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Trace Paley-Wiener theorem for reductive p-adic groups. (English) Zbl 0634.22011
Let 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 $$E=\ell_{\infty}$$.
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
##### MathOverflow Questions:
Simple trace formula with different spectral footprint?
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##### References:
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