# Gelfand–Graev representation

In representation theory, a branch of mathematics, the **Gelfand–Graev representation** is a representation of a reductive group over a finite field introduced by Gelfand & Graev (1962), induced from a non-degenerate character of a Sylow subgroup.

The Gelfand–Graev representation is reducible and decomposes as the sum of irreducible representations, each of multiplicity at most 1. The irreducible representations occurring in the Gelfand–Graev representation are called **regular representations**. These are the analogues for finite groups of representations with a Whittaker model.

## References

- Carter, Roger W. (1985),
*Finite groups of Lie type. Conjugacy classes and complex characters.*, Pure and Applied Mathematics (New York), New York: John Wiley & Sons, ISBN 978-0-471-90554-7, MR 0794307 - Gelfand, I. M.; Graev, M. I. (1962), "Construction of irreducible representations of simple algebraic groups over a finite field",
*Doklady Akademii Nauk SSSR*,**147**: 529–532, ISSN 0002-3264, MR 0148765 English translation in volume 2 of Gelfand's collected works.

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