POLYNOMIALS ATTACHED TO REPRESENTATIONS OF G ( r , p , n )
نویسنده
چکیده
The rational Cherednik algebra H is a certain algebra of differential-reflection operators attached to a complex reflection group. There is a category O of modules for this algebra which is a highest weight category. For the infinite family G(r, p, n) of complex reflection groups, the algebra H contains a subalgebra isomorphic to a (generalized) degenerate affine Hecke algebra, and our strategy is to study the standard modules in category O by means of this subalgebra. We use the Okounkov-Vershik approach to the representations of G(r, p, n) to compute the spectra of the standard modules in O with respect to the polynomial subalgebra of the affine Hecke algebra. The eigenbasis consists of a generalization of the non-symmetric Jack polynomials. As an application, we show that when the parameters are chosen “coprime to the Coxeter number of G(r, p, n)”, category O has an especially simple structure, with exactly one non-semisimple block. In the final section we compute the norms of the generalized Jack polynomials with respect to the contravariant form. This formula determines the radical of the standard modules in the cases in which the Jack polynomials are all well defined.
منابع مشابه
JACK POLYNOMIALS ATTACHED TO REPRESENTATIONS OF G(r, p, n)
The rational Cherednik algebra H is a certain algebra of differential-reflection operators attached to a complex reflection group. There is a category O of modules for this algebra which is a highest weight category. For the infinite family G(r, p, n) of complex reflection groups, the algebra H contains a subalgebra isomorphic to a (generalized) degenerate affine Hecke algebra, and our strategy...
متن کاملCo-centralizing generalized derivations acting on multilinear polynomials in prime rings
Let $R$ be a noncommutative prime ring of characteristic different from $2$, $U$ the Utumi quotient ring of $R$, $C$ $(=Z(U))$ the extended centroid of $R$. Let $0neq ain R$ and $f(x_1,ldots,x_n)$ a multilinear polynomial over $C$ which is noncentral valued on $R$. Suppose that $G$ and $H$ are two nonzero generalized derivations of $R$ such that $a(H(f(x))f(x)-f(x)G(f(x)))in ...
متن کاملThe Representations and Positive Type Functions of Some Homogenous Spaces
‎For a homogeneous spaces ‎$‎G/H‎$‎, we show that the convolution on $L^1(G/H)$ is the same as convolution on $L^1(K)$, where $G$ is semidirect product of a closed subgroup $H$ and a normal subgroup $K $ of ‎$‎G‎$‎. ‎Also we prove that there exists a one to one correspondence between nondegenerat $ast$-representations of $L^1(G/H)$ and representations of ...
متن کاملQUASI-PERMUTATION REPRESENTATIONS OF SUZtTKI GROUP
By a quasi-permutation matrix we mean a square matrix over the complex field C with non-negative integral trace. Thus every permutation matrix over C is a quasipermutation matrix. For a given finite group G, let p(G) denote the minimal degree of a faithful permutation representation of G (or of a faithful representation of G by permutation matrices), let q(G) denote the minimal degree of a fai...
متن کاملRecurrences and explicit formulae for the expansion and connection coefficients in series of the product of two classical discrete orthogonal polynomials
Suppose that for an arbitrary function $f(x,y)$ of two discrete variables, we have the formal expansions. [f(x,y)=sumlimits_{m,n=0}^{infty }a_{m,n},P_{m}(x)P_{n}(y),] $$ x^{m}P_{j}(x)=sumlimits_{n=0}^{2m}a_{m,,n}(j)P_{j+m-n}(x),$$ we find the coefficients $b_{i,j}^{(p,q,ell ,,r)}$ in the expansion $$ x^{ell }y^{r},nabla _{x}^{p}nabla _{y}^{q},f(x,y)=x^{ell }y^{r}f^{(p,q)}(x,y) =sumli...
متن کامل