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**Harish-Chandra homomorphisms for \({\mathfrak p}\)-adic groups.**
*(English)*
Zbl 0593.22014

Regional Conference Series in Mathematics 59. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-0709-9). xi, 76 p. (1985).

This beautiful work proposes an analogue of the Harish-Chandra homomorphism for \({\mathfrak p}\)-groups, with special emphasis on GL(n). As so often happens with \({\mathfrak p}\)-adic groups, the word ”analogue” must be taken in a subjective sense, having perhaps more to do with the ends it will serve than with the construction itself.

The Harish-Chandra homomorphism for a real reductive Lie group G is a homomorphism between the centres of the universal enveloping algebras of the Lie algebra of G and the Levi component of a parabolic subalgebra. For \({\mathfrak p}\)-adic groups, alternatives must be found for the centres of the universal enveloping algebras.

The first chapter is a brief but elegant presentation of the representation theory of GL(n) of a finite field, adapted to the purpose at hand. The treatment quickly reduces to the following situation: suppose P is a parabolic subgroup of \(G=GL(n, {\mathbb{F}}_ q)\) whose Levi component is the product of n/k copies of GL(k), and suppose we are inducing a representation whose restrictions to the GL(k) factors of the Levi component are all equivalent. The decomposition of the induced representation is described by a generalized Hecke algebra involving P:P double cosets.

The author shows that this algebra is naturally isomorphic to the Hecke algebra of Borel: Borel double cosets in \(G^ 0=GL(n/k, {\mathbb{F}}_{q^ k})\), and so is able to relate representations of G induced from P with representations of \(G^ 0\) induced from the Borel subgroup. Moreover the algebra isomorphism is relatively easy to describe, and the resulting correspondence for representations is compatible with Plancherel measures, induction, and matrix coefficient asymptotics.

The remainder of the book is devoted to generalizing this construction to \({\mathfrak p}\)-adic groups, especially GL(n). The author remarks that much of it can be extended to other groups, at least when the tori involved are only tamely ramified.

The author is hesitant about formulating the result in full generality. He is optimistic that this will be done eventually, but details remain unclear. In light of the description above for finite fields it is hardly surprising, but nontheless tantalizing, when he observes that the final version may involve Hecke algebras of endoscopic groups rather than of subgroups.

The Harish-Chandra homomorphism for a real reductive Lie group G is a homomorphism between the centres of the universal enveloping algebras of the Lie algebra of G and the Levi component of a parabolic subalgebra. For \({\mathfrak p}\)-adic groups, alternatives must be found for the centres of the universal enveloping algebras.

The first chapter is a brief but elegant presentation of the representation theory of GL(n) of a finite field, adapted to the purpose at hand. The treatment quickly reduces to the following situation: suppose P is a parabolic subgroup of \(G=GL(n, {\mathbb{F}}_ q)\) whose Levi component is the product of n/k copies of GL(k), and suppose we are inducing a representation whose restrictions to the GL(k) factors of the Levi component are all equivalent. The decomposition of the induced representation is described by a generalized Hecke algebra involving P:P double cosets.

The author shows that this algebra is naturally isomorphic to the Hecke algebra of Borel: Borel double cosets in \(G^ 0=GL(n/k, {\mathbb{F}}_{q^ k})\), and so is able to relate representations of G induced from P with representations of \(G^ 0\) induced from the Borel subgroup. Moreover the algebra isomorphism is relatively easy to describe, and the resulting correspondence for representations is compatible with Plancherel measures, induction, and matrix coefficient asymptotics.

The remainder of the book is devoted to generalizing this construction to \({\mathfrak p}\)-adic groups, especially GL(n). The author remarks that much of it can be extended to other groups, at least when the tori involved are only tamely ramified.

The author is hesitant about formulating the result in full generality. He is optimistic that this will be done eventually, but details remain unclear. In light of the description above for finite fields it is hardly surprising, but nontheless tantalizing, when he observes that the final version may involve Hecke algebras of endoscopic groups rather than of subgroups.

Reviewer: J.Repka

### MSC:

22E50 | Representations of Lie and linear algebraic groups over local fields |

11F70 | Representation-theoretic methods; automorphic representations over local and global fields |

22-02 | Research exposition (monographs, survey articles) pertaining to topological groups |

22D30 | Induced representations for locally compact groups |

22E46 | Semisimple Lie groups and their representations |

16S34 | Group rings |

16W99 | Associative rings and algebras with additional structure |