Geometry and Integrability in Mathematical Physics GIMP'06

May 15 - 19, 2006, Moscow, Russia

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Sergei Khoroshkin

ITEP, Moscow

Mickelsson algebras, Zhelobenko operators and Yangians

Let $\g=\n_-+\h+\n$ be a reductive Lie algebra with a fixed Cartan decomposition and $\A$ an associative algebra, which contains universal enveloping algebra $U(\g)$ or its $q$-analog $U_q(\g)$. The Mickelsson algebra $S^\n(\A)$ is the quotient of the normalizer $N(\A\n)$ of the ideal $\A\n$ over the ideal $\A\n$. The Mickelsson algebra $S^\n(\A)$ acts in the space $V^\n$ of $\n$-invariant vectors of any $\A$-module $V$. This construction performs a reduction of a representation of $\A$ over the action of $U(\g)$ and can be viewed as a counterpart of hamiltonian reduction.

The structure of Mickelsson algebra simplifies after the localization over certain multiplicative subset of $U(\h)$. The corresponding algebra $Z^\n(\A)$ is generated by a finite-dimensional space of generators, which obey quadratic-linear relations. These generators can be defined with a help of extremal projector of Asherova-Smirnov-Tolstoy. Zhelobenko suggested a construction of another generators of Mickelsson algebra by means of a family of special operators, which form a cocycle on the Weyl group $W(\g)$.

In \cite{KO}, we used Zhelobenko operators for the construction of a family of automorphisms of Mickelsson algebra $Z^\n(\A)$, which form a representation of the braid group. If $\A$ is a smash product of $U(\g)$ and symmetric algebra $S(V)$ of a $U(\g)$-module $V$, then we get another derivation of the dynamical Weyl group by Tarasov-Varchenko-Etingof.

In \cite{KN1} and \cite{KN2}, we apply the constructed automorphisms of Mickelsson algebras to representations of Yangian. It is known due to Drinfeld, how to construct representations of Yangian $Y(gl_n)$, starting from modules over degenerate affine Hecke algebra ${\mathcal H}_n$. On the other hand, there is a functor from the category of highest weight $gl_m$ to the category of finite-dimensional ${\mathcal H}_n$-modules, due to Cherednick, Tsuchiya-Suzuki-Arakawa.

The composition of these functors is related to the following construction, closely relation to so called centralizer's construction by G.Olshansky. For a $gl_m$-module $V$ we define an action of the Yangian $Y(gl_n)$ in the space $\E_m(V)=V\ot S(\CC^{ m}\ot\CC^{ n})$, commuting with diagonal action of $gl_m$. It can be shown that coinvariants $\E_m(M_\lambda)_{\n_-}^\mu$, of the weight $\mu\in\h^*$ where $M_\lambda$ is a Verma module over $gl_m$, are isomorphic to a tensor product of symmetric powers of vector representations of $Y(gl_n)$, evaluated at the points $\mu_k-k+1$, namely $\E_m(M_\lambda)_{\n_-}^\mu \approx S^{\lambda_1-\mu_1}_{\mu_1}\ot\cdots S^{\lambda_m-\mu_m}_{\mu_m-m+1}$. The action of the Yangian $Y(gl_n)$ factors through the action of the Mickelsson algebra, related to a tensor product $$ \A=U(gl_m)\ot PD (\CC^{ m}\ot\CC^{ n}). $$ of $U(gl_m)$ and polynomial differential operators over $\CC^{ m}\ot\CC^{n}$.

Then the automorphism of the Mickelsson algebra, related to the longest element of the Weyl group of $gl_m$, intertwins $Y(gl_n)$-modules $S^{\lambda_1-\mu_1}_{\mu_1}\ot\cdots S^{\lambda_m-\mu_m}_{\mu_m-m+1}$ and $S^{\lambda_m-\mu_m}_{\mu_m-m+1}\ot\cdots S^{\lambda_1-\mu_1}_{\mu_1}$. The image of this intertwining operator is an irreducible $Y(gl_n)$-module, related to a skew Young dyagram $\lambda-\mu$.