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Seminar on analytic theory of differential equations
April 8, 2015 14:00, 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 one-dimensional 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 non-negative 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 $\nabla\circ V=V\circ\frac d{dx}$.

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