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In 1979, Vogan proposed a generalised $\tau$-invariant for characterisingprimitive ideals in enveloping algebras. Via a known dictionary this translatesto an invariant of left cells of finite Weyl groups. Although it is not acomplete invariant, it is extremely useful in describing left cells. Here, wepropose a general framework for defining such invariants which also applies toHecke algebras with unequal parameters.

A generalised $\tau$-invariant for the unequal parameter case