Matrices for affine Hecke modules
Let \(H\) be the affine Hecke algebra associated with a reductive Lie group \(G\). Then \(H\) is generated by two subalgebras \(H_0\) and \(A\), where \(A\) is the coordinate ring of a complex torus \(T\), which is a maximal torus in \(G\), and \(H_0\) is generated by operators \(T_s\), where \(s\) runs over a fixed set \(\Sigma\) of simple reflections in the Weyl group \(W\) of \(G\), satisfying the usual braid relations along with the relation \((T_s-q_s)(T_s+1)=0\) for parameters \(q_s>0\). The restriction of any finite-dimensional \(H\)-module \(E\) to the commutative algebra \(A\) decomposes as \(E=\bigoplus_{\tau\in T}E_\tau\), where \(E_\tau\) is the weight space consisting of the vectors in \(E\) annihilated by some power of the maximal ideal \(m_\tau\) of \(A\). This paper is an extension of an earlier work done by the author [in Forum Math. 9, No. 4, 457-516 (1997; Zbl 0882.22020)]. The main result is an explicit formula for the weight spaces in a simple \(H\)-module with standard singularity. This formula is valid for any parameter set \(\{q_s\}\).
scientific article; zbMATH DE number 1526436
scientific article