# zbMATH — the first resource for mathematics

A unitarity criterion for p-adic groups. (English) Zbl 0676.22012
Let (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.
Reviewer: A.Deitmar

##### MSC:
 22E50 Representations of Lie and linear algebraic groups over local fields 11F33 Congruences for modular and $$p$$-adic modular forms
Full Text:
##### References:
  [A] Arthur, J.: On some problems suggested by the trace formula. In: Herb, R., Kudla, S., Lipsman, R., Rosenberg, J. (eds.) Lie group representations II. Proceedings, Maryland 1982/83 (Lect. Notes Math., vol. 1041, pp. 1-49) Berlin Heidelberg New York: Springer 1984  [B] Borel, A.: Admissible representations of a semisimple group over a local field with fixed vectors under an Iwahori subgroup. Invent. Math.35, 233-259 (1976) · Zbl 0334.22012  [BW] Borel, A., Wallach, N.: Continuous cohomology, discrete subgroups and representations of reductive groups. Ann. Math. Stud.94, Princeton University Press, 1980 · Zbl 0443.22010  [C] Casselman, W.: A new non-unitarity argument forp-adic representations. J. Fac. Sci., Univ. of Tokyo28, 907-928 (1981) · Zbl 0519.22011  [G] Ginsburg, V.: Deligne-Langlands conjecture and representations of Affine Hecke algebras, preprint  [HC] Harish-Chandra: Harmonic analysis onp-adic group. Proc. Symp. Pure Math.26, 167-192 (1973)  [KL1] Kazhdan, D., Lusztig, G.: EquivariantK-theory and representations of Hecke algebras II. Invent. Math.80, 209-231 (1985) · Zbl 0613.22003  [KL 2] Kazhdan, D., Lusztig, G.: Proof of the Deligne-Langlands conjecture for Hecke algebras. Invent. Math.87, 153-215 (1987) · Zbl 0613.22004  [Ln] Langlands, R.: Problems in the theory of automorphic forms. In: Peterson, F.P. (ed.): The Steenrod algebra and its applications. Proceedings, Columbus, 1970 (Lect. Notes Math., vol. 170, pp. 18-86) Berlin Heidelberg New York: Springer 1970 · Zbl 0225.14022  [Ls1] Lusztig, G.: On a theorem of Benson and Curtis. J. Algebra71, 490-498 (1981) · Zbl 0465.20042  [Ls2] Lusztig, G.: Some examples of square integrable representations of semisimplep-adic groups. TAMS277, 623-653 (1983) · Zbl 0526.22015  [Ls3] Lusztig, G.: Cells in affine Weyl groups IV (Preprint)  [Ls4] Lusztig, G.: Cuspidal local systems and graded Hecke algebras (in preparation)  [R] Rogawski, J.: On modules over the Hecke algebra of ap-adic group. Invent. Math.79, 443-465 (1985) · Zbl 0579.20037  [S] Silberger, A.: The Langlands quotient theorem forp-adic groups. Ann.236, 95-104 (1978) · Zbl 0362.20029  [T] Tate, J.: Number theoretic background. Proc. Symp. Pure Math.33, 3-26 (1979) · Zbl 0422.12007  [V1] Vogan, D.: Representations of real reductive Lie groups, Boston Basel Stuttgart: Birkhäuser 1981 · Zbl 0469.22012  [V2] Vogan, D.: Unitarizability of certain series of representations. Ann. Math.120, 141-187 (1984) · Zbl 0561.22010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.