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N. Wallach has considered an element of the group algebra of the symmetric group S n which is the sum of an n-cycle, an (n − 1)-cycle, . . . , a 2-cycle and the identity. He showed that multiplication by this element has eigenvalues 0, 1, 2, . . . , n −2, n. We prove an analogous result in which the group algebra of S n is replaced by the corresponding Hecke algebra.

A q-analogue of an identity of N. Wallach