\$, i.e. determinant one matrices with entries in the quantum torus. By following the confluences of the Painlev\'e equations, we produce the corresponding confluences of the Cherednik algebra and their embeddings in \$Mat(2,\mathbb T_q)\$. Finally, by following the confluences of the spherical sub-algebra of the Cherednik algebra in its basic representation (i.e. the representation on the space of symmetric Laurent polynomials) we obtain a relation between Painlev\'e equations and some members of the q-Askey scheme.
View original: http://arxiv.org/abs/1307.6140