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On formal dimensions for reductive p-adic groups. (English) Zbl 0732.22007
Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Pt. I: Papers in representation theory, Pap. Workshop L-Functions, Number Theory, Harmonic Anal., Tel-Aviv/Isr. 1989, Isr. Math. Conf. Proc. 2, 225-266 (1990).
[For the entire collection see Zbl 0698.00020.]
Let G be the group of F-rational points of a connected reductive linear group over a local non-Archimedean field F with residual characteristic p (a reductive p-adic group), with split rank \(r=r_ G\). H(G), the Hecke algebra of G, is the algebra of complex-valued, locally constant, compactly supported functions on G. Consider the category \({\mathcal C}_ G\) of algebraic complex finitely generated representations of G; this is isomorphic to the category of unital modules on H(G). The main theorem is that if \(V\in {\mathcal C}_ G\) and dg is a choice of Haar measure for G, then there exists d(V,dg)\(\in {\mathcal C}_ c^{\infty}(G)^*\) such that if \(\Gamma\subset G\) is a discrete, cocompact, torsion-free subgroup, then \[ vol(G/\Gamma,dg)=\sum_{i}(-1)^ i\dim Ext^ i(V,L^ 2(G/\Gamma)); \] the sum is \(>0\) if V is projective. When char F\(=0\), such \(\Gamma\) exist. As a consequence, there is a uniquely defined element r(v)\(\in H(G)/[H(G),H(G)]\) such that \[ Tr \pi (r(v))=\sum_{i}Tr(Ext^ i(V,W),1) \] for all irreducible (\(\pi\),W); one defines r(V)(1) to be the rank of V. If char \(F\neq 0\) one gets similar results by using a trick of Kazhdan. These formal degrees are those of square integrable representations when the latter exist. Thus they generalize the usual definition. They are all rational.

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