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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.

##### MSC:
 2.2e+36 Analysis on $$p$$-adic Lie groups