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Buildings in the representation theory over local number fields. (Gebäude in der Darstellungstheorie über lokalen Zahlkörpern.) (German) Zbl 0872.11028
The paper starts off from the $$p$$-adic counterpart of semisimple real Lie groups modulo maximal compact subgroups, i.e, of symmetric spaces. As for an example, on the one hand one has $$SL_2(\mathbb{R})/SO(2)\simeq\mathbb{H}$$, and on the other, $$SL_2(\mathbb{Q}_p)/SL_2(\mathbb{Z}_p)$$, say, which, contrary to $$\mathbb{H}$$, does not have an interesting topology yet. A second difference is that in the real case all maximal compact subgroups are conjugate, but no longer so in the $$p$$-adic situation. There arises the problem of classifying the conjugacy classes and of understanding the geometry involved. This is tackled by means of the Bruhat-Tits-building $$X$$, which is a topological space carrying a cell structure, a metric and a compatible $$G$$-action, where now $$G$$ is a connected reductive algebraic group over $$\mathbb{Q}_p$$. The inner structure of $$G$$ is reflected in its action on $$X$$: (1) At each vertex $$x$$ of $$X$$ there exists a model of $$G$$ over $$\mathbb{Z}_p$$, i.e., a certain group scheme $$G_x$$ over $$\mathbb{Z}_p$$ such that e.g. the structure of $$X$$ in a neighbourhood of $$x$$ is given by the structure of the finite group $$G_x(\mathbb{F}_p)$$. (2) The apartments of $$X$$ are isometric to $$\mathbb{R}^d$$ with $$d$$ denoting the semisimple $$\mathbb{Q}_p$$-rank of $$G$$. Illustrative examples and pictures are provided $$(G=SL_2,SL_3,Sp_4)$$.
The main topic of the paper regards a description of the smooth representation theory of $$G$$ by means of the building $$X$$; this is a report (without proofs) on joint work with Stuhler. The interest in the representation theory of reductive $$p$$-adic groups comes from the Langlands programme, which asks for a parametrization of the finite-dimensional representations of the absolute Galois group $$G_{\mathbb{Q}_p}$$ of $$\mathbb{Q}_p$$ in terms of the smooth irreducible representations of $$Gl_n(\mathbb{Q}_p)$$, $$n\geq 1$$. As has been shown by C. J. Bushnell and P. C. Kutzko [Ann. Math. Stud. 129 (1993; Zbl 0787.22016)], the smooth representations of $$GL_n(\mathbb{Q}_p)$$ can be characterized by the representations of the maximal compact subgroups. Here now it is shown that for general (semisimple) $$G$$ the building $$X$$ can be used to define, given a smooth representation $$V$$ of finite length, a certain sheaf $$\widetilde V$$ on $$X$$ which, in turn, gives back $$V$$. Finally, a trace formula in terms of the cohomology of $$\widetilde V$$ is stated that allows for the computation of the values of the function $$\theta_V$$ on the elliptic part of $$G$$, which appears in the character function to $$V:\text{Tr}(\varphi,V)= \int_G\varphi(g)\theta_V(g)dg$$ (for all elements $$\varphi$$ of the Hecke algebra of $$G$$).

##### MSC:
 11F70 Representation-theoretic methods; automorphic representations over local and global fields 22E50 Representations of Lie and linear algebraic groups over local fields