Trace Paley-Wiener theorem for reductive p-adic groups.

*(English)*Zbl 0634.22011Let F be a nonarchimedean local field, \({\mathbb{G}}\) be a reductive F-group, and \(G={\mathbb{G}}(F)\). Let R(G) be the Grothendieck group of G, with basis Irr(G), the set of equivalence classes of smooth irreducible representations of G. Let H(G) be the Hecke algebra \(C_ c^{\infty}(G)\) under convolution. Then each h in H(G) defines a linear form \(f_ h: R(G)\to {\mathbb{C}}\) by \(f_ h(\pi)=trace \pi (h)\). Let M be a standard Levi subgroup of G. The group \(\Psi\) (M) of unramified characters of M has a natural structure of a complex algebraic group. The form \(f=f_ h\) for h in H(G) satisfies: (i) For any standard Levi subgroup M and \(\sigma\) in Irr(M), the function \(\psi \mapsto f(i_{GM}(\psi \sigma))\) is regular on the complex algebraic variety \(\psi\) (M). (ii) There is a compact open subgroup K of G for which f is non-zero only on the G-modules E which have a nontrivial space \(E^ K\) of K-invariant vectors. The Trace Paley-Wiener Theorem is that the conditions (i) and (ii) characterize the trace forms \(\{f_ h:\) \(h\in H(G)\}.\)

This theorem describes the image of the morphism tr: H(G)\(\to R^*(G)\). Theorem B of the article reviewed above shows that ker tr\(=[H(G),H(G)]\). Let \(i_{GM}: R(M)\to R(G)\) be the induction functor and let \(R_ I(G)\) be the subgroup of R(G) spanned by \(\{i_{GM}(\sigma)|\) M a proper Levi subgroup of G, \(\sigma\in Irr(M)\}\). An irreducible G-module will be called discrete if its class is not contained in \(R_ I(G)\). An infinitesimal character is called discrete if it is the infinitesimal character of some discrete G-module. A linear form on R(G) is said to be discrete if it vanishes on \(R_ I(G)\). Let \(r_{MG}: R(G)\to R(M)\) be the Jacquet functor. A combinatorial lemma is proved using the operators \(T_ M=i_{GM}\circ r_{MG}\) on R(G), that there exist rational constants \(c_ M\), for M proper, such that for any linear form f on R(G), the form f-\(\sum_ Mc_ Mr^*_{MG}i^*_{GM}(f)\) is discrete.

Let \(\Theta\) (G) be the complex algebraic variety of infinitesimal characters of G, i.e., the set of cuspidal pairs (M,\(\rho)\) up to conjugation. Results are reviewed on the Bernstein center, realized as the algebra of regular functions on \(\Theta\) (G). It is shown that in each component of \(\Theta\) (G), the set of discrete infinitesimal characters consists of a finite number of \(\Psi\) (G)-orbits. This finiteness result and the combinatorial lemma allow the indquence of coefficient functionals), then E has the maximal support property.

(ii) Each separable, countably order complete lattice-subspace X of the locally solid lattice Banach space E with the maximal support property has an order-positive Schauder basis. This yields for \(X=E\) that if E is order complete, E has an order-positive Schauder basis \((e_ n,f_ n)\) if and only if E has the maximal support property.

Finally, the author applies these results to the case \(E=\ell_{\infty}\).

This theorem describes the image of the morphism tr: H(G)\(\to R^*(G)\). Theorem B of the article reviewed above shows that ker tr\(=[H(G),H(G)]\). Let \(i_{GM}: R(M)\to R(G)\) be the induction functor and let \(R_ I(G)\) be the subgroup of R(G) spanned by \(\{i_{GM}(\sigma)|\) M a proper Levi subgroup of G, \(\sigma\in Irr(M)\}\). An irreducible G-module will be called discrete if its class is not contained in \(R_ I(G)\). An infinitesimal character is called discrete if it is the infinitesimal character of some discrete G-module. A linear form on R(G) is said to be discrete if it vanishes on \(R_ I(G)\). Let \(r_{MG}: R(G)\to R(M)\) be the Jacquet functor. A combinatorial lemma is proved using the operators \(T_ M=i_{GM}\circ r_{MG}\) on R(G), that there exist rational constants \(c_ M\), for M proper, such that for any linear form f on R(G), the form f-\(\sum_ Mc_ Mr^*_{MG}i^*_{GM}(f)\) is discrete.

Let \(\Theta\) (G) be the complex algebraic variety of infinitesimal characters of G, i.e., the set of cuspidal pairs (M,\(\rho)\) up to conjugation. Results are reviewed on the Bernstein center, realized as the algebra of regular functions on \(\Theta\) (G). It is shown that in each component of \(\Theta\) (G), the set of discrete infinitesimal characters consists of a finite number of \(\Psi\) (G)-orbits. This finiteness result and the combinatorial lemma allow the indquence of coefficient functionals), then E has the maximal support property.

(ii) Each separable, countably order complete lattice-subspace X of the locally solid lattice Banach space E with the maximal support property has an order-positive Schauder basis. This yields for \(X=E\) that if E is order complete, E has an order-positive Schauder basis \((e_ n,f_ n)\) if and only if E has the maximal support property.

Finally, the author applies these results to the case \(E=\ell_{\infty}\).

Reviewer: C.B.Huijsmans

##### MSC:

22E35 | Analysis on \(p\)-adic Lie groups |

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

43A80 | Analysis on other specific Lie groups |

##### Keywords:

nonarchimedean local field; Grothendieck group; smooth irreducible representations; Hecke algebra; Levi subgroup; unramified characters; Paley-Wiener Theorem; Jacquet functor; complex algebraic variety of infinitesimal characters; maximal support property; order-positive Schauder basis
PDF
BibTeX
XML
Cite

\textit{J. Bernstein} et al., J. Anal. Math. 47, 180--192 (1986; Zbl 0634.22011)

Full Text:
DOI

##### References:

[1] | J. N. Bernstein and P. Deligne,Le ”centre” de Bernstein, inReprĂ©sentations des groupes reductifs sur un corps local, Hermann, Paris, 1985. |

[2] | J. Bernstein and A. Zelevinsky,Induced representations of reductive p-adic groups I, Ann. Sci. Ec. Norm. Super.10 (1977), 441–472. · Zbl 0412.22015 |

[3] | A. Borel and N. Wallach,Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Ann. of Math. Studies, Princeton Univ. Press, 1980. · Zbl 0443.22010 |

[4] | W. Casselman,Characters and Jacquet modules, Math. Ann.230 (1977), 101–105. · Zbl 0337.22019 · doi:10.1007/BF01370657 |

[5] | J. Dixmier,Algebres Enveloppantes, Gauthier-Villars, 1974. |

[6] | D. Kazhdan,Cuspital geometry of p-adic groups, J. Analyse Math.47 (1986), 1–36 (this issue). · Zbl 0634.22009 · doi:10.1007/BF02792530 |

[7] | A. Zelevinsky,Induced representations of reductive p-adic groups II, Ann. Sci. Ec. Norm. Super.13 (1980), 165–210. · Zbl 0441.22014 |

[8] | D. Kazhdan,Representations of groups over close local fields, J. Analyse Math.47 (1986), 175–179 (this issue). · Zbl 0634.22010 · doi:10.1007/BF02792537 |

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.