

Seminar on analytic theory of differential equations
April 8, 2015 14:00–15:30, Moscow, Steklov Mathematical Institute, Room 440 (8 Gubkina)






The Dunkl operators and integrability
S. P. Khekalo^{} 
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Abstract:
In the talk I'll explain which analogs of the classical onedimensional Dunkl operator can be intertwinned in the Cherednik algebra with an operator of differentiation or integration. The corollaries will be discussed. The key notions are the following.
A reflection in $\mathbb R$: $s[f](x)=f(x)$;
The Dunkl operator in $\mathbb R$: $\nabla=\frac d{dx}\frac{k}{x}s$, where $k$ is a nonnegative integer.
An analog of Dunkl operator on $\mathbb R$: $\nabla_{\omega}=\frac d{dx}(\ln \omega(x))'s$, where $\omega$ is an even function;
The Cherednik algebra: $A=\langle 1, x, d/dx, s \rangle$;
$V$ interwins $\nabla$ and $\frac d{dx}$, if $\nabla\circ V=V\circ\frac d{dx}$.

