A unitarity criterion for p-adic groups.

*(English)*Zbl 0676.22012Let (G,B) denote a Hecke pair. The map that sends V to \(V^ B\), the subspace of B fixed vectors, induces an equivalence of categories from the G-modules that are generated by their B fixed vectors and the modules over the Hecke algebra \({\mathcal H}={\mathcal H}(G,B)\). For V irreducible it is clear that V is hermitian if and only if \(V^ B\) is hermitian and that V unitary implies \(V^ B\) unitary but it is wrong in general that \(V^ B\) unitary implies V unitary. But for \(G={\mathcal G}(F)\), the F-rational points of a reductive group over the p-adic field F the last implication is expected to hold.

Now let \({\mathcal G}\) be split over F with connected center. In this situation D. Kazhdan and G. Lusztig [Invent. Math. 87, 153- 215 (1987; Zbl 0613.22004)] gave a classification of irreducible G- modules with Iwahori fixed vectors. This classification parametrizes these representations essentially by those homomorphisms \(\Phi\) of the Weil-Deligne group of F into the L-group of G, that are trivial on the inertia group. A module corresponding to \(\Phi\) with \(\Phi\) (Frobenius) hyperbolic is called real.

The authors show that a p-adic analog of the signature theorem due to D. Vogan holds. The role of K-types is played by the representations of the Weyl group Hecke algebra \({\mathcal H}_ W\). The signature theorem is the main tool for the proof of the main theorem of the paper which says that a real irreducible hermitian G-module V is unitary if and only if \(V^ B\) is unitary as Hecke module. Besides the signature theorem, the main theorem and a lot of related assertions the paper contains a well written survey on the results of Kazhdan and Lusztig as well as the representation theory of \({\mathcal H}_ W\) due to Springer.

Now let \({\mathcal G}\) be split over F with connected center. In this situation D. Kazhdan and G. Lusztig [Invent. Math. 87, 153- 215 (1987; Zbl 0613.22004)] gave a classification of irreducible G- modules with Iwahori fixed vectors. This classification parametrizes these representations essentially by those homomorphisms \(\Phi\) of the Weil-Deligne group of F into the L-group of G, that are trivial on the inertia group. A module corresponding to \(\Phi\) with \(\Phi\) (Frobenius) hyperbolic is called real.

The authors show that a p-adic analog of the signature theorem due to D. Vogan holds. The role of K-types is played by the representations of the Weyl group Hecke algebra \({\mathcal H}_ W\). The signature theorem is the main tool for the proof of the main theorem of the paper which says that a real irreducible hermitian G-module V is unitary if and only if \(V^ B\) is unitary as Hecke module. Besides the signature theorem, the main theorem and a lot of related assertions the paper contains a well written survey on the results of Kazhdan and Lusztig as well as the representation theory of \({\mathcal H}_ W\) due to Springer.

Reviewer: A.Deitmar

##### MSC:

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

11F33 | Congruences for modular and \(p\)-adic modular forms |

##### Keywords:

Hecke algebra; reductive group; p-adic field; irreducible G-modules; representations; inertia group; signature theorem; K-types; real irreducible hermitian G-module
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\textit{D. Barbasch} and \textit{A. Moy}, Invent. Math. 98, No. 1, 19--37 (1988; Zbl 0676.22012)

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##### References:

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