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A mixed hook-length formula for affine Hecke algebras. (English) Zbl 1065.20010

Summary: Let \(\widehat H_l\) be the affine Hecke algebra corresponding to the group \(\text{GL}_l\) over a \(p\)-adic field with residue field of cardinality \(q\). We regard \(\widehat H_l\) as an associative algebra over the field \(\mathbb{C}(q)\). Consider the \(\widehat H_{l+m}\)-module \(W\) induced from the tensor product of the evaluation modules over the algebras \(\widehat H_l\) and \(\widehat H_m\). The module \(W\) depends on two partitions \(\lambda\) of \(l\) and \(\mu\) of \(m\), and on two non-zero elements of the field \(\mathbb{C}(q)\). There is a canonical operator \(J\) acting on \(W\), it corresponds to the trigonometric \(R\)-matrix. The algebra \(\widehat H_{l+m}\) contains the finite dimensional Hecke algebra \(H_{l+m}\) as a subalgebra, and the operator \(J\) commutes with the action of this subalgebra on \(W\). Under this action, \(W\) decomposes into irreducible subspaces according to the Littlewood-Richardson rule. We compute the eigenvalues of \(J\), corresponding to certain multiplicity-free irreducible components of \(W\). In particular, we give a formula for the ratio of two eigenvalues of \(J\), corresponding to the “highest” and the “lowest” components. As an application, we derive the well known \(q\)-analogue of the hook-length formula for the number of standard tableaux of shape \(\lambda\).

MSC:

20C08 Hecke algebras and their representations
05E10 Combinatorial aspects of representation theory
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