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Representation theory and sheaves on the Bruhat-Tits building. (English) Zbl 0892.22012
The Bruhat-Tits building \(\mathcal I\) of a connected reductive algebraic group \(G\) over a \(p\)-adic field \(K\) displays in a geometric way the inner structure of the locally compact group \(G(K)\) of the set of \(K\)-points of \(G\) like the classification of the maximal subgroups of \(G(K)\) or the theory of its parahoric subgroups. One might consider \(\mathcal I\) as a kind of skeleton of an analogue of a real symmetric space because of that it clearly appears as an important tool in the study of the theory of smooth representations of \(G(K)\).
In the paper under review, the authors develop a systematic and conceptual theory which gives a functorial way to pass from smooth representations of \(G(K)\) to equivariant objects on \(\mathcal I\). In a cohomological theory, they also associate (again functorially) \(G(K)\)-equivariant sheaves on \(\mathcal I\) with smooth representations of \(G(K)\). Their main result in this aspect is the computation of the cohomology with compact support of the sheaves coming from an irreducible smooth representation of \(G(K)\).
For any facet \(F\) of \(\mathcal I\) the authors construct certain decreasing filtrations \({\mathcal P}_F\supset U_F^{(0)}\supset U_F^{(1)}\supset\cdots\supset U_F^{(e)}\supset\cdots\) of its pointwise stabilizer \({\mathcal P}_F\) in \(G(K)\) by compact open subgroups \(U_F^{(e)}\) and establish some nice properties of that filtration (especially they study how the groups \(U_F^{(e)}\) behave if the facet \(F\) is moved along in the geodesic in the building \(\mathcal I\)). Similar filtrations appear in papers of Prasad and Raghunathan and of Moy and Prasad.
Let \(e\) be a fixed “level”. A homological functor \(\gamma_e\) is constructed from smooth representations of \(G(K)\) to \(G(K)\)-equivariant coefficient systems on \(\mathcal I\): the coefficient system \(\gamma_e(V)\) corresponding to a representation \(V\) of \(G(K)\) is formed by associating with a facet \(F\) the subspace of the \(U_G^{(e)}\)-invariant vectors in \(V\). One of the main results says that at least for any finitely generated smooth representation \(V\) of \(G(K)\) it is possible to choose the level \(e\) large enough so that the chain complex \(\gamma_e(V)\) is an exact resolution of \(V\) in the category of all smooth representations of \(G(K)\).
The authors also define a general theory of Euler-Poincaré functions for finite length smooth representations of \(G(K)\) which are pseudo-coefficients and such that their elliptic orbital integrals coincide with the Harish-Chandra character of the given representation. This leads to a Hopf-Lefschetz type trace formula for the Harish-Chandra character of an elliptic element which, combined with results of Kazhdan, also leads to a proof of the general orthogonality formula for Harish-Chandra characters which was conjectured by Kazhdan.
Since the polysimplicial structure of the building \(\mathcal I\) is locally finite it is possible to associate with the coefficient system \(\gamma_e(V)\) also a complex of cochains with finite support. Let \(\chi\) denote a fixed character of the connected component of the center of \(G\), let \(\text{Alg}_\chi(G)\) be the category of all those smooth representations of \(G(K)\) on which that component acts through \(\chi\), and let \({\mathcal H}_\chi\) be the \(\chi\)-Hecke algebra of \(G(K)\). If \(V\) is an admissible representation in \(\text{Alg}_\chi(G)\) then the functor \(\operatorname{Hom}_G(\cdot,{\mathcal H}_\chi)\) transforms the chain complex of \(\gamma_e(V)\) into the cochain complex of \(\gamma_e(\widetilde V)\) where \(\widetilde V\) is the contragredient representation of \(V\). In case \(V\) is of finite length and \(e\) is large enough the chain complex of \(\gamma_e(V)\) is a projective resolution of \(V\) in \(\text{Alg}_\chi(G)\). It follows that the cochain complex \(\gamma_e(\widetilde V)\) computes the Ext-groups \({\mathcal E}^\star(V):=\text{Ext}^\star_{\text{Alg}_\chi(G)}(V,{\mathcal H}_\chi)\). It is proven that for an arbitrary irreducible smooth representation \(V\) the groups \({\mathcal E}^\star(V)\) vanish except in a single degree \(d(V)\), that \({\mathcal E}^{d(V)}\) again is an irreducible smooth representation, and moreover \({\mathcal E}^{d(V)}({\mathcal E}^{d(V)}(V))=V\).
The sheaves under consideration are extended to the Borel-Serre compactification of \(\mathcal I\) in such a way that the cohomology at the boundary becomes computable (since the stabilizers of boundary points are parabolic subgroups this can be achieved by using the Jacquet modules of the representations as the stalks at the boundary points), then the cohomology at the boundary is calculated by adapting the strategy of Deligne and Lusztig for reductive groups over finite fields. The induced functor on the category of finite length smooth representations of \(G(K)\) coincides with the duality functor defined using the spherical building only by the reviewer [Trans. Am. Math. Soc. 347, No. 6, 2179-2189 (1995; Zbl 0827.22005); Erratum, ibid. 348, 4687-4690 (1996; Zbl 0827.22005)]. The fact that the duality functor preserves irreducibility was in the case \(G=\text{GL}_N\) a conjecture due to Zelevinsky (for \(\text{GL}_N\) this conjecture has been proved in 1994 by Procter in a completely different way by using the explicit construction given by Bushnell and Kutzko of the admissible dual of that group).
Reviewer: A.M.Aubert (Paris)

MSC:
22E50 Representations of Lie and linear algebraic groups over local fields
18F30 Grothendieck groups (category-theoretic aspects)
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